SFCM-R: A novel algorithm for the hamiltonian sequence problem
SSFCM-R: A novel algorithm for the hamiltonian sequenceproblem
C´ıcero A. de Lima ∗ June 19, 2019
Abstract
A hamiltonian sequence is a path walk P that can be a hamiltonian path or hamiltoniancircuit. Determining whether such hamiltonian sequence exists in a given graph G = ( V, E )is a NP-Complete problem. In this paper, a novel algorithm for hamiltonian sequenceproblem is proposed. The proposed algorithm assumes that G has potential forbiddenminors that prevent a potential hamiltonian sequence P (cid:48) from being a hamiltonian sequence.The algorithm’s goal is to degenerate such potential forbidden minors in a two-phraseprocess. In first phrase, the algorithm passes through G in order to construct a potentialhamiltonian sequence P (cid:48) with the aim of degenerating these potential forbidden minors.The algorithm, in turn, tries to reconstruct P (cid:48) in second phrase by using a goal-orientedapproach. A hamiltonian sequence is a path walk P that can be a hamiltonian path or hamiltonian circuit.Determining whether such sequence exists in a given graph is a NP-Complete problem ([6]; [4]).Several algorithms have been proposed to find hamiltonian sequences in a graph G . For example,Held and Karp ([5]) proposed an algorithm that runs in O ( n n ) to compute a hamiltonian pathby using dynamic programming. In 2014, Bj¨orklund ([1]) proposed a randomized algorithm thatruns in O (1 . n to compute a hamiltonian circuit in undirected graphs.The currently best known exact algorithm for the hamiltonian sequence problem runs in O ∗ (2 n − Θ( √ n/ log n ) ) ([2]). Despite the progress made in the hamiltonian sequence problem,a substantial improvement in the area of exact algorithms for this problem remains an openproblem. Unfortunately, exact algorithms for the hamiltonian sequence problem, which is de-termining if hamiltonian path or hamiltonian circuit exists in a graph G , still run in exponentialtime complexity.In this paper, a novel algorithm is proposed to solve the hamiltonian sequence problem. Thegoal of the proposed algorithm is to construct a potential hamiltonian sequence P (cid:48) , assumingthat G may have potential forbidden minors that prevent a potential hamiltonian sequence P (cid:48) from being a hamiltonian sequence. Thus, these potential forbidden minors need to bedegenerated in some state k in a two-phrase process by using a goal-oriented approach. Ouralgorithm outputs a valid hamiltonian sequence by reconstructing P (cid:48) , or aborts itself, if it isforced to use probability instead of the proposed goal-oriented approach. Hence, this studypresents new techniques to solve the hamiltonian sequence problem.The rest of the paper is organized as follows. In section 2 we list some technical conventionsand provide a solid foundation for a better understanding of this paper. Finally, in section 3 wepresent the details of the proposed algorithm and prove its correctness. In this section, some concepts about graph theory are described. Also, this section provides aconcise background needed for a better understanding of the paper. ∗ email: [email protected], orcid: 0000-0002-3117-3065 a r X i v : . [ c s . D S ] J un graph G = ( V, E ) consists of a set V of vertices and a set E of edges. The vertex setof a graph G is refereed to as V ( G ), its edge set as E ( G ). The number of elements of any set X is written as | X | . Each edge e ∈ E is undirected and joins two vertices u, v ∈ V , denotedby e = uv . To represent adjacency between two vertices we use the notation u ∼ v . u ∼ S isused to represent the adjacency between u and at least one vertex w ∈ S , S ⊇ V . The set ofneighbors of a vertex v ∈ V ( G ) is denoted by N ( v ). G [ U ] is a subgraph of G induced by U ⊇ V that contains all the edges xy ∈ E with x, y ∈ U . G − U is a subgraph obtained from G bydeleting all the vertices U ∩ V and their incident edges. If | X | = 1 and X = { v } , we write G − v rather than G − { v } . ω ( G ) is the number of components of a graph G . If v is an articulationpoint then we will have ω ( G − v ) > ω ( G ). The graph G \ e is obtained from G by contractingthe edge e and replace its endpoints x , y with new vertex v e , which becomes adjacent to all theformer neighbors of x or y . Formally, the resulting graph G \ e is a graph ( V (cid:48) , E (cid:48) ) with vertexset V (cid:48) = { ( V \ x, y ) ∪ v e } and an edge set E (cid:48) = { vw ∈ E | v, w ∩ x, y = ∅} ∪ { v e w | xw ∈ E \ { e } or yw ∈ E \ { e }} ([9]) .A minor of a graph G is any subgraph obtainable from G by means of a sequence of vertexand edge deletions and edge contractions. A class or a family F of graphs G contain all graphs G that posses some common characterization. Many families of graphs are minor-closed, thatis, for every G in F every minor G (cid:48) of G also belongs to F . Every minor-closed families hasa finite set X of excluded minors. ([9]) For example, a major step towards deciding whether agiven graph is planar is provided by Kuratowski’s theorem which states that if G in P where P is the family of planar graphs, then G contains no minor belongs to X = { K , K , } ([9])Many methods were studied to test the planarity of a graph G. One interesting method todetermine if a graph G is planar was proposed by Schmidt. This method incrementally buildsplanar embeddings of every 3-connected component of G . Schmidt studied a far-reaching gen-eralization of canonical orderings to non-planar graphs of Lee Mondshein’s PhD thesis ([7]) andproposed an algorithm that computes the called Mondshein sequence in O ( m ) ([10]). Mondsheinsequence generalizes canonical orderings and became later and independently known under thename non-separating ear decomposition ([10]). Definition 1.
An ear decomposition of a 2-connected graph G = ( V, E ) is a decomposition G = ( P , P , ...P k ) such that P is a cycle and every P i , ≤ i ≤ k is a path that intersects P ∪ P ∪ ... ∪ P i − in exactly its endspoints. Each P i is called an ear. ([10]) Mondshein proposed to order the vertices of a graph in a sequence that, for any i , the verticesfrom 1 to i induce essencially a 2-connected graph while the remaining vertices from i + 1 to n induce a connected graph. For conciseness, we will stick with following short ear-based definitionof mondshein sequence. ([10]) Definition 2.
Let G be a graph with edge ru . Assuming that ru ∪ tr is part of the outer faceof G. A Mondshein sequence avoiding ru is an ear decomposition D of G such that (1) r ∈ P ,(2) P birth ( u ) is the last long ear, contains u as its only inner vertex and doesn’t contain ru and(3) D is non-separating. ([10]) An ear decomposition D that satisfies the conditions (1) (2) and (3) is said to avoid ru , so ru is forced to be added last in D , right after the ear containing u as an inner vertex ([10]).If we negate the constraints (1) (2) and (3), we form the forbidden condition of the Schmidt’salgorithm, seeing that such algorithm can’t ignore them. Otherwise, the Schmidt’s algorithmfails to produce a valid output. Definition 3.
A forbidden condition F of an algorithm A is a set F = { f ...f n } of sufficientconditions that makes A fail to produce a valid output. Before continuing, let
Validator be a generic hamiltonian sequence validator function thatoutputs true if P = v i ..v k with 1 ≤ i ≤ k is a hamiltonian sequence of G by performingsubsequent G − v i operations. 2 lgorithm 1 Hamiltonian sequence validator
Input: G = ( V, E ) , P = v i ..v k Output: true , false function Validator output ← false for each v i ∈ P do if ω ( G − v i ) > ω ( G ) then break G − v i if | V | = 0 then output ← true return outputConditions like | P | (cid:54) = | V ( G ) | or v i being an articulation point makes Validator ouput false .Unfortunately, such invalid conditions are useful only to test if P is a hamiltonian sequence ornot. There’s some sufficient conditions available for a graph to posses a hamiltonian sequence([3]; [8]) but there’s no known non-exhaustive algorithm for hamiltonian sequence characteriza-tion test that constructs a valid hamitonian sequence by performing subsequent G − v i operationsand throwing an error, if G doesn’t have any hamiltonian sequence. Likewise, there’s no knownforbidden condition for the hamiltonian sequence problem. At the same time, find a hamiltoniansequence P by relying on exhaustive methods is not feasible. The lack of a known forbiddencondition for hamiltonian sequence characterization test motivated this research.In this paper, a novel algorithm called SFCM-R is proposed to solve the hamiltonian sequenceproblem in a different way. SFCM-R is a type of what we call Syncronization-based ForbiddenCondition Mirroring (SFCM) algorithm, which is formally defined as follows.
Definition 4.
Let G = ( V, E ) be a graph. The Synchronization-based Forbidden ConditionMirroring (SFCM) algorithm is an algorithm with a configuration g : W × F → A , that consistsof: (1) a finite set of scenes W = W i ...W n , W = G , W i ≡ ... ≡ W n , ≤ i ≤ n associated toa finite set of synchronizable forbidden conditions F = F i ...F n ; and (2) a pair ( W i , F i ) , with W i ∈ W and F i ∈ F , associated to each mirrorable algorithm in A = A i ...A n . Definition 5. (Synchronizable forbidden condition) If F i ∈ F and F k ∈ F of A i ∈ A and A k ∈ A , respectively, are conceptually equivalent, then both F i ∈ F and F k ∈ F are synchronizableforbidden conditions that will be synchronized eventually when both A i and A k are executed. Definition 6. (Mirrorable algorithm) If F i ∈ F and F k ∈ F are synchronizable forbiddenconditions, then both A i ∈ A and A k ∈ A are conceptually equivalent mirrorable algorithms thatwill be mirrored eventually when both A i and A k are executed. Before continuing, a trivial example of how the proposed algorithm works in practice ispresented for a better understanding of this paper. Let’s convert the Schmidt’s algorithm toa SFCM algorithm that we call SFCM-S algorithm. Let W = G be a 2-connected scene thatSchmidt’s algorithm takes as input and W be a scene called Schmidt Scene that SFCM-S takesas input. The description of Schmidt scene is as follows. Schmidt Scene
Each P k ∈ D , with D being an ear decomposition of G , is a component andthe ru edge is a forbidden ru-component that needs to be degenerated in some state k .Notice that ru-component is a potential forbidden minor of Schmidt Scene that needs to beadded last by SFCM-S in order to not make such algorithm fail to produce a valid output. Asthe ru edge could be also considered a potential forbidden minor of Schmidt’s algorithm, all weneed to do is to make the SFCM-S imitate the behaviour of Schmidt’s algorithm so that theforbidden conditions of both algorithms will be completely synchronized eventually.As the only difference between SFCM-S and Schmidt’s algorithm is that they’re conceptuallyequivalent , they will be also completely mirrored eventually.3n this paper, we use a variation of the same approach to construct a hamiltonian sequencepath P . In this case, W = G is the scene that an unknown non-exhaustive hamiltonian sequencecharacterization test, that performs subsequent G − v i operations, takes as input. W is thescene called minimal scene that the proposed algorithm for hamiltonian sequence problem, thatwe call SFCM-R, takes as input. Such unknown non-exhaustive algorithm will be called realscene algorithm or RS-R. Throughout this paper, we also refer to an exhaustive version of realscene algorithm as RS-E.We assume that every state of SFCM-R has potential forbidden minors that make suchalgorithm fail to produce a valid hamilton sequence. In addition, we also assume at first that,if W has a hamiltonian sequence, these potential forbidden minors will be degenerated in somestate k of SFCM-R.In summary, the goal of the SFCM-R is to synchronize the forbidden condition of SFCM-Rand the forbidden condition of an unknown non-exhaustive hamiltonian sequence characteri-zation test by using an imitation process. The minimal scene description is based on invalidconditions that make Validator have false as output on minimal state . In other words, suchinvalid conditions belong only to minimal scene , not to real scene or simply G , which is thescene that RS-R takes as input. In this section, SFCM-R is explained in detail. Before continuing, we need to define formally theminimal scene and some important functions. The minimal scene is formally defined as follows.
Definition 7.
A minimal scene is a rooted graph H = ( V, E, v , L, Ω) with a set L = ( L w ) w ∈ V of labels associated with each vertex w , a root vertex v and an ordered set Ω = ( τ i ...τ n ) of tiers τ i ⊃ H . Let H = ( V, E, v , L, Ω) be a minimal scene and G = ( V, E ) be a real scene. Let u and v be vertices such that v, u ∈ V . By convention, v is the current state’s vertex and u , a potentialsuccessor u ∈ N ( v ). V − v is performed whenever a vertex u is defined as a successor of v . Itwill be written as v → u = T . When u is defined as an invalid successor, it will be written as v → u = F . P u is a path from v i to u . As we need to find a vertex u such that v → u = T holds for u , SFCM-R analyses a subscene H (cid:48) ⊃ H that will be denoted as tier . Definition 8.
Let H = ( V, E, v , L, Ω) be a minimal scene. A tier τ i is a subscene τ i ⊃ H suchthat τ i = H [ V − X i ] , X i = ( S ∪ ... ∪ S i +1 ) , S = { v } and S k being a set of nodes with depth k of a breadth-first search transversal tree of H . A set Ω of tiers is defined by the function called maximum-induction . When maximum-induction outputs a valid set Ω, the next step is to get the vertices labelled according tothe function
Lv-label that outputs a set L = ( L w ) w ∈ V , which is the set of labels associatedwith each vertex w . By convention, v LABEL is a vertex labelled as v LABEL and N v LABEL ( w )represents a set of vertices w (cid:48) ∈ N ( w ) labelled as v LABEL . H (cid:48) v is the root of H (cid:48) ⊇ H and H (cid:48) v is the v of its current state. The notation H v LABEL represents a set of all vertices labelled as v LABEL . 4 lgorithm 2
Maximum induction of H Input: H = ( V, E, v , L, Ω) Output:
Set Ω = ( τ i ...τ n ) of tiers function Maximum-induction Ω ← ∅ A ← { v } X ← { v } repeat B ← ∅ for each v ∈ A do for each u ∈ { N ( v ) − X } do B = B ∪ { u } if B (cid:54) = ∅ then X ← X ∪ B if { V − X } (cid:54) = ∅ then Ω ← Ω ∪ H [ V − X ] A ← B else if | V | (cid:54) = | X | then throw error until | V | = | X | return ΩNotice that a tier τ i ∈ Ω can potentially have τ i ≡ H ≡ G in some state k of SFCM-R,RS-R, and RS-E. Because of that, we need to identify all the stumbling blocks that may happento break a potential hamiltonian sequence P on each tier and degenerate them. These pointsare denoted as hamiltonian breakpoints or simply v B , because SFCM-R assumes that they’repotential forbidden minors that prevent P from being a hamiltonian sequence due to the factthat a tier can potentially have τ i ≡ H (cid:48) holding for τ i , with H (cid:48) ⊃ H being the scene H (cid:48) ⊃ H ofthe current state of SFCM-R.Before continuing, we’ll briefly describe what each label means. Let w be a vertex w ∈ V .If w is an articulation point of τ ∈ Ω, it will labelled as v A or minimal articulation vertex. If τ ∈ Ω and d ( w ) = 1 it will labelled as minimal leaf or v L . Every w (cid:48) ∈ N ( v B ), w (cid:48) (cid:54) = v B willbe labelled as minimal degeneration vertex or v D . Every w / ∈ { v D , v B } such that w (cid:54) = v B and N v D ( w ) ≥ minimal intersection vertex or v I . Every non-labelled vertex will be labelledas v N . A vertex labelled as v A or v L is a v B vertex. On the other hand, a vertex labelled as v A v N is not considered a hamiltonian breakpoint.5 lgorithm 3 L v labelling Input: H = ( V, E, v , L, Ω) Output: L function Lv-label L ← ∅ for each τ ∈ Ω do X ← every w (cid:48) such that ω ( τ − w (cid:48) ) > ω ( τ ) for each w ∈ V ( τ ) do if d ( w ) (cid:54) = 2 in H then if w ∈ X then L w ← L w ∪ { v A } if d ( v ) = 1 in τ then L w ← L w ∪ { v L } else if w ∈ X then L w ← L w ∪ { v A v N } for each w labelled as v B ∈ V ( H ) do if L w = { v A , v L } then L w ← L w − { v L } for each w ∈ N ( v B ) do if w (cid:54) = v B then L w ← { v D } for each non-labelled w do if | N v D ( w ) | ≥ then L w ← L w ∪ { v I } for each non-labelled w do L w ← L w ∪ { v N } return L Now, a concise description of minimal scene is finally provided below.
Minimal Scene
Every v A vertex is an articulation point of H and every v L vertex is apotential articulation point. In addition, every v B vertex is part of an isolated C v B componentsuch that C v B = v B ∪ N v D ( v B ). Thus, if we have H − v B then we will have | C v B | − v D -components. v I vertices are potential intersection points between C v B components. v I , v N and v A v N vertices aren’t part of any C v B component. The function A v ( v B , H ) defined bellowreturns T if v B is a virtual articulation of H or F , otherwise. v B is a virtual articulation onlyif | N v D ( v B ) | ≥ A v ( v B , H ) = (cid:26) T, | N v D ( v B ) | ≥ F, otherwise (cid:27) (1)The term virtual indicates that some definition belongs only to minimal scene, not to realscene. Thus, we’ll define an additional function A ( w, H ) that returns true if ω ( H − w ) > ω ( H ).For conciseness, Every real articulation point of H is labelled as v H and C v H = v H ∪ N ( v H )represents a C v H component.In order to keep a valid state, every C v B needs to be mapped and degenerated in some state k of first or second phrase. If v B could be degenerated in a state k , then v B is called b-treatable .We need to assume at first that ∀ v B ∈ V ( H ), there exists a state k which v B will be b-treatable . Definition 9.
Let w be a vertex with { v B } ∈ L w . Let v, w and z be vertices of H . A b-treatable v B or v BT is a vertex w reachable from v through a path P = v...z with z ∼ w such that wehave { v B } / ∈ L w when we recalculate its label in subscene H − P If v B could be degenerated, v B cant’ be considered a v B b-consistent anymore.6 efinition 10. A v B b-consistent or v BC is a consistent v B that can’t be degenerated in currentstate. As we don’t know a detailed description about the unknown forbidden condition of RS-R,we will stick with a conceptually equivalent definition of hamiltonian sequence problem thatrelates real scene to minimal scene explicitly. The v B path problem is as follows. Definition 11. ( v B path problem) Given a scene H = ( V, E, v , L, Ω) , is there a simple path P that visits all vertices with P such that P = P v Bi ...P v Bk ∪ P u with | P | = | V ( H ) | ? As the v B path problem is similar to the hamiltonian sequence problem, it must be NP-complete. In the minimal scene, if we have a path P = P v B − v B that degenerates C v B in H − P ,such P will be part of another P v B (cid:48) fragment. In first phrase of SFCM-R, we pass through H withthe aim of degenerating C v B components in order to create a potential hamiltonian sequence L e , which is a sequence of path fragments.It means that the following theorem, which is a sufficient condition to make Validator output false , will be partially ignored in first phrase. Such phrase is called mapping phrase,which is represented by
Mapping function (see Sect. 3.1).
Theorem 1.
Let G = ( V, E ) be a graph. If P = v i ...v k with ≤ i ≤ k , ≤ k ≤ | V | ishamiltonian sequence of G , then v i → v i +1 = T holds for v only and only if ω ( G − v i ) ≤ ω ( G ) .Proof. If we have ω ( G − v i ) > ω ( G ), at least one component is not reachable from u = v i +1 .Therefore, | P | (cid:54) = | V | holds for P , which is not a hamiltonian sequence, since at least one vertexis not reachable from u = v i +1 .Because of that, both first and second phrase of SFCM-R need to enforce basic constraintsrelated to real scene in order to not ignore Theorem 1 completely. Before continuing, we willdefine two properties that a subscene H (cid:48) ⊃ H may have. Property 1. ( | H n | property) The property | H n | indicates that H (cid:48) is a component of H (cid:48) ⊃ H [ V − H v H ] that has n vertices w ∈ Z with Z = { w (cid:48) ∈ V ( H [ V ]) : ( | N ( w (cid:48) ) ∩ H v H | ≥ ∧ ( w (cid:48) (cid:54) = v H ) } . The value of | H n | is equal to α = (cid:12)(cid:12)(cid:83) w ∈ Z { N ( w ) ∩ H v H } (cid:12)(cid:12) ; | H nv H | returns a set β = (cid:83) w ∈ Z { N ( w ) ∩ H v H } . Property 2. ( | H c | property) The property | H c | = F indicates that H (cid:48) ⊃ H [ V − H v H ] isa creatable component of H [ V ] , which implies that it still doesn’t exist in H [ V ] . | H c | = T indicates that H (cid:48) is a component that exists in H [ V ] . The two basic constraints are as follows.
Constraint 1. If H v H (cid:54) = 0 , H can’t have a creatable component H (cid:48) ⊃ H [ V − H v H ] with | H n | = 1 , { V ( H (cid:48) ) ∩ { v ∪ N ( v ) ∪ v ∪ N ( v ) }} = ∅ and | H c | = F .Why. If H (cid:48) is created and reached by either v or v , ω ( G − w ) > ω ( G ) may hold for G − w with w ∈ { v, v } . Such situation is invalid since it can potentially make SFCM-R ignore Theorem 1completely. SFCM-R assumes that every w is reachable from either v or v , without ignoringTheorem 1 completely. Constraint 2. If H v H (cid:54) = 0 , H [ V − H v H ] can’t have a component H (cid:48) ⊇ H [ V − H v H ] with v ∈ V ( H (cid:48) ) , | H n | = 0 and | H c | = T .Why. In this case, v can’t reach other components. Such situation is invalid since it canmake SFCM-R ignore Theorem 1 completely. SFCM-R assumes that every w is reachable fromeither v or v , without ignoring Theorem 1 completely.In addition, SFCM-R can’t have an exponential complexity. Otherwise, we’re implicitlytrying to solve this problem by imitating RS-E. Such situation is clearly invalid seeing thatSFCM-R needs to try to imitate the behaviour of RS-R. That’s why every v → u = T choicemust be goal-oriented in both two phrases of SFCM-R. In other words, both phrases must begoal-oriented. Throughout this paper, we prove that both phrases are imitating the behaviourof RS-R. Such proofs shall be presented with an appropriate background. (see Sect. 3.1.3 and3.4). 7 efinition 12. Goal-oriented choice is a non-probabilistic v → u = T choice that involvesminimal scene directly and real scene partially. As SFCM-R passes through H instead of G , we’re considering minimal scene directly. Inaddition, the real scene is considered partially since some basic constraints related to RS-R areevaluated by SFCM-R. Throughout this paper, constraints followed by an intuitive descriptionshall be presented in this order by convention. All the goal-oriented strategies developed throughthis research shall be presented along with an appropriate background (see Sect. 3.3.2).Notice that we can assume that RS-R generates only consistent C v H components in orderto construct a valid hamiltonian sequence (if it exists), or throws an error when G doesn’t haveany hamiltonian sequence in order to abort itself. For that reason, we represent the real scenealgorithm as follows. Algorithm 4
Non-mirrorable RS-R algorithm
Input: G = ( V, E ) , v , P Output:
Hamiltonian sequence P function Hamiltonian-sequence A ← N ( v ) X ← every w (cid:48) such that A ( w (cid:48) , H ) = T for each u ∈ A do if u ∈ X or v → u = F then X ← X ∪ { u } A ← A − X if A (cid:54) = ∅ then v → u = T with u ∈ A P ← P ∪ { u } Hamiltonian-sequence( G , u , P ) else P ← { v } ∪ P if | P | (cid:54) = | V ( H ) | then throw error return PAs this version of RS-R doesn’t have any explicitly relationship with the proposed minimalscene, it needs to be modified to properly represent a mirrorable real scene algorithm, which isthe real scene algorithm we want to directly mirror in reconstruction phrase. Such modificationshall be presented with an appropriate background.For conciseness, we use RS-R to represent the non-mirrorable RS-R algorithm in order toavoid confusion unless the term mirrorable is explicitly written. The reason is that the cor-rectness of SFCM-R implies that both non-mirrorable RS-R algorithm and mirrorable RS-Ralgorithm are conceptually equivalent mirrorable algorithms, which consequently implies thatthere’s no specific reason to differentiate one from another throughout this paper.In summary, the main goal of SFCM-R is to imitate the behaviour of RS-R in order to avoidusing probability, which is a known behaviour of RS-E. That’s why the second phrase, that iscalled reconstruction phrase, which is represented by the function
Reconstruct , aborts theprocess if it’s forced to use probability while reconstructing P . Such reconstruction process isexplained in section 3.2. In this section, the mapping phrase is explained. This phrase outputs a non-synchronizedhamiltonian sequence that is called L e set. Such set is used by reconstruction phrase, which triesto reconstruct a hamiltonian sequence by modifying L e in order to output a valid hamiltoniansequence (if it exists). The mapping task is done by the Mapping function. This functiontakes both H = ( V, E, v , L, Ω) and G = ( V (cid:48) , E (cid:48) ) as input by reference along with additionalparameters ( L e , v , η , ε , m , κ , S ) by reference and keeps calling itself recursively until reachingits base case. 8 efinition 13. Let H = ( V, E, v , L, Ω) be a minimal scene. A non-synchronized hamiltoniansequence is a sequence L e = ( e i ...e n ) , L e ⊇ E ( H ) of path fragments. By convention, the ( x, y ) notation will be used to represent non-synchronized edges xy cre-ated by Mapping . ( w, (cid:3) ) is a non-synchronized edge e ∈ L e with w ∈ e . { L e ∩ ( w, (cid:3) ) } is anordered set that contains each e ∈ L e with w ∈ e . The mapping task performed by Mapping has the following structure:
Base case (1) | V ( H ) | = 0 or (2) ε > η forms the base case of recursion. If base case is reachedby first condition, then we assume at first that every v BC will be v BT in some state k Degeneration state
The current state of
Mapping , in which the main operations are asfollows: (1) perform V − v ; (2) perform v → u = T ; (3) perform a recursive Mapping call; and(4) throw an error exception.In degeneration state, some constraint must make v → v LABEL = T hold for at least one v LABEL . If we don’t have any u = v LABEL with v → v LABEL = T , we have to undo one stepand try an alternative choice in current scene until ε > η , with ε being the local error counterof Mapping and η being the local error counter limit of Mapping .The
Sync-Error procedure is called by
Mapping whenever it finds an inconsistency. Suchprocedure increments both ε and κ by reference, with κ being a global error counter of Mapping .If ε > η , the current subscene must be discarded by
Mapping and the degeneration state ischanged to another v in an earlier valid subscene. On the other hand, if κ > m , with m beingthe global error counter limit of Mapping , the mapping process must be aborted.
Algorithm 5
Pre-synchronization error handler
Input: H = ( V, E, v , L, Ω) , η , ε , m , κ , throw-error procedure Sync-Error ε ← ε + 1 κ ← κ + 1 if κ > m then Abort mapping process else if ε > η then Discard H if throw-error = true then throw errorEvery constraint must be checked into H [ V − v ]. In order to check if a constraint holdsfor u = v LABEL , Mapping must update the labelling of H by calling Lv-label ( H [ V − v ])function only. Some constraints have nested constraints that induce H (cid:48) = H [ V − v ] by a set U ⊃ V . These nested constraints also need to be checked into H (cid:48) [ U ] by calling Lv-label ( H (cid:48) [ U ])function only.The only case that requires the current subscene to be completely changed is when H [ V − v ] v H (cid:54) = ∅ . In this case, we have to perform a Context Change (CC) operation into a new subscene H (cid:48) ⊃ H , due to the fact that v H must be reachable by v or v , without ignoring Theorem 1completely. Because of that, a creatable component H (cid:48) ⊃ H [ V − H v H − v ] such that | H n | = 1, { V ( H (cid:48) ) ∩ { N ( v ) ∪ v }} (cid:54) = ∅ , , | H c | = F needs to be explicitly created by Mapping since theminimal scene is not aware of the existence of real articulation points. We call this creatablecomponent
H(cid:63) . After
H(cid:63) is created, it need to be configured by the following operations: V ( H (cid:48) ) = V ( H (cid:48) ) ∪ { v , | H nv H |} , H (cid:48) v = v , H (cid:48) v = | H nv H | . When it’s processed by Mapping ,the current labelling of H becomes obsolete. Because of that, H also needs to perform a CCoperation.Notice that an edge v v is added temporarily whenever Mapping make a CC operation,which is done by the function
Context-change , in order to make both
Lv-Label ( H ) and Maximum-induction ( H ) work correctly. That’s because such v will act like a vertex u that9as chosen by v → v = T in an imaginary state with H [ V − v ] v H = ∅ , which makes H(cid:63) andthe degeneration state behave like v = v in maximal H ≡ G , v ∈ V ( H ). Algorithm 6
Context Change (CC) Operation
Input: H = ( V, E, v , L, Ω) , w ∈ V ( H ), v ∈ V ( H ) Output:
Scene H = ( V, E, v , L, Ω) function Context-change if First context change of H then for each y ∈ V ( H ) do y.LAST ← null y.SP LIT ← true H v ← w edge created ← f alse e ← null if w (cid:54) = v and w / ∈ N ( v ) then e ← vw E ( H ) ← E ( H ) ∪ { e } edge created ← true Ω ← Maximum-induction ( H ) L ← Lv-Label ( H ) if edge created then E ( H ) ← E ( H ) − { e } return H The constraints considered in this phrase are defined as follows.
Constraint 3. v → v D = T , if v D ∼ v A and N v A ( v A ) = ∅ .Why. As v A is considered an isolated component by minimal scene, it can’t influence thelabelling of any v A (cid:48) directly. Constraint 4. v → v D = T , if we have at least one v AT for H [ V − v D − P ] with P being a v H -path P = w i ...w k generated by H [ V − v D ] such that w = v D , ≤ i ≤ k , and d ( w ) = 2 with w ∈ P such that w (cid:54) = w .Why. In this case, v D is part of a degeneration process. As P is a mandatory path ofsubdivisions, H [ V − v D − P ] is performed in order to check if w k also behaves like v D since v = w k will hold for v eventually. Constraint 5. v → v L = T , if we have at least one v AT for H [ V − v L − P ] with P being a v H -path P = w i ...w k generated by H [ V − v L ] such that w = v L , ≤ i ≤ k , and d ( w ) = 2 with w ∈ P such that w (cid:54) = w .Why. A v L is a leaf on its minimal state, that can act like a v D (cid:48) with d ( v D (cid:48) ) = 1 thatdegenerates a C v A such that v D (cid:48) ∈ C v A . In this case, v L is behaving like a leaf w of RS-R suchthat w ∈ C v H instead of a v B vertex since it’s part of a degeneration process. Constraint 6. v → v D = T , if v D ∼ v A and v D / ∈ τ ∧ ( A ( v A , τ ) = T ) ∧ ( | N v A ( v D ) | = 1) .Why. In this case, v D doesn’t influence the labelling of any v A vertex directly since v D / ∈ τ and | N v A ( v D ) | = 1. Constraint 7. v → v D = T , If v D ∼ v L ∧ v D → v L = T .Why. If v D ∼ v L and v D → v L = T we assume that v D → v L = T may be the next choice. Constraint 8. v → v D = T , if there exists a v H -path P = w i ...w k , w k ∼ v L , generated by H [ V − v D ] such that: (1) w = v D , ≤ i ≤ k , d ( w ) = 2 with w ∈ P such that w (cid:54) = w ; and (2) w k → v L = T in H [ V − v D − P ] . hy. If there exists P , which is a mandatory path of subdivisions, we check if w k → v L = T holds for v L since v = w with w = v L will hold for v eventually. Constraint 9. v → v D = T , If v D ∼ v A , and (1) A v ( v A , H ) = F in H or (2) A v ( v A , H ) = F in H [ V − v D ] .Why. In this case, we have 0 ≤ | N v D ( v A ) | ≤ v A is not a consistent virtualarticulation since we have 0 ≤ | C v A | − ≤ Constraint 10. v → v D = T , if | H v A | = 0 .Why. If | H v A | = 0 and | H v D | (cid:54) = 0, then | H v L | (cid:54) = 0. In such state, Mapping tries to make v L behave like leafs w of real scene such that w ∈ C v H . Constraint 11.
If there’s no other valid choice for v , we have v → v A v N = T , v → v I = T ,and v → v N = T .Why. Vertices labelled as v I , v N and v A v N aren’t part of any C v B directly. The goal of mapping phrase is to output a valid L e set ready to be reconstructed in nextphrase. As a consequence, if Mapping generates an inconsistent v H that prevents L e frombeing a hamiltonian sequence, Reconstruct will be able to degenerate such inconsistency andgenerate another v H (cid:48) to change parts of L e until we have a valid hamiltonian sequence (if itexists) by correcting parts of mapping process. We call this process C v H attaching or minimalscene attachment , because inconsistent C v H components are degenerated by considering minimalscene directly and real scene partially. Such process is done in reconstruction phrase by using agoal-oriented approach (see Sect. 3.3). Definition 14. A C v H attaching is when we choose a vertex u with u ∈ C v H before C v H makesa scene H (cid:48) ⊇ H be inconsistent in current state of SFCM-R. A C v H is attached when: (1) H (cid:48) v H = ∅ holds for H (cid:48) − u ; or (2) a v H -path P generated by H (cid:48) − u that doesn’t generate anyinconsistency in H (cid:48) − P . Definition 15. A v H -path is a path P = P v Hi ...P v Hk , generated by H [ V − v ] with H [ V − v ] v H (cid:54) = ∅ , such that ≤ i ≤ k , ≤ k ≤ | V | , in which every v H i reaches v H i +1 properly. The key to constructing a valid L e is take into account the priority order of each choice. Thepriority order plays an important role in this phrase since it will contribute to make the mappingphrase imitate the behaviour of RS-R. The priority order relies on the label of u . If the priorityis n times higher than an arbitrary constant i , it will be denoted as v i + nLABEL . The highestpriority is to make a v BC be v BT . So we will have v Li +4 and v Di +3 . v L has the highest prioritybecause it can potentially make | H v A | increase since it’s considered a potential real articulationpoint according to minimal scene’s description. If Mapping can’t make any v BC be v BT in itscurrent state, we want to perform a CC operation instead of undoing states. Thus, we will have v A v N i +2 since these vertices can generate v H articulations with a considerable probability dueto d ( v A v N ) = 2. If we don’t have any C v B ∼ v , we have v N i +1 in order to make Mapping reachdifferent regions of H . The lowest priority is for v I . So we have v I i for vertices labelled as v I .Notice that we don’t have any constraint that makes v → v A = T hold for v , since it candisconnect the minimal scene according to its description. Even so, we will have v → v A = T in some state k of SFCM-R if Reconstruct outputs a valid hamiltonian sequence. It meansthat the constraints related to vertices labelled as v A can’t be evaluated directly in this phrase. In this section, the pseudocode of
Mapping is explained. Every line number mentioned in thissection refers to the pseudocode of
Mapping , which is as follows.11 lgorithm 7
Mapping of H Input: H = ( V, E, v , L, Ω) , G = ( V (cid:48) , E (cid:48) ), L e , v , η , ε , m , κ , S Output:
Set L e = e ...e n of non-synchronized edges function Mapping if | V ( H ) | (cid:54) = 1 then if v.SP LIT and H [ V − v ] v H (cid:54) = ∅ then if constraint 1 or 2 doesn’t hold for H [ V − v ] then Sync-Error( H , η , ε , m , κ , true ) H − v Set and configure
H(cid:63) in H [ V − H v H ] if H(cid:63) was set and configured then try H(cid:63) ← Context-change(
H(cid:63) , H(cid:63) v , H(cid:63) v ) Mapping(
H(cid:63) , G , L e , H(cid:63) v , η , 0, m , κ , S ) Update H if w ∈ V ( H ) with w ≡ H(cid:63) v then v.SP LIT ← false Restore v and w ∈ V ( H ) with w ≡ H(cid:63) v v ← w H ← Context-change( H , v , v ) Mapping( H , G , L e , v , η , ε , m , κ , S ) if N ( v ) = ∅ in H then H − v catch error Sync-Error( H , η , ε , m , κ , true ) else if N ( v ) (cid:54) = ∅ in H then found ← false v.SP LIT ← false if v.U is not set then L ← Lv-Label ( H [ V − v ]) v.U ← every v LABEL ∈ N ( v ) such that v → v LABEL = T in H [ V − v ] while there exists a non-visited u do u ← a non-visited u ∈ v.U with highest priority chosen randomly try Select( G , v , u , L e , S ) Mapping( H , G , L e , u , η , ε , m , κ , S ) found ← true break catch error Sync-Error( H , η , ε , m , κ , false ) Undo modifications in H , L e , and S if found = false then Sync-Error( H , η , ε , m , κ , true ) else H − v else Select( G , v , v , L e , S ) return L e Firstly, a
Context-change ( v , v , H ) call is needed to calculate Ω and L of H such that | H v H | = 0, H v = v , H v = v before the first Mapping call. When
Mapping is called, if H [ V − v ] v H (cid:54) = ∅ , Mapping must remove v from V ( H ) in order to create a valid H (cid:63) componentwith | H c | = T (line 7). In addition, every w ∈ V ( H(cid:63) ) must be deep copies of w ∈ V ( H ) because12e treat vertices as objects in order facilitate the understanding of the proposed pseudocode.Its important to mention that Mapping needs to call
Context-change function before
Mapping call itself recursively if
H(cid:63) was set (lines 10 and 17). The new v is set to w ≡ H(cid:63) v with w ∈ V ( H ) (line 16). Every vertex x ≡ x (cid:48) , x ∈ V ( H ), x (cid:48) ∈ V ( H(cid:63) ), x (cid:48) ∈ S that wasremoved from H(cid:63) by a v → u = T operation made by Select function (lines 33 and 45) mustbe also removed from H before a CC operation (line 12), including H(cid:63) v , despite to the factthat is restored in H(cid:63) . This rule doesn’t apply for vertices removed from H when N ( v ) = ∅ (lines 20 and 43) since such v may be part of another H(cid:63) component in different recursive calls.If w ∈ V ( H ) with w ≡ H(cid:63) v , the split property v.SP LIT is set to false (line 14). In thiscase, H(cid:63) v was not explicitly reached by any v → u = T operation made by Select . As we’reignoring Theorem 1 partially, we need to force v to be changed in next recursion call in orderto make Mapping create a new
H(cid:63) since a new
H(cid:63) v may happen to be explicitly reached bya v → u = T operation made by Select in a new
H(cid:63) .If H [ V − v ] v H = ∅ , we need to follow the constraints and priorities mentioned earlier to set v.U , which is the set of possible successors of v (lines 27 to 29), and set v.SP LIT to false (line26). In this case, If v → v LABEL = T holds for at least one u = v LABEL , we must: (1) perform v → u = T ; (2) perform S ← S ∪ v in order to update H properly; (3) add a non-synchronizededge ( v.LAST, v ) to L e ; and (4) perform u.LAST ← v . Mapping needs to call
Select in orderto do such operations by reference when the context remains unchanged (lines 33 and 45).
Observation 1.
Notice that, as v → u = T performs V − v by convention, G = ( V (cid:48) , E (cid:48) ) isnot changed. The reason is that we use G to make Mapping keep track of adjacency between v.LAST and v in the maximal H ≡ G . If v → v LABEL = F happens to hold for w = v LABEL with w ∈ v.U due to an error thrownby Sync-Error (line 38),
Mapping must undo modifications made in H , L e and S to restoreits state before choosing a new unvisited u ∈ v.U as successor (line 39). On the other hand,if v → v LABEL = F holds for every v LABEL ∈ v.U , we need to undo the last step and try analternative, incrementing both κ and ε by calling Sync-Error (line 41). Every error found inmapping phrase must increment κ and ε . If ε > η , the current subscene H must be discardedby Mapping , that needs to perform undo operations to choose another v in an earlier validsubscene. On the other hand, if κ > m , the process must be aborted. In this phrase, a vertexcan’t have more than two incident edges since L e must be an ordered sequence of path fragments.Therefore, Select must remove the first element of S = { L e ∩ ( w, (cid:3) ) } from L e by reference if | S | > Select is as follows.
Algorithm 8
Non-synchronized edge handler
Input: G = ( V (cid:48) , E (cid:48) ), v , u , L e , S procedure Select v → u = T S ← S ∪ v if v = v then Restore v if v.LAST ∈ N ( v ) in G and v (cid:54) = u then L e ← L e ∪ ( v.LAST, v ) Remove the first element of S = { L e ∩ ( w, (cid:3) ) } from L e if | S | > u.LAST ← v In addition,
Mapping must never remove w = v from H except in two cases. The firstcase is before a CC operation that makes w (cid:54) = v hold for w (line 17). The second case is when N ( v ) = ∅ and v = v (lines 20 and 43). Also, Mapping can’t have v → v = T with v (cid:54) = v unless v ∼ v , d ( v ) = 1, d ( v ) = 1 and | V ( H ) | = 2. Such restriction imitates the way that RS-Rreaches v from v . This section is dedicated to the proof of correctness of mapping phrase. It’s important tomention that SFCM-R can only use goal-oriented choices. Because of that, we need to prove13hat
Mapping is goal-oriented. Consider the following lemmas.
Lemma 1.
Let H = ( V, E, v , L, Ω) be a scene. | H v A ∩ H [ V − v ] v H | ≤ | H [ V − v ] v H | holds forevery H with | V | > and H v A ∩ H [ V − v ] v H (cid:54) = ∅ .Proof. Let H = ( V, E, v , L, Ω) be a scene such that H is a minimal hamiltonian graph, E = w i w i +1 ...w n − w n and w i = w n . Suppose that | V | = 4. For every H [ V − v ] with v ∈ V ( H ),we have | H [ V − v ] v H | = | V | −
3, which is the maximum value possible of | H [ V − v ] v H | . If wecall Context-Change ( v , v , H ) with v being an arbitrary vertex w ∈ V , the first τ ∈ Ωwill have | τ v H | = 0 since | V ( τ ) | = 1. Let’s add a vertex w (cid:48) and an edge w (cid:48) w i to H , set w (cid:48) to v and call Context-Change ( v , v , H ). As V ( τ ) = V ( H ) − N ( v ) − v holds for first tier τ ∈ Ω, | V | − | τ v H | . Notice that: (1) if we had d ( v ) = 2, d ( w ) = 1 would hold for every w ∈ V ( τ ); and (2) if we had d ( v ) = 3, | Ω | = 0. Thus, | H v A | = 0holds for H when | V | ≤
4. Now suppose that | V | >
4. Let’s connect w i with every vertexexcept v = w (cid:48) and call Context-Change ( v , v , H ) again. In this case, H v A = H [ V − v ] v H will hold for H with v = v since d ( w ) > H holds for every w ∈ τ v H . Notice that ifwe remove at least one edge w i x with x (cid:54) = v and call Context-Change ( v , v , H ) again, H v A < H [ V − v ] v H will hold for H with v = v since d ( x ) = 2 in H holds for x ∈ τ v H inthis case. Therefore, | H v A ∩ H [ V − v ] v H | ≤ | H [ V − v ] v H | holds for every H with | V | > H v A ∩ H [ V − v ] v H (cid:54) = ∅ . Lemma 2.
Let H = ( V, E, v , L, Ω) be a scene. ε such that ε < | V | aborts the mapping taskprocess only if at least one valid u for every v found is known.Proof. If H [ V − v ] v H = 0 and at least one valid u for every v found is known in mapping task, ε = | N ( v ) | will force Mapping to abort the process when there’s only invalid u vertices for v .If H [ V − v ] v H (cid:54) = 0, ε such that ε ≤ | N ( v ) | aborts the mapping task when there’s only invalid u vertices for v , since at least one invalid u ∈ N ( v ) may be part of different H(cid:63) components setby
Mapping . As | N ( v ) | ≤ | V | − ε such that ε < | V | aborts the mapping task process only ifat least one valid u for every v found is known.The following theorem we want to prove states that Mapping is goal-oriented with η = | V | and m = | V | −| V | , even when it doesn’t reach its base case. As a consequence, Mapping mayrequire some attempts with a different vertex set as v to reach its base case in order to outputa set L e that maps the majority of the vertices w ∈ V ( H ) (if it exists). Observation 2.
The proof of the following theorem assumes that
Mapping takes as input aconnected H = ( V, E, v , L, Ω) with H v H = ∅ . The reason is that Mapping needs to enforceconstraints 1 and 2, which doesn’t imply that SFCM-R will fail when H v H (cid:54) = ∅ and also doesn’timply that Mapping needs to take a connected H = ( V, E, v , L, Ω) with H v H = ∅ in order to begoal-oriented. Thus, if we need to reconstruct a hamiltonian path in a scene H with H v H (cid:54) = ∅ ,we need to reconstruct multiple hamiltonian path fragments for each H (cid:48) ⊃ H , | H n | = 1 generatedby H [ H − H v H ] separately in different instances of SFCM-R. Because
Mapping is goal-oriented by the following theorem, even when it doesn’t reach itsbase case, SFCM-R also assumes that both
Mapping and RS-R have pre-synchronized forbiddenconditions.
Theorem 2.
Mapping is goal-oriented with η = | V | and m = | V | −| V | .Proof. Let H = ( V, E, v , L, Ω) be a connected minimal scene with H v H = ∅ that Mapping takes as input, and (cid:98) F (cid:101) be the unknown negated forbidden condition of RS-R.As not every C v A will happen to be C v H , then v Li +4 and v Di +3 can potentially cancelthe appearance of non-mandatory C v H components and consequently retard ε growth rate since v A ∈ V ( τ i ) is a potential v H of Mapping . As (cid:98) F (cid:101) also cancels the appearance of non-mandatory C v H components, Mapping is imitating RS-R by giving the degeneration process a high priority.Even if C v A happens to be a C v H , such C v A will not influence the labelling of any other v A (cid:48) directly since such C v H forces Mapping to perform a CC operation. (cid:98) F (cid:101) also forces RS-Rto perform a CC-like operation in order to pass through a potential forbidden minor X ⊃ H ,directly or indirectly. As (cid:98) F (cid:101) is optimal, X can be used by (cid:98) F (cid:101) to decide whether the real scene14as a hamiltonian sequence. In this specific case, such X is an inconsistent component with0 ≤ | H n | ≤ | H c | ∈ { T, F } in a state x , in which RS-R decides to abort itself. Therefore,if H v H (cid:54) = ∅ in RS-R context, then v i ∈ P holds for every valid v i found with P such that P = P (cid:48) ∪ X (cid:48) with P (cid:48) being a v H -path and X (cid:48) ⊇ V ( X ).As P can be split into potential independent forbidden minors in RS-R context by (cid:98) F (cid:101) , Mapping is imitating RS-R by:(1) forcing C v A components to be isolated through the degeneration process; and(2) performing a CC operation in case of H v H [ V − v ] (cid:54) = ∅ when both constraints 1 and 2 holdfor H [ V − v ].Notice that as (cid:98) F (cid:101) detects both potential non-mandatory C v H components and potentialindependent forbidden minors that don’t exist in current minimal scene context of real scene,directly or indirectly, Mapping is still imitating RS-R when a vertex w (cid:54) = v A happens to be v H without ignoring both constraints 1 and 2.If we have u = v D and u ∈ N ( v L ), Mapping is forcing such v L to be a leaf of real scene.Notice that if z = v L , z ∼ v happens to be a real leaf with d ( z ) = 1, Mapping can prevent z from being a potential independent forbidden minor X since d ( z ) = 1. Even so, z couldpotentially create non-mandatory C v H components. As (cid:98) F (cid:101) cancels the appearance of a leaf inorder to prevent it from creating non-mandatory C v H components, Mapping is also imitatingRS-R by v Li +4 because:(1) not every v L will turn to be a leaf; and(2) u = v L can also cancel the appearance of non-mandatory C v H components and potentialindependent forbidden minor X by either preventing v L from being a real leaf or degener-ating C v A components.Thus, v Li +4 can also retard ε growth rate. In addition, as (cid:98) F (cid:101) needs to ensure that at leastone v ∼ v L will reach v L by v → v L = T due to the fact that v L is a potential leaf, Mapping is imitating RS-R by giving v L the highest priority.If u = v A v N due to v A v N i +2 , and z (cid:48) = v A v N , z (cid:48) ∼ v happens to be a v H with d ( z (cid:48) ) = 2, Mapping can prevent z (cid:48) from being a potential independent forbidden minor X since d ( z (cid:48) ) = 1will hold for u = z (cid:48) when Mapping is passing through z (cid:48) . Even so, it could potentially generatenon-mandatory C v H components due to d ( z (cid:48) ) = 2. As:(1) these non-mandatory C v H components can be degenerated by v Li +4 and v Di +3 ; and(2) not every v A v N will turn to be a v H with d ( v H ) = 2; Mapping is imitating RS-R by giving v A v N an intermediary priority in order to prevent v A v N from generating non-mandatory C v H components.If we have u = v I due to v I i , v I can delay the appearance of C v H components by forcing Mapping to cancel the appearance of non-mandatory C v H components since u = v I prevents v I from being transformed into a v A . In addition, v I can also retard ε growth rate by forcing Mapping to give the degeneration process a higher priority due to v I ∼ v D , v Di +3 and v Li +4 .Notice that v I can also retard ε growth rate by maximizing the following equation, whichis the sum of abs ( | V ( A i ) | − | V ( B i ) | ) from state i = 0 to current state x , with A i = H(cid:63) beingthe component set in line 7 of in state i of Mapping , and B i = H i [ V ( H i ) − { V ( A i ) − A i v } ], B i v ≡ A i v in the same state i .maximize x (cid:88) i =0 abs ( | V ( A i ) | − | V ( B i ) | )subject to A i v ≡ B i v (2)15he reason is that v I can potentially reduce the local connectivity l ( w, v ) of at least one w ∈ C v B where v I ∼ C v B . If so, (cid:80) xi =0 abs ( | V ( A i ) | − | V ( B i ) | ) tends to be maximized by u = v I ,specially when v I forces at least one v D ∼ v I of such C v B to be a subdivision of H , which couldincrease the success rate of CC operations made by Mapping when v ∈ C v B , seeing that:(1) v I has the lowest priority; and(2) not every w ∈ C v B with v I ∼ C v B will have its local connectivity l ( w, v ) reduced, becauseof the higher priority given to degeneration process.Notice that ε such that ε > | V | suggests that the current scene H (cid:48) ⊇ H of Mapping hasregions R of vertices with a small local connectivity l ( w, v ), w ∈ R . As Mapping minimizesindirectly the appearance of C v H components by decreasing both | V | , | H v A | , and consequently | H v I | , the appearance of such regions can be minimized. That’s because the appearance ofmandatory C v H components is maximized by minimizing the following equation, which is thesummation from state i = 0 to current state x of an equation that ,by Lemma 1, relates themaximization of | H [ V − v ] v H | to | H v A | . As a consequence, Mapping can make the success rateof CC operations increase, and retard ε growth rate through its degeneration process.minimize x (cid:88) i =0 | H i [ V − v ] v H | − | H i vA ∩ H i [ V − v ] v H | subject to H i vA ∩ H i [ V − v ] v H (cid:54) = ∅ (3)The success rate of CC operations also can be increased by u = v D with v D ∈ C v A when:(1) A v ( v A , H ) = F in H ; or (2) A v ( v A , H ) = F in H [ V − v D ]. The reason is that such v A can potentially create both independent potential forbidden minors X and non-mandatory C v H components, with v D ∈ X and v D ∈ C v H . When Mapping passes through such independentpotential forbidden minors X and non-mandatory C v H components before passes through v A , itcould cancel the appearance of them, and consequently make the success rate of CC operationsincrease when 0 ≤ | C v A − | ≤
1. If so, such v A will behave like an isolated component. As (cid:98) F (cid:101) also cancels the appearance of both independent potential forbidden minors X and non-mandatory C v H components, Mapping is imitating RS-R in this case, even if such v A is notexplicitly independent in minimal scene.If we have u = v N due to v N i +1 , we can also increase the success rate of CC operations, sinceit doesn’t influence any C v B to be C v H directly. Because of that, it can prevent | H v A | and | H v I | from growing, which delays the appearance of C v H components. Even if v D ∼ v N , v D ∈ C v B ,both v Di +3 and v Li +4 can prevent w ∈ C v B from having l ( w, v ) reduced. Thus, v N can alsoretard ε growth rate.In addition, notice that even if Mapping generates non-mandatory C v H components inregions R of vertices with small local connectivity l ( w, v ), w ∈ R , no error is thrown when v or v has none or more than one different vertices as successor unless constraint 1 or 2 doesn’t holdfor H [ V − v ]. Such flexibility also makes the success rate of CC operations increase and canretard ε growth rate. Furthermore, Mapping can throw an error with ε being very small when H has regions with a small connectivity, since Mapping doesn’t make v → u = T operationswhen H [ V − v ] v H (cid:54) = 0.Even so, (cid:98) F (cid:101) can’t ignore minimal scene constraints completely. If RS-R ignores minimalscene constraints completely, we have:(1) at least one v B ∈ V ( τ i ) in every scenario with H v B (cid:54) = ∅ would happen to be an inconsistencyof real scene in at least one of its states. If so, (cid:98) F (cid:101) in every scenario with H v B (cid:54) = ∅ wouldbe ignoring Theorem 1 completely in at least one state of RS-R, which is invalid.(2) at least one v ∈ V ( H ) in every scenario with H v B = ∅ would happen to have v → u = F ,for every u ∼ v , in at least one of its states, even when Theorem 1 is not being ignoredcompletely, which is invalid.Thus, Mapping ignoring Theorem 1 partially is not a sufficient condition to prove that
Mapping is not imitating RS-R. 16s every constraint of
Mapping can potentially retard ε growth rate, Mapping can po-tentially distort its potentially-exponential error rate curve. (cid:98) F (cid:101) also distorts the potentially-exponential error rate curve of RS-R, which is represented by the number of times that v → u = F holds for u , since (cid:98) F (cid:101) predicts, directly or indirectly, when the error rate curve of RS-Rwill grow exponentially in order to make RS-R abort itself. As a consequence, ε growth ratemust be distorted by Mapping in order to make ε converge to k such that k < | V | in order toprevent it from aborting itself, which is not a sufficient condition to prove that Mapping is notimitating RS-R.If ε happens to converge to k such that k ≥ | V | , Mapping would be failing to make ε growthrate retard. In this case, Mapping would be using probability explicitly when it doesn’t discardits current scene since:(1) by Lemma 2, it doesn’t know at least one valid u for a v in the worst case scenario; and(2) it is tending to ignore Theorem 1 completely as every constraint is failing to make ε growthrate retard.When Mapping discard its current scene due to ε converging to k such that k ≥ | V | , itis still imitating RS-R. The reason is that we can assume that (cid:98) F (cid:101) needs to construct a validhamiltonian sequence fragment starting from u by calling Hamiltonian-Sequence recursivelyin order to check if v → u = F holds for u , directly or indirectly, since RS-R performs only v → u = T operations. If (cid:98) F (cid:101) can’t construct such valid hamiltonian sequence fragment startingfrom u , it’ll also discard G without aborting RS-R in order to return v → u = F to its caller,that in turn, either increments its error count by one or makes v → u = F hold for the remaining u . If v → u = F holds for every u ∼ v and (cid:98) F (cid:101) makes RS-R throw a non-catchable exception, (cid:98) F (cid:101) is predicting when its error rate curve distortion is about to be degenerated in order toabort RS-R.If Lemma 2 holds for Mapping , m = ϑ , with ϑ = | V | −| V | being the number of times that Mapping checks if v → u = T holds for every u found when it is not aborted in the worst casescenario. That’s because, f Lemma 2 holds for Mapping , for each vertex v i found by Mapping with i such that 1 ≤ i ≤ | V | , Mapping needs to check if v i → u = T holds for u ∼ v i at most | V | − i times.If m > ϑ , Mapping would be failing to retard ε growth rate. In this case, Mapping wouldbe using probability explicitly if it doesn’t abort itself since:(1) By Lemma 2, at least one v would have an unknown successor; and(2) It is tending to ignore Theorem 1 completely since every constraint is failing to make ε growth rate retard.However, when Mapping aborts itself due to m > ϑ , Mapping is still imitating RS-R sinceit enforces the stop condition of RS-R by aborting itself, seeing that the first instance of RS-Ralso checks if v → u = T holds for every u found | V | −| V | times when it is not aborted in theworst case scenario. As a consequence, m = ϑ must hold for m in order to prevent Mapping from aborting itself, which is not a sufficient condition to prove that
Mapping is not imitatingRS-R.In addition, notice that
Mapping can produce an incomplete L e , without aborting itself andwithout reaching its base case, when Sync-Error throws a non-catchable exception. Even so,
Mapping is still imitating the behaviour of RS-R since (cid:98) F (cid:101) can abort RS-R without visitingevery vertex from real scene when v → u = F holds for every u ∼ v . Furthermore, we canassume that:(1) (cid:98) F (cid:101) can change the first v = y of the first Hamiltonian-Sequence call, when y is pre-venting (cid:98) F (cid:101) from constructing a valid hamiltonian sequence S in order to not make RS-Rfail to produce a valid output, with S = v i ...v k such that | S | = | V | , 1 ≤ k ≤ | V | , 1 ≤ i ≤ k , v (cid:54) = y ; or 172) (cid:98) F (cid:101) can split H into different components with | H n | = 1 when it wants to create a hamilto-nian path S = S ∪ S such that S = v...r , S = v...r (cid:48) , r ∈ V ( H ), r (cid:48) ∈ V ( H ). In this case,when r or r (cid:48) is reached, (cid:98) F (cid:101) creates a new instance of RS-R to reach the remaining deadend, which consequently forces the current instance RS-R to reach its base case instead oftrying to enforce constraints 1 and 2.Therefore, Mapping producing an incomplete L e is not a sufficient condition to prove that Mapping is not imitating RS-R.As
Mapping imitates RS-R, even when it not reach its base case,
Mapping ignoring The-orem 1 partially is not a sufficient condition to make
Mapping imitate RS-E. Thus, it suggeststhat:(1) (cid:98) F (cid:101) can generate at least one hamiltonian sequence S = e i e i +1 ...e k − e k (if it exists) of H such that { S ∩ L e } (cid:54) = ∅ ; and(2) (cid:98) F (cid:101) can also generate at least one path S (cid:48) (cid:54) = ∅ , { S (cid:48) ∩ L e } (cid:54) = ∅ , that makes RS-R not enforceconstraints 1 and 2 explicitly in at least one of its states when (cid:98) F (cid:101) wants to either:- make RS-R abort itself in the absence of at least one constructable hamiltonian sequence S = e i e i +1 ...e k − e k ; or- construct a hamiltonian path S = S ∪ S such that S = v...r , S = v...r (cid:48) , r ∈ V ( H ), r (cid:48) ∈ V ( H ) by creating a new instance of RS-R to reach r when r (cid:48) is reached (or vice-versa)in order to force the current instance of RS-R to reach its base case instead of trying toenforce constraints 1 and 2.Thus, Mapping is goal-oriented with η = | V | and m = | V | −| V | . In this section, the reconstruction phrase is explained. The reconstruction task is done bythe
Reconstruct function, that takes following parameters as input by reference: H =( V, E, v , L, Ω) , L e , H ∗ , φ , P x , P x . The edge φ is a non-synchronized edge ( x x ) ∈ L e where x and x are initially the last vertices of two expandable paths P (cid:48) = ( x ) and P (cid:48)(cid:48) = ( x ),respectively. In addition, we need to assume that x = v and x = v in this phrase in orderto check if both constrains 1 and 2 hold for H [ V − v ]. P x = P (cid:48) will be the current path we’reexpanding and P x = P (cid:48)(cid:48) , the other path. As for every u , v must be added to either P x or P x , x and x must be properly updated in order to represent the last vertices of P x and P x ,respectively.The term expansion call is used throughout this paper whenever we make a recursive callto Reconstruct . Every expansion call restores the initial state of both H and L e . Someconventions are used in this section. The synchronized edges will be written as [ v, u ]. The edge[ w, (cid:3) ] is a synchronized edge e ∈ L e with w ∈ e . Definition 16.
A synchronized edge is either: (1) a non-synchronized edge ( v, u ) that gotconverted to [ v, u ] by Reconstruct ; or (2) an edge [ v, u ] added to L e by Reconstruct . The notation d ∗ ( x ) is used to represent the degree of a vertex x of a scene H ∗ , which is aclone of H (cid:48) ⊇ H scene of the current state of Reconstruct , such that V ( H ∗ ) = V ( H (cid:48) ) and E ( H ∗ ) = L e ∩ E ( H (cid:48) ). P v ( u ) function is used by Reconstruct to pass through H by using paths of H ∗ , startingfrom v ∈ { x , x } until it reaches z = u such that d ∗ ( z ) = 1. During this process, it performssuccessive H − v operation, converts edges from ( v, u ) to [ v, u ], and updates P v . When z isreached, it returns z . [ v, u ] cannot be removed from H unless by undoing operations performedby Reconstruct . 18 .2.1 Goal
The goal of reconstruction phrase is to reconstruct a hamiltonian sequence (if it exists) by passingthrough H in order to attach inconsistent C v H components. If such hamiltonian sequence isreconstructed, H ∗ will be a path graph corresponding to a valid hamiltonian sequence of themaximal H ≡ G . In order to do that, some edges may need to be added to L e to merge acomponent H ∗(cid:48) with v ∈ V ( H ∗(cid:48) ) to another component H ∗(cid:48)(cid:48) so that P v ( u ) can reach vertices u ∈ V ( H ∗(cid:48)(cid:48) ) properly.Notice that if Reconstruct passes through H (cid:48) in a scene H (cid:48)(cid:48) , with H (cid:48) being a scene ina state k of Mapping and H (cid:48)(cid:48) being the current scene of Reconstruct such that V ( H (cid:48) ) ∩ V ( H (cid:48)(cid:48) ) (cid:54) = ∅ , some edges ( v, u ) ∈ L e could be removed from H to make both constraints 1 and2 hold for H [ V − v ]. However, this is not a sufficient condition to prove that Mapping is notimitating the behaviour of RS-R. (see Sect. 3.4). Therefore,
Reconstruct can make bothconstraints 1 and 2 hold for H [ V − v ], even if some edges are removed from L e .The problem is that Reconstruct must decide when to abort the reconstruction process.Because of that, the non-existence of a sequence of C v H attachments that needs to be made inorder to convert L e to a hamiltonian sequence is part of the forbidden condition of SFCM-R.That’s because the following is an immediate corollary of Theorem 2. Corollary 1. If Mapping outputs L e , such set will be formed by path fragments that generate inRS-R context both (1) potential independent forbidden minors and (2) potential non-mandatory C v H components. It means that if such sequence of C v H attachments exists for the current L e , and it is notproperly enforced by Reconstruct , then it can be considered a possible sufficient conditionto make the mirrorable real scene algorithm , which is modified version of RS-R that we wantto mirror in this phrase, fail to produce a valid output. We call the output of modified RS-R hamiltonian sequence given L e , because it takes a non-synchronized hamiltonian sequence L e = L e − S as input, which S being a set of non-synchronized edges removed from L e by Reconstruct . Definition 17.
Let H = ( V, E, v , L, Ω) be a minimal scene. A hamiltonian sequence given L e is a simple path P = v i ...v k with ≤ i ≤ k of H , that visits all vertices, such that P ∩ { w ∈ P : |{ L e ∩ ( w, (cid:3) ) }| ≥ } (cid:54) = ∅ The modified version of real scene algorithm is as follows.19 lgorithm 9
Mirrorable RS-R algorithm
Input: H = ( V, E, v , L, Ω) , P x , P x , v ∈ { x , x } , L e Output:
Hamiltonian sequence P (cid:48) function Hamiltonian-sequence A ← N ( v ) U ← { u ∈ A : ( v, u ) ∈ L e } X ← ∅ if constraints 1 or 2 doesn’t hold for H [ V − v ] then A ← ∅ for each u ∈ A do if ( v, u ) ∈ U then if constraints 1 or 2 doesn’t hold for H [ V − { v, u } ] then X ← X ∪ { u } L e ← L e − ( v, u ) else A ← { u } X ← ∅ break else if v → u = F then X ← X ∪ { u } A ← A − X if A (cid:54) = ∅ then v → u = T with u ∈ A if ( v, u ) ∈ L e then Convert ( v, u ) to [ v, u ] else Remove an edge ( u, (cid:3) ) from L e if |{ L e ∩ ( u, (cid:3) ) }| > L e ← L e ∪ [ v, u ] Update P v u ← w ∈ { x , x } Hamiltonian-sequence( H , P x , P x , u , L e ) else if | P x ∪ P x | (cid:54) = | V ( H ) | then throw error B ← P x in reverse order P (cid:48) ← B ∪ P x return P (cid:48) In addition,
Reconstruct may have inconsistent subscenes H (cid:48) ⊃ H with non-attachable C v H components. It means that if we try to attach every inconsistent C v H by modifying L e aggressively, we could end up with SFCM-R imitating RS-E. Remember that SFCM-R mustnot use exhaustive methods to reconstruct the hamiltonian sequence since we want to mirrora non-exhaustive algorithm. Therefore, we need to use a goal-oriented approach in order toattach inconsistent C v H properly without relying on probability and find a valid sequence of C v H attachments. (see Sect. 3.3) In this section, the pseudocode of
Reconstruct is explained. Every line number mentionedin this section refers to the pseudocode of
Reconstruct . Initially,
Reconstruct takes thefollowing parameters as input: H = ( V, E, v , L, Ω) , H ∗ , φ = ( v , (cid:3) ), L e , P x = { x } and P x = { x } , with x = v ∈ φ and x = { w ∈ φ : w (cid:54) = v } . Reconstruct passes through H by using paths of H ∗ , performs subsequent H − v operationsby expanding P x or P x paths alternatively with v such that v ∈ { x , x } (line 7), and connectscomponents of H ∗ by adding a synchronized edge [ v, u ] (line 27). During this process, it needs20o remove some inconsistent edges ( v, u ) ∈ L e in its current state considering the following cases.I. The first case is when we have ( v, v H ).II. The second case is when H v H (cid:54) = ∅ and ( v, u ) doesn’t enforce both constraints 1 and 2.III. The third case is when ( v, x ) or ( v, x ), since both P x and P x are concatenated to formthe output of mirrorable RS-R algorithm.Notice that I or II could be ignored in hamiltonian path context since both P x and P x canhave non-adjacent dead ends. As Reconstruct considers these two cases inconsistencies, weneed to use specific goal-oriented strategies if we want to reconstruct a hamiltonian path (seeSect. 3.3.3).
Algorithm 10
Reconstruction of a hamiltonian sequence given L e (Simplified) Input: H = ( V, E, v , L, Ω) , H ∗ , L e , φ , P x , P x Output:
Set L e of synchronized edges function Reconstruct S ← ( ∅ ) while reconstruction of L e is not done do try if constraint 1 or 2 doesn’t hold for H [ V − v ] then throw error v ← P v ( u ) with d ∗ ( u ) = 1 S ← ( ∅ ) S ← ( ∅ ) S ← ( ∅ ) for each non-synchronized e ∈ L e do if w ∈ e with w being a valid non-visited w ∼ v then if d ∗ ( w ) = 1 then S ← S ∪ e if d ∗ ( w ) = 2 then S ← S ∪ e for each non-mapped w with d ∗ ( w ) = 0, w ∼ v do e ← ( w, w ) L e ← L e ∪ { e, e } S ← S ∪ e S ← S ∪ S S ← S ∪ S if S (cid:54) = ∅ then u ← w with ( w, (cid:3) ) ∈ S if v → u = T then L e ← L e − S L e ← L e ∪ [ v, u ] v ← q ∈ { x , x } else throw error catch error L e ← L e − S Undo k states Use goal-oriented strategies return L e If Reconstruct finds a valid v with d ∗ ( v ) = 1, the next step is to choose w ∼ v (line 24),which will be the successor of v , by using the following conventions in an ordered manner.1. If S (cid:54) = ∅ , choose w (cid:48) ∼ v of the first element ( w (cid:48) , (cid:3) ) ∈ S .21. If S = ∅ , remove the first element y = ( w (cid:48)(cid:48) , (cid:3) ) ∈ S from L e in order to make d ∗ ( w (cid:48)(cid:48) ) = 1, w (cid:48)(cid:48) ∼ v hold for w (cid:48)(cid:48) , then choose w (cid:48)(cid:48) such that z = { L e ∩ ( w (cid:48)(cid:48) , (cid:3) ) } − y , w (cid:48)(cid:48) ∈ z .3. If z is removed from L e because of I, II or III in next state, perform L e ∪ y and remove z from L e instead of y , Then, choose w (cid:48)(cid:48) such that y = { L e ∩ ( w (cid:48)(cid:48) , (cid:3) ) } − z , w (cid:48)(cid:48) ∈ y . Observation 3.
Whenever a goal-oriented strategy removes either ( v, (cid:3) ) or [ v, (cid:3) ] from L e , andmakes d ∗ ( v ) = 1 hold for v , Reconstruct must use the conventions of this section in order tochoose a non-visited u Notice that
Reconstruct temporarily changes d ∗ ( w ) when d ∗ ( w ) = 0 (line 19) in order toforce P v to use the aforementioned conventions even when w is an non-mapped vertex.If an inconsistency is found during this process, an error needs to be thrown by Recon-struct (lines 6 and 30). Every inconsistent C v H component must be attached by goal-orientedstrategies (lines 34). Because of that, Reconstruct undoes modifications in H , L e , P x , and P x (line 33), in order to go back to an earlier v state to be able to use some goal-oriented strat-egy to attach inconsistent C v H components. The reconstruction process continues until eitherthe reconstruction of L e is done (line 3) or a goal-oriented strategy aborts the reconstructionprocess.As an example of hamiltonian sequence reconstructed by SFCM-R, Figure 3.2.2 shows anarbitrary graph H mapped by Mapping function with v = 23 on the left side. On the right side,we can see the non-synchronized hamiltonian sequence of H reconstructed by Reconstruct .Figure 1: Example of minimal scene mapping with v = v = 0 (on the left side). Hamiltoniancircuit reconstructed with φ = (0 , x = 23, v = x = 0, µ x = 81 ,
48% (on the right side)In this figure, purple edges represent synchronized edges added by
Reconstruct to connectcomponents of H ∗ . The red edges represent non-synchronized edges that got converted tosynchronized edges by Reconstruct . The green edges represent synchronized edges [ v, w ]that were added to L e in order to attach an inconsistent C v H component with w ∈ C v H . x isthe final state of reconstruction process. In this section, the goal-oriented approach is presented and can be used in a non-probabilisticgoal-oriented implementation of reconstruction phrase. The main goal of using a goal-orientedapproach is to prevent SFCM-R from imitating RS-E during the reconstruction process. Beforecontinuing, we define a structure that we use to help
Reconstruct to make goal-orientedchoices. Such structure will be called real-scene perception network (RSPN) , and we use it tostore informations related to goal-oriented strategies.
Definition 18.
Real scene perception network (RSPN) is a directed tree-like goal-orientednetwork that starts at
RSPN node, which has the following children set { A , C , J , N } , where = { a i ...a n } is the the attachment node, C = { c i ...c n } is the current state node, J = { j i ...j k } is the ordering node, and N = { n i ...n k } is the region node. It’s very important to store some informations about goal-oriented strategies since the onlydifference between an expansion process from another is the way we pass through H by usingedges e ∈ L e , which can lead to the creation of different attachable C v H . Because Recon-struct has conventions to pass through H by using paths of H ∗ , RSPN and strategies can beuseful to change L e relying on knowledge related to real scene instead of probability in order togive such conventions more flexibility.Before continuing, two rates need to be defined. Definition 19.
The negativity rate γ is the sum of f γ ( x = 0 , a γ i ) from states i = 0 to currentstate z and represents the rate of how likely is the current state z of reconstruction process to beinconsistent. f γ ( x, a γ i ) = 1(1 − a γ i ) √ π e − x (1 − a γ i ) 0 ≤ a γ i < , x ≤ γ = z (cid:88) i =0 f γ ( x = 0 , a γ i ) (5) Definition 20.
The tolerance rate t is the sum of degree of tolerance over γ from states i = 0 to current state z of reconstruction process. t = z (cid:88) i =0 f γ ( x = 0 , a γ i ) + t i (6)As Reconstruct undoes k states to attach inconsistent C v H components, γ growth ratemust be adjusted whenever a specific strategy fails to attach a C v H properly. A tolerance policy (cid:98) T (cid:101) is needed to adjust γ and t in order to select and trigger a goal-oriented strategy in anappropriate moment. (cid:98) T (cid:101) must also prevent SFCM-R from imitating RS-E by making, whatwe call curve distortion ring (cid:110) ( γ, t ), be disintegrated in some state of Reconstruct . (cid:110) ( γ, t )is disintegrated when it returns F . (cid:110) ( γ, t ) = (cid:26) T, if t − γ > F, otherwise (cid:27) (7)The disintegration of (cid:110) ( γ, t ) made by (cid:98) T (cid:101) is used to make Reconstruct perform a newexpansion call. These expansion calls, in turn, makes SFCM-R be more prone to degenerateitself in case of successive negative events that makes
Reconstruct be tending to imitateRS-E explicitly, which is invalid (refer to section 3.3.2 to understand how this process works).Therefore, (cid:98) T (cid:101) needs to adjust a γ i and t i of every state i by using a set of actions in order toaccomplish the aforementioned goals. 23 lgorithm 11 Tolerance policy (cid:98) T (cid:101) s ← current state of Reconstruct Adjust t s and a γ s Update RSPN if needed S ← ∅ { set of goal-oriented strategies } Populate S while s is inconsistent do for each goal-oriented strategy s (cid:48) ∈ S do Trigger s (cid:48) inside Reconstruct environment s ← current state of Reconstruct
Adjust t s and a γ s Update RSPN if needed if s is consistent then break Populate S Go back to
Reconstruct environmentTherefore, one of the main goals of (cid:98) T (cid:101) is to keep a balance between: (1) retarding the growthrate of both f ( x, a γ i ) and γ by triggering goal-oriented strategies that attach inconsistent C v H components properly; and (2) not retarding the growth rate of both f ( x, a γ i ) and γ when somegoal-oriented strategy fails to attach inconsistent C v H components; in order to Reconstruct be able to continue to reconstruction process without imitating RS-E. Later in this paper, wewill prove that a potential hamiltonian sequence can be reconstructed by SFCM-R if (cid:98) T (cid:101) isoptimizable (see Sect. 3.4). Definition 21.
Let H = ( V, E, v , L, Ω) be a minimal scene. A tolerance policy (cid:98) T (cid:101) is opti-mizable if it computes the following constrained optimization problem, with S being a set thatcontains every inconsistent state i found with (cid:110) ( γ i , t i ) = F , without making Reconstruct failto produce a valid output while behaving like a non-exhaustive algorithm. arg min t i (cid:88) i ∈ S ( t i − γ i ) subject to: t i , γ i ∈ R , γ i > t i (8) To understand how this process can prevent SFCM-R from imitating RS-E, we can intuitivelythink of the retardation of γ growth rate process as the following simplified quantum-inspiredprocess. In this process, we assume that a distortion ring (cid:110) has N distortion particles p + , thatcan behave like their own anti-distortion particles p − , and vice-versa. When p − and p + collide,they annihilate each other.Let N (cid:110) be the number of distortion particles p + expected to be observed in distortion ring (cid:110) , and N (cid:110) be an unknown non-observed number of anti-distortion particles p − in distortionring (cid:110) . In addition, let E T ( x ) = E + ( x ) + E − ( x ) be the sum of Electromagnetic (EM) wavesemitted in (cid:110) as a function of time, E + ( x ) be imaginary EM waves that are expected to beemitted from observed p + particles as a function of time, and E − ( x ) be imaginary EM waveswith opposite charge that are expected to be emitted from observable p − particles as a functionof time. To simplify, we assume that N (cid:110) is equivalent to the positive amplitude peak of E + ( x )and N (cid:110) is equivalent to the positive amplitude peak of E − ( x ).The idea here is to consider a consistent state s of reconstruction process p + particles, aninconsistent state s (cid:48) of reconstruction process p − particles, and Reconstruct the observer ofboth p − and p + . The following equations are used in this explanation. E + ( x ) = sen ( βx ) (9) E − ( x ) = δ γ sen ( β x T ( x ) = E + ( x ) + E − ( x ) (11) α, β = 2 (12)In addition, we use the following sigmoid function to represent a γ i , which is the γ growthrate in state i . For conciseness, we assume that (cid:98) T (cid:101) updates both t i and a γ i whenever δ γ ischanged to avoid repetition. a γ i = f ( δ γ ) = (cid:32)
11 + e − ( δ γ ) (cid:33) ≤ f ( δ γ ) < (cid:110) is represented by a circle whose center is the point ( C (cid:110) ,
0) , as illustratedin next figure. C (cid:110) is represented by the following equation. C (cid:110) = 4 πβ (14)The force of (cid:110) (or simply F (cid:110) ) is equal to the amplitude A of the second inner wave of (cid:110) from E T ( x ). If A > (cid:110) (in blue) with F × = A . If A is a negative peak amplitude or A = 0, we have an observableanti-distortion ring (cid:110) (in red) with F (cid:110) = − A .By convention, we use a dashed blue line for E + ( x ), a solid red line for E − ( x ), and a solidblue line for E T ( x ). Figure 2 shows (cid:110) with F (cid:110) > E + ( x ) > (cid:110) with the F (cid:110) > E + > E + ( x ) = E T ( x ).Before continuing, consider the following corollary of Theorem 2 Corollary 2.
Mapping performs an error curve distortion conceptually equivalent to (cid:98) F (cid:101) , inorder to distort its potentially-equivalent error rate curve, even when Sync-Error throws anon-catchable error or makes
Mapping abort itself.
By Corollary 2, we can assume that, neglecting some technical complexities, the observablecollision rate between p − and p + , can be maximized in a way that it favours p + over p − . In otherwords, we want to observe N (cid:110) > N (cid:110) in order to be sure that we have a consistent observable (cid:110) .Because of that, E + ( x ) represents the EM waves emitted from a total of R T = N (cid:110) − N (cid:110) , R T > p − and p + particles, even if the observed R T happens to changedue to the principle of superposition of states in quantum mechanics as Reconstruct passesthrough H .An ideal scenario is represented in Figure 3.Figure 3: Ideal distortion ring (cid:110) with E T ( x ) > δ γ = 025n this scenario, the curve of f γ ( x, a γ i ) is illustrated in Figure 4. The blue line from s to γ = s n represents the curvature of the distorted curve that Mapping created, that is expectedto exists by Corollary 2, given an optimizable (cid:98) T (cid:101) and a scene H with at least one hamiltoniansequence.As the observable collision rate between p − and p + can be maximized in a way that itfavours p + over p − , an optimizable (cid:98) T (cid:101) assumes that Reconstruct is expected to terminateits execution by observing (cid:110) and projecting a curvature of a non-exponential curve between s and s n . In other words, Reconstruct is expected to terminate its execution without abortingitself, with a small N (cid:110) . Because of that, an optimizable (cid:98) T (cid:101) considers the curve of f γ ( x, a γ i )the curvature of a potential error rate curve of a Reconstruct instance that runs withoutaborting itself in the worst case scenario.Figure 4: Curve of f ( x, a γ i ) created by (cid:110) , with δ γ = 0, E T ( x ) > E T ( x ) after the observation of particles p − in a state with δ γ = α . Noticethat the increase of E − ( x ) was not enough for the amplitudes peaks of E − ( x ) to be greaterthan the amplitude peaks of E + ( x ) and E T ( x ).Figure 5: Distortion ring (cid:110) observed with E − ( x ) < E + ( x ) and δ γ = α The f γ ( x, a γ i ) curve in the scenario is illustrated in Figure 6. Such curve is representingthe curvature of a non-exponential curve, which is a desired curvature since we want to imitateRS-R. Figure 6: Curve of f ( x, a γ i ) created by (cid:110) , when E − ( x ) < E + ( x ) and δ γ = α Therefore, if
Reconstruct finds a valid hamiltonian sequence in a state with E − ( x )
Reconstruct observed that the maximization of collisionrate between p − and p + doesn’t favoured p + over p − , since (cid:110) was spotted behaving like a (cid:110) . Inother words, the expected imbalance between p + and p − was not observed, which means that (cid:110) is disintegrated. Because of that, the non-exponential curvature of f γ ( x, a γ i ) became unstable.As f γ ( x, a γ i ) became unstable, (cid:98) T (cid:101) could also make the real error rate function of Recon-struct collapse to + ∞ in order to force the current instance of Reconstruct to ”jump” intoan imaginary state of RS-E and, at the same time, make f ( x, a γ i ) project a curvature of anexponential curve. Such curve is illustrated in the following figure.Figure 8: Curve of f ( x, a γ i ) created by (cid:110) with δ γ = α As a consequence, (cid:98) T (cid:101) can: (1) negate the definitions of E − ( x ) and E + ( x ); and (2) make E T ( x ) collapse to E + ( x ) by setting δ γ = 0. If so, we will have an ideal anti-distortion ringwith almost the same inner structure of the observed anti-distortion ring (cid:110) showed in Figure 7.Figure 9 shows such ideal anti-distortion ring.Figure 9: An ideal anti-distortion ring (cid:110) with δ γ = 0In this scenario, (cid:98) T (cid:101) can force a new expansion call, which makes SFCM-R be more proneto degenerate itself in case of successive negative events that make Reconstruct be tendingto imitate RS-E explicitly. This is a desired behaviour of (cid:98) T (cid:101) since SFCM-R can’t imitate RS-Eexplicitly.In conclusion, (cid:98) T (cid:101) is essentially taking advantage of Corollary 2 since it implies that theobservable collision rate between p − and p + can be maximized in a way that it favours p + over p − , given an optimizable (cid:98) T (cid:101) and a scene H with at least one hamiltonian sequence.27herefore, the main goal of (cid:98) T (cid:101) is to force Reconstruct to imitate RS-R in order to minimizethe observation of p − particles, which represent inconsistent states, and consequently try toprevent Reconstruct from behaving like RS-E explicitly.
In this section, we present the goal-oriented strategies that SFCM-R needs to use to reconstructa hamiltonian sequence. We call them general goal-oriented strategies due to the fact that theycan be used to reconstruct both hamiltonian paths or hamiltonian circuits. The goal-orientedproposed in this section are primarily focused on keeping H connected while preventing SFCM-R from imitating RS-E. Because of that, we assume that every goal-oriented strategy presentedin this section is enforcing both constraints 1 and 2. Please refer to section 3.3.3 to see specificstrategies for hamiltonian path, that allow P x and P x to have non-adjacent dead ends whenit’s needed. Observation 4.
The strategies proposed in section 3.3.2 and 3.3.3 don’t have necessarily anorder of activation. It depends on how (cid:98) T (cid:101) is implemented, and specific signs that suggest thata specific strategy should be triggered by (cid:98) T (cid:101) in Reconstruct environment.
Before continuing, we need to define some conventions. Every inconsistency v H ∈ H v H mustbe added to current state node C when (cid:110) ( γ, t ) = T . If (cid:110) ( γ, t ) = F , every v H ∈ H v H must beadded as a child of v H i node in expansion call i . Such v H i node is called static v H articulation and it must be child of J . Every C v H attached by adding an edge [ v, w ] to L e with w ∈ C v H ,must be added to attachment node A .The first strategy is to have φ = ( v H , (cid:3) ), with v H being a v H added to v H i of J , for everynew expansion call made when (cid:110) ( γ, t ) = F . As an example, the figure bellow shows the node J of RSPN. We can see on the left side an expansion call k − j = v H = { w , w , w } that was created in expansion call k −
3, and another node j = v H = { w } that was created inexpansion call k − J in expansion call k with a node j = v H = { w } that was created in expansion call k −
1. In such case, J was updated since w can’t be partof two ordering constraints at the same time due to the fact that every vertex is visited once inhamiltonian sequence context.Therefore, w was removed from v H in expansion call k −
1. As w and w are the uniquenodes of v H and v H respectively, we can enforce the ordering between w and w in expansioncall k . In this case, j = v H and j = v H are active . Such enforcement could result in non-synchronized edge removal operations in current expansion. In addition, if j i = v H i has onlyone child, j i = v H i can’t be changed anymore.Figure 10: RSPN’s node J of expansion call k − k (on right side)By Strategy 1, if SFCM-R runs in exponential time, we’ll no longer have a consistent mini-mal scene mapping. Such situation forces Reconstruct to choose by probability. As
Recon-struct can’t choose by probability, (cid:98) T (cid:101) will be forced to make SFCM-R abort itself since γ willgrow exponentially by using the following strategy. Therefore, this strategy forces the numberof expansion calls to not grow exponentially. Strategy 1.
Make a new expansion call i with φ = ( v H , (cid:3) ) such that v H ∈ H v H when (cid:110) ( γ, t ) = F and add every vertex v H ∈ H v H to a child j i = v H i of node J . Update J and enforce orderingbetween j i = v H i and j k = v H k with k > i , if both are active. If (1) such ordering can’t beenforced or (2) v H i = ∅ , γ must grow exponentially in order to make SFCM-R abort itself. Mapping and RS-R have pre-synchronized forbidden conditions. It means that (cid:98) T (cid:101) must avoid making SFCM-R abort itselfby Strategy 1. Also notice that high peaks of γ can theoretically make J store an inconsistentordering as the number of expansion calls grows. Even if it happens, J can’t be changedarbitrarily.Therefore, (cid:98) T (cid:101) must try to retard γ growth rate faster instead of making SFCM-R abortitself, in order to: (1) prevent a new expansion call; or (2) add another inconsistent H v H set to J that either postpones the activation of static v H points or causes less non-synchronized edgeremoval operations when (cid:110) ( γ, t ) = F . Strategy 2.
Make a new expansion call i with φ = ( v H , (cid:3) ) such that v H ∈ H v H when (cid:110) ( γ, t ) = F with H v H being a set that either postpones the activation of static v H points or causes lessnon-synchronized removal operations. (cid:98) T (cid:101) can also prevent the number of expansion calls from growing exponentially by preventing Reconstruct from making expansion calls to expand the same P x twice. Thus, we have thefollowing strategy. Strategy 3.
Every expansion call must have a different x ∈ φ As mentioned earlier, each expansion call i generates a static j i = v H i that must be addedto node J of RSPN. However, SFCM-R needs to assume that the exactness rate is enoughfor reconstruction process since Reconstruct must use paths of H ∗ , which is goal-oriented byTheorem 2, to pass through H . The exactness rate µ x is the rate of how many non-synchronizededges got converted to synchronized edges from state i = 0 to current state x . The more edgesare removed from L e , the lower is the exactness rate µ x . The λ i function outputs a set of ( v, (cid:3) )edges that was removed from L e in state i . S is the number of edges e ∈ L e before reconstructionphrase. µ x = (cid:32) − x (cid:88) i =0 | λ i | S (cid:33) (15)As we need to assume that the exactness rate is enough for reconstruction process, we wantto restart the process considering P x as P x before making a new expansion call when γ > t .In this case, we have a path swap since P x becomes P x and vice-versa. Strategy 4.
Before making a new expansion call, make a path swap in order to restart theprocess starting from P x path instead of P x path. In addition, we need to use a lazy approach in order to assume that the exactness rate isenough for reconstruction process. As an example, if we undo k states to attach some incon-sistent C v H , we need to assume that such C v H will be properly attached without analysing theconsequences of such attachment in its region. Strategy 5.
Any inconsistency correction must be made by using a lazy approach.
The negativity rate can be also used when
Reconstruct connects components of H ∗ byadding [ v, u ] successively in non-mapped regions, with u such that d ∗ ( u ) = 0. In this case, Re-construct is tending to ignore H ∗ paths completely and consequently imitate RS-E, speciallywhen d ∗ ( u ) = 0 holds for every non-visited u in the absence of inconsistent C v H componentsthat need to be attached. Because of that, γ growth rate must be increased in this case.In addition, the number of times that Reconstruct can do it must be limited by a variablethat is decreased as γ growth rate is increased. This strategy is particularly useful to makeSFCM-R degenerate itself when Reconstruct takes an incomplete L e as input, that wasproduced by Mapping without reaching its base.
Strategy 6. If d ∗ ( u ) = 0 holds for every non-visited u and Reconstruct successively con-nects components of H ∗ by adding [ v, u ] with d ∗ ( u ) = 0 , γ growth rate must be increased. If Reconstruct connects such components of H ∗ in the absence of inconsistent C v H componentsthat need to be attached, γ growth rate must get increased drastically. trategy 7. The number of times that
Reconstruct connects components of H ∗ by adding [ v, u ] successively, with u such that d ∗ ( u ) = 0 must be limited by a variable that is decreased as γ growth rate is increased. Observation 5.
Consider the following corollary of Theorem 2.
Corollary 3. If (cid:98) T (cid:101) is optimizable and Reconstruct wants to reconstruct a hamiltonian path, (cid:98) T (cid:101) may need to compute P x in a new instance of SFCM-R when P x reaches a dead-end if Reconstruct takes an incomplete L e as input, with L e being a non-synchronized hamiltoniansequence that was produced by Mapping without reaching its base case.By Corollary 3 , (cid:98) T (cid:101) may need to create a new instance of SFCM-R in order to prevent itselffrom using Strategy 6 or 7 to degenerate the current instance of SFCM-R in a wrong moment,due to the fact that the reconstruction of P x in a different instance of SFCM-R could make thecurrent instance of SFCM-R reach its base case instead of trying to enforce constraints 1 and 2. We can also use γ to make k increase or decrease. For example, if Reconstruct triesto attach an inconsistent v H stored in node C of RSPN by undoing k states in order to adda synchronized-edge [ v, w ] such that w ∈ C v H and ( v, w ) / ∈ L e and every attachment attemptkeeps generating another inconsistencies for every non-visited w ∈ N ( v ) found, then γ growthrate and k can be increased at the same time to prevent SFCM-R from imitating RS-E.As a result, Reconstruct undoes k (cid:48) states such that k (cid:48) > k in order to not visit allneighbours of v . Therefore, k must be proportional to γ growth rate assuming that region R is treatable by expanding P x or P x . Such relationship between k and γ growth rate helps Reconstruct to attach frequently inconsistent C v H components. On the other hand, if wecan’t find any attachable C v H component by undoing k states due to a high peak of γ , wecan just delete the synchronized edge that is generating them, since they could happen to beattachable later. Strategy 8.
Undo k states until we find the first inconsistent C v H stored in C node attachablethrough w with w such that S = { w ∈ N ( v ) : ( w ∼ C v H ) (cid:54) = ∅ ∧ ( w was not visited ) } , S (cid:54) = ∅ , andremove the inconsistent [ v, u ] edge from L e . Then, choose a non-visited w with w ∈ S , and adda synchronized-edge [ v, w ] such that w ∈ C v H and ( v, w ) / ∈ L e . If no attachable C v H componentis found in any previous states, due to a high peak of γ , then increase γ growth rate, remove theinconsistent [ v, u ] and go back to the former v = y in order to choose another non-visited u ∼ y . Strategy 9.
The variable k must be proportional to γ growth rate assuming that region R istreatable by expanding P x or P x . The node A can have some properties node to make Reconstruct keep track of an incon-sistent region R that Reconstruct wants to correct by triggering a strategy. The total cost needed to attach an inconsistent C v H and its appearance frequency can be used by (cid:98) T (cid:101) to detectif SFCM-R is tending to behave like RS-E. The total cost needed to attach an inconsistent C v H can be represented by the following equation, where: ∆ γ ( s,a i ) = γ s − − γ s ; γ s − is the value of γ of state s − γ s is the value of γ of an inconsistent state s ∈ S where a i = C v H , a i ∈ A ,appeared as inconsistency; and p ( s, a ) is an extra cost directly proportional to the appearancefrequency of a i = C v H in s ∈ S . A .cost ( a i ) = (cid:88) s ∈ S ∆ γ ( s, a i ) + p ( s, a i ) (16)Because of that, (cid:98) T (cid:101) needs to make γ growth rate increase as both the total cost needed toattach an inconsistent C v H of R , and its appearance frequency, tends to increase.Thus, we have the following strategy. Strategy 10.
Make γ growth rate increase, as A .cost ( a i ) gets increased. Furthermore, we can also use the negativity rate along with attached C v H stored in A tochange the variable k . Thus, A can be used by Reconstruct to keep track of specific regions30n current expansion, serving as an extra parameter to change k . As an example, we can undo k states until we find an arbitrary C v H that was attached in current Reconstruct call.As mentioned before, we need to store inconsistent C v H components in node C before us-ing any attaching strategy. However, SFCM-R can’t imitate RS-E by trying to attach themaggressively. Thus, the following strategies could be useful to prevent SFCM-R from imitatingRS-E. Strategy 11.
Avoid adding new C v H (cid:48) to C node until we have at least one well-succeeded C v H attaching. Strategy 12.
If attachment attempts always generates new C (cid:48) v H components, γ growth ratemust be increased drastically. In such case, try to attach additional v H (cid:48) components by addingthem to C node and giving them a higher priority. Also, the number of C v H of C node can be limited by a variable that is decreased as γ growthrate is increased. Such strategy forces Reconstruct to not try to attach C v H componentsaggressively when we have successive peaks of γ . Strategy 13.
The number of C v H components considered by current state must be limited by avariable that is decreased as γ growth rate is increased. Notice that once we have a valid attachable C v H , the remaining C v H components can’t bechosen by probability. As the choice of remaining C (cid:48) v H components must be explicitly tied to agoal-oriented strategy, (cid:98) T (cid:101) can remove these v H vertices from C since SFCM-R assumes that µ x is enough for reconstruction process. Strategy 14.
Remove every C v H from C node for every v after a valid attachable C v H is found. As mentioned earlier, if we try to attach every C v H aggressively we can end up with SFCM-Rimitating RS-E, since we can have subscenes with only invalid C v H components. In other words,there is no guarantee that every C v H found in every R of vertices will be consistent withoutmaking any expansion call. Also, SFCM-R assumes that every vertex w is reachable through v or v . It means that there may exist components C v H only attachable though P x . In bothcases, P x is overlapping P x since an inconsistent region R can happen to be consistent byeither: (1) making a path swap in order to expand P x to correct inconsistencies; or (2) makinga new expansion with a different x ∈ φ . Definition 22.
A path overlapping in a region R of vertices is when: (1) P x needs to passthrough R to attach or cancel the appearance of inconsistent C v H components found by expanding P x ; or (2) φ needs to be changed in order to attach or cancel the appearance of inconsistenciesfound by expanding P x or P x . A path overlapping can occur in many cases. For example, if H − P x generates a non-reachable component H (cid:48)(cid:48) with V ( H (cid:48)(cid:48) ) ∩ { x , x } = ∅ , H (cid:48)(cid:48) is clearly invalid in both hamilto-nian circuit and hamiltonian path context. Also, we can have, in hamiltonian circuit context, A ( x , H ) = T holding for x by expanding P x , or even worse, successive peaks of γ in a region R . If we have successive peaks of γ in a region R , there may exist a C v H component frequentlyinconsistent by expanding P x , suggesting that it may be attachable by expanding P x . Anothersign of path overlapping is when A ( x , H ) = T holds for x in hamiltonian circuit context, and H − P x generates a component H (cid:48) with x ∈ V ( H (cid:48) ) and | V ( H (cid:48) ) | being very small. This signsuggests that such H (cid:48) can’t be generated by P x .In such cases, the path overlapping correction strategies can be useful since we may finddifferent C v H components by expanding P x that can degenerate such inconsistencies withoutmaking new expansion calls. Therefore, we have the following strategy. Strategy 15.
If we have a path overlapping in some R in P x , undo k states and make a pathswap, so that we can pass through R by expanding P x . If path overlapping is corrected, makeanother path swap to continue the reconstruction process through former P x . As an alternative, instead of making a path swap to continue this process through former P x , we can continue through P x without making a path swap.31 trategy 16. If we have a path overlapping in some R , undo k states and make a path swap,so that we can pass through R by expanding P x . Continue through P x until we have anotherpath overlapping. Also, we can continue this process through P x until we have a new inconsistent C v H ∼ x in either current P x state or earlier states with a different v = x . If such C v H is found, weundo the states created after the path swap and then, make another path swap to go back to P x in order to attach C v H . The goal here is to generate new inconsistent C v H components tobe attached by P x and change P x without relying on probability. Strategy 17.
If we have a path overlapping in some R , undo k states and make a path swap, sothat we can pass through R by expanding P x . If path overlapping is corrected, continue through P x until we have new inconsistent C v H ∼ x in either current P x state or earlier states witha different v = x . If such C v H is found, undo the states created after the path swap and then,make another path swap to go back to former P x in order to attach such C v H . As we’re ignoring u = v A vertices in Mapping , we can have sequences of creatable com-ponents H (cid:48) ⊃ H with | H n | = 2, | H c | = F when Reconstruct is passing through a potential v H -path. If Reconstruct needs to attach an inconsistent C v H of a potential v H -path, wecould choose an attachable C v H of one of its endpoints in order to not make γ growth rateget increased drastically. Such endpoints will be C v H components that appear as inconsistencyfrequently. Strategy 18.
Undo k states until we find the first attachable C v H of an endpoint of a potential v H -path instead of making γ growth rate increase drastically. Before continuing, we need to define the last type of vertex mentioned in this paper, thatwill be called C v H generators or simply v G . Definition 23. ( C v H generator) Let H = ( V, E, v , L, Ω) be a minimal scene. A vertex w ∈ V is a C v H generator when | H [ V − w ] v H | > | H [ V ] v H | . From a technical point of view, v G is not C v H . On the other hand, if we consider v G asan inconsistent C v H , v H = v G , we can degenerate it so that the unwanted C v H components arenot created by v G . Also, we can degenerate it by considering such unwanted C v H componentsinconsistencies if we want to change the inconsistent v H -path that v G is about to create. As v G is not an explicit C v H , this kind of event must make γ growth rate increase but it’s particularlyuseful in very specific cases.As an example, let w be a vertex that for every H (cid:48) ⊇ H , H (cid:48) − w generates two potential v H -paths starting from w . It means that there’s only one way to reach w without having P x and P x being paths with non-adjacent dead ends. If w needs to be attached as C v H , using alazy approach here could make γ growth rate increase. So we have to assume that either v G , orsuch unwanted C v H components created by v G , are inconsistencies in order to attach w properly.Another example is when we have unwanted C v H components preventing (cid:98) T (cid:101) from makingminimal scene attachments through Strategy 8. If these unwanted C v H are properly attached, (cid:98) T (cid:101) can prevent itself from using path overlapping correction strategies. As a result, this strategycan make Reconstruct undo a small number of states, which can retard γ growth rate andconsequently postpone the need of a new expansion call.We can also use this strategy to enforce the ordering constraints of ordering node J , or whenwe have signs that suggests that there exists hidden region ordering constraints. A possible signof hidden region ordering constraints is when Reconstruct finds itself using path overlappingcorrection strategies that generate always almost the same C v H components from P x and P x with no significant progress. In this case, Reconstruct would just make a new expansion calldue to a high peak of γ in order to enforce such ordering by using ordering constraints of node J . However, (cid:98) T (cid:101) can try to use this strategy before making a new expansion call when thesecomponents are about to force either P x or P x to create a wrong region ordering. Strategy 19.
If there’s unwanted C v H components created by v G , assume that v G or suchunwanted C v H components are inconsistent C v H components that need to be attached, make γ growth rate increase and try to attach these inconsistencies.
32e can also store valid sequences of minimal scene attachments in the region node N ofRSPN whenever we find inconsistencies that cause successive peaks of γ . In this case, a usefulstrategy is to create a temporary expansion call with φ = ( w, (cid:3) ) with w being frequently partof non-attachable C v H in current expansion, store a valid sequence of attached C v H componentsin N and enforce this sequence of attachments through P x by using a lazy approach locally. Itmeans that (cid:98) T (cid:101) will not enforce this sequence at first. It must enforce parts of such sequence ofattachments progressively only if it finds successive peaks of γ .The goal of this strategy is to minimize the number of expansions calls since we’re enforcinga known valid sequence of C v H . It’s important to mention that in order to enforce such ordering,these C v H components need to appear as inconsistency explicitly. Therefore, such strategy is anextra parameter to change k . As an example, Reconstruct can undo k states until it finds avertex v ∈ C v H with C v H being part of a valid sequence of attachments. Strategy 20.
If we have high peaks of γ in a region R of vertices, then create a temporaryexpansion call with φ = ( w, (cid:3) ) with w being frequently part of non-attachable C v H componentsin current expansion in order to find and store a valid sequence of attached C v H components in N . Next, enforce this sequence of attachments C v H through P x or P x progressively by using alazy approach locally. If this strategy fails, γ growth rate must be increased drastically. As this strategy doesn’t assume that µ x is enough to reconstruct the hamiltonian sequencein region R , it must be used only in very specific cases. As an example, such strategy couldbe used when (cid:98) T (cid:101) is about to abort the reconstruction process or detects that the number ofexpansion calls is increasing very fast with no significant progress whenever Reconstruct triesto pass through such region.
In this section, we present specific goal-oriented strategies that SFCM-R needs to use to recon-struct a hamiltonian path. As mentioned earlier, the goal-oriented strategies of section 3.3.2 arefocused on keeping H connected, considering v → v H = T as an inconsistency. However, P x and P x may have non-adjacent dead ends in hamiltonian path. In this case, we can have up toone v → v H = T . In other words, we can have 0 ≤ ∆( H ) ≤ − d with ∆( H ) being the numberof creatable components H (cid:48) ⊃ H , V ( H (cid:48) ) ∩ { x , x } = ∅ , | H n | = 1, | H c | = F , and d = 0 being avariable that is incremented when x or x reaches a dead end.Notice that the same strategies can be used in hamiltonian path context. In this context, wecan ignore at least two C v H attaching operations. If these C v H components happen to be non-reachable by P x or P x , just enforce the attachment of such invalid C v H by using goal-strategiesof section 3.3.2 and continue the reconstruction process. Strategy 21.
In hamiltonian path context, Allow ∆( H ) components to exist, assuming thatthese components are reachable by x or x . As an alternative strategy, we can enforce H to have H v H = ∅ until we have only non-attachable C v H components. When it happens, allow one v → v H = T and split the scene H in two different subscenes H (cid:48) and H (cid:48)(cid:48) with x ∈ V ( H (cid:48) ) and x ∈ V ( H (cid:48)(cid:48) ). In this case, x of H will be the x of H (cid:48) and x of H will be the x of H (cid:48)(cid:48) . The x of H (cid:48) and H (cid:48)(cid:48) will be the root ofa creatable component with | H n | = 1, | H c | = F (if one exists) of H (cid:48) and H (cid:48)(cid:48) , respectively. Strategy 22.
In hamiltonian path context, enforce H v H = ∅ until we have only non-attachable C v H components. If we enforce H v H = ∅ until we have only non-attachable C v H components, we can findpossible mandatory dead ends of hamiltonian path. As an example, the figure bellow shows aRSPN with j = v H being an empty child of J . The reason is that the vertices w and w ,which were added to J in expansion call k −
4, were added to J again in expansion call k − J when (cid:110) ( γ, t ) = F . It means that if we pass through w or w in expansion call k , ∆( H )could get increased by (cid:98) T (cid:101) at any moment since (cid:98) T (cid:101) failed to prevent v H from being empty.33igure 11: RSPN’s node J of expansion call k − k (on the right side)with v H being a mandatory dead end.When there’s an empty j i = v H i node, such v H i can be considered active. In figure 3, w and w forms together the only possible choice of v H , which represents a possible mandatorydead end. So if we have H v H ∩ { w , w } (cid:54) = ∅ , or a creatable component H (cid:48) ⊃ H with 1 ≤ | H n | ≤ | H c | = F , { w , w } ∩ V ( H (cid:48) ) (cid:54) = ∅ , SFCM-R can just ignore these attachments at first since itmust assume that ∆( H ) could get increased by (cid:98) T (cid:101) at any moment since (cid:98) T (cid:101) failed to prevent v H from being empty. This section is dedicated to the proof of correctness of
Reconstruct , which consequentlyproves the correctness of SFCM-R algorithm. In this section, the unknown negated forbiddencondition of RS-R is refereed to as (cid:98) F (cid:101) . Before continuing, consider the following corollaries ofTheorem 2. Corollary 4. (cid:98) F (cid:101) can’t ignore the constraints of SFCM-R completely. Corollary 5.
If RS-R ran without aborting itself, its potentially-exponential error rate curvewas completely distorted by (cid:98) F (cid:101) in its final state. Now, we will prove the following theorem, which states that
Reconstruct is goal-orientedwith at most | V | − x ∈ φ , are made by using an optimizabletolerance policy (cid:98) T (cid:101) . Theorem 3.
Reconstruct is goal-oriented if at most | V | − expansion calls, with a different x ∈ φ , are made by using an optimizable tolerance policy (cid:98) T (cid:101) .Proof. Let H = ( V, E, v , L, Ω) be a scene, and (cid:98) T (cid:101) be an optimizable tolerance policy. As Mapping ignores Theorem 1 partially and
Reconstruct passes through H by using paths of H ∗ , Reconstruct is goal-oriented only if its error rate curve, which is the curve of γ , doesn’tdegenerate the error rate curve distortion made by Mapping (Corollary 2) while enforcing bothconstraints 1 and 2.Let F v = P v with v ∈ { x , x } , be a forbidden sequence of H ∗ that makes the current ofstate of Reconstruct be inconsistent in H [ V − F v ]. Let Z = H v H be an inconsistent H v H generated by F v . If F v is found by Reconstruct , (cid:98) T (cid:101) (along with the proposed goal-orientedstrategies and variants) makes Reconstruct either:(1) degenerate F v by undoing k states in order to attach a C v H component such that v H ∈ Z through x or x ; or(2) perform a new expansion call with φ such that φ = ( v H , (cid:3) ) and v H ∈ Z in order todegenerate F v ordering by accessing Z before F v .Notice that:(I) Reconstruct imitates
Mapping , which is a goal-oriented by Theorem 2, in order todegenerate F v since it minimizes the appearance of non-mandatory C v H components byattaching them successfully, while using of paths H ∗ to pass through H , which couldmake SFCM-R ignore its own constraints partially to imitate RS-R, that also can ignoreSFCM-R constraints partially by Corollary 4.34II) By Corollary 1, the existence of non-mandatory C v H components and potential isolatedforbidden minors doesn’t imply that Reconstruct is ignoring (cid:98) F (cid:101) by imitating Map-ping ;(III) F v is not degenerated by imitating RS-E explicitly due to both (cid:98) T (cid:101) , and restrictionsrelated to the proposed goal-oriented strategies (and variants) that forces ∆( H ) to beconsistent while preventing Reconstruct from imitating RS-E explicitly;(IV) Assuming that (cid:98) F (cid:101) makes a recursive Hamiltonian-Sequence call to check if v → u = T hold for u , due to fact that RS-R performs only v → u = T operations, (cid:98) F (cid:101) can discard thescene G of successive recursive calls without aborting RS-R in order to return v → u = F to their callers. Each caller, in turn, either increments its error count by one or makes v → u = F hold for the remaining u . Thus, Reconstruct is imitating RS-R when
Reconstruct is undoing k states in order to attach a C v H component such that v H ∈ Z ;(V) We can also assume that (cid:98) F (cid:101) can also change the first v = y of the first Hamiltonian-Sequence call, when y is preventing (cid:98) F (cid:101) from constructing a valid hamiltonian sequence S in order to not make RS-R fail to produce a valid output, with S = v i ...v k such that | S | = | V | , 1 ≤ k ≤ | V | ,1 ≤ i ≤ k , v (cid:54) = y . Thus, Reconstruct is imitating RS-R when
Reconstruct is performing a new expansion call with φ such that φ = ( v H , (cid:3) ) and v H ∈ Z in order to degenerate F v ordering by accessing Z before F v .To illustrate (I), (II), (III), (IV) and (V), assume that RS-R, SFCM-R and RS-E are thermo-dynamic closed isolated systems in a row, defined by S RS-R = S (SFCM-R , x i , x (cid:48) j ), S SFCM-R = S (RS-R , x i , x j ), and S RS-E = S (RS-E , x i , x (cid:48)(cid:48) j ), respectively. S ( A, x i = w i , z j ) is a linear combi-nation of Gaussian kernels, which illustrates non-overlapping homeomorphic imaginary surfacesin different dimensions. S ( A, x i = w i , z j ) = n (cid:88) j =1 (cid:102) j e −(cid:107) x i − z j (cid:107) , w i ∈ V, (cid:102) j ≥ , n = | V | (17) S SF CM − R in the middle illustrates the following quantum superposition as we want. S SFCM-R = c | S RS-R (cid:105) + c | S RS-E (cid:105) (18) c = max (0 , t − γ ) (19) c = 1 − c (20)Let T S SFCM − R , T S RS − R and T S RS − E be γ , ∇ and ∞ , respectively, with T S being the temper-ature of S at equilibrium and ∇ being an imaginary variable.In this context, we set γ as follows because the hidden variable (cid:102) j , that corresponds to thetemperature at x j , is uniform in every x j when S SF CM − R is at equilibrium. In this sense, as F v represents an inconsistency of SFCM-R, F v makes c j and (cid:102) j increase. γ = (cid:88) (cid:102) j | V | = (cid:102) j (21) (cid:102) j = γ + c j , c j ≥ S RS − E and S SF CM − R are essen-tially disputing the following minimax-based game, which tests the effectiveness of S SF CM − R on minimizing its disorder (entropy) as γ growth rate increases by using a systematic method,which forces T S SFCM − R to approach T S RS − R instead of T S RS − E in order to (cid:98) T (cid:101) not be moreprone to abort S SF CM − R . ϑ = min (cid:102) j max γ (cid:97) ( (cid:102) j , γ ) = 1 n (cid:88) j ∈ V ( (cid:102) j − γ ) , ≤ γ ≤ (cid:102) j , γ < t, n = | V | (23)35n other words, S SF CM − R win ϑ only and only if it forces itself to not collapse to S RS − E successive times due to c = 0, which can maximize the entropy of S SF CM − R by T S SFCM − R approaching ∞ . Figure 12 illustrates an scenario where ϑ became unfair for S SF CM − R .Figure 12: Illustration of an approximated ”anti-exponential” curve h ( x ) of S SF CM − R decayingexponentially from ( x i , A i ) , A i = max (0 , t i − γ i ) = t i − γ i to ( x (cid:48) i , y i ), y i = max (0 , t i − γ i ) = 0, x (cid:48) i ∈ [ x i ...x (cid:48) i ], as a function of the number of expansion calls, with A i representing each positiveamplitude peak of h ( x ), resulting on the entropy of S SF CM − R being maximized by T S SFCM − R approaching ∞ . The functions used in this figure are as follows. f ( x ) = e − bx (blue dashed line) (24) g ( x ) = − ( f ( x )) (dashed red line) (25) h ( x ) = f ( x ) cos ( (cid:112) w − b ) (blue line) (26)In fact, by Theorem 2 , (cid:98) T (cid:101) must compute the value of s ∈ Φ( s ) in order to not abort S SF CM − R due to c = p , with Φ( s ) : S → S , S = v H i ...v
H k being a set that maps attachable C v H components in S to subsets of S . If such computation is possible, then the existence of s implies that ϑ is unfair for S R − SE because of S SF CM − R reaching x ∈ [ p ...p [ is guaranteed by (cid:98) T (cid:101) because of (I), (II), (III), (IV) and (V). Likewise, the non-existence of s implies that ϑ isunfair for S SF CM − R because of S SF CM − R reaching x = p is guaranteed by (cid:98) T (cid:101) .It’s worth mentioning that (cid:98) F (cid:101) also needs to minimize the entropy of S RS − R by using asystematic method instead of using a probabilistic approach in order to not fail to produce avalid output, directly or indirectly, since T S RS − R needs to collapse to either ∇ or ∞ precisely. s ∈ Φ( s ) = ⇒ (min (cid:102) j max γ (cid:97) ( (cid:102) j , γ ) is unfair for S R − SE ) (27)Therefore, (I), (II), (III), (IV) and (V) imply that the existence of F v is not a sufficientcondition to make Reconstruct degenerate the error rate distortion made by
Mapping andimitate RS-E.Now, let X i be a set Z added to J node in expansion call i that doesn’t active any static v H . As Reconstruct passes through H by using H ∗ paths, we have:(1) by Corollary 1, paths of H ∗ can generate potential independent forbidden minors; and(2) by Corollary 4, (cid:98) F (cid:101) can’t ignore the constraints of SFCM-R completely.36hus, we can assume that, if there exists X i = j i and X k = j k , i > k , then there existstwo ordered fragments S i and S k of a potential hamiltonian sequence, such that S i ∩ X i (cid:54) = ∅ , S k ∩ X k (cid:54) = ∅ that (cid:98) T (cid:101) is forced to create due to (cid:110) = F in order to degenerate X i and X k .Otherwise, Reconstruct would need to imitate RS-E explicitly, since it should have usedprobability to imitate RS-E in order to avoid both X i and X k , instead of using the proposedgoal-oriented strategies (and variants) along with (cid:98) T (cid:101) to postpone the creation of both X i and X k as well as the activation of both X i and X k , which is invalid because of S SF CM − R avoiding ϑ . Notice we can also assume that RS-R also uses an optimizable tolerance policy, directly orindirectly, since:(1) by Corollary 5, (cid:98) F (cid:101) must distort the potentially-exponential error rate curve of RS-R, whichis represented by the number of times that v → u = F holds for u , by using a systematicapproach in order to make RS-R fail to produce a valid output; and(2) by Corollary 4, as (cid:98) F (cid:101) can ignore the constraints of SFCM-R partially, we can assumethat,directly or indirectly, (cid:98) F (cid:101) can tolerate a small γ growth rate in order to considerSFCM-R constraints progressively as a mean to minimize the entropy of S RS − R .Otherwise, RS-R would also need to imitate RS-E explicitly, since it wouldn’t predict opti-mally if its error rate curve distortion would be degenerated in order to abort itself, which isinvalid.As (cid:98) F (cid:101) needs to map the ordering constraints related to potential independent forbiddenminors by using RSPN in order to not make RS-R imitate RS-E, Reconstruct is imitatingRS-R due to the fact that
Reconstruct needs to pass through X i before X k by using (cid:98) T (cid:101) .However, if Reconstruct happens to pass through X k before X i , there may exist a hiddenregion X l such that l > k , that updates X k in way that the ordering X i ...X k remains preserved.As X l can be created by (cid:98) T (cid:101) in any subsequent expansion call, such event is not a sufficientcondition to prove that Reconstruct ignores X i ...X k ordering unless both j i = v H i and j k = v H k are active. In this case, if v → u = T with u ∈ X k and X i ∩ V ( H ) (cid:54) = ∅ , ∆( H ) couldget increased by (cid:98) T (cid:101) at any moment since:(1) Reconstruct can’t create any X l in subsequent expansion calls; and(2) (cid:98) T (cid:101) failed to postpone the creation of both v H i and v H k and the activation of both v H i and v H k , while preventing
Reconstruct from imitating RS-E.Because of that, (cid:98) T (cid:101) is forced to delete some edges e ∈ L e to enforce the ordering of activestatic v H points in order to make ∆( H ) be consistent. In this case, Reconstruct is still usingpaths of H ∗ even if some of edges are removed from L e , which means that the existence of suchremoval operations is not a sufficient condition to prove that the error rate distortion made by Mapping is degenerated by
Reconstruct . In addition, by Corollary 4, (cid:98) F (cid:101) can’t ignore theconstraints of SFCM-R completely.However, if Reconstruct is not able to add [ v, u ], for at least one v , in an arbitrary region R because of such ordering, Reconstruct can’t pass through such region unless by usingprobability. In such state,
Reconstruct is aborted by (cid:98) T (cid:101) since the error rate curve distortionmade by Mapping (Corollary 2) is about to be degenerated, which makes (cid:98) T (cid:101) trigger Strategy1 to disintegrate the curve distortion ring (cid:110) in order to make γ grow exponentially.That’s because Reconstruct would need to imitate the behaviour of RS-E explicitly byignoring L e as well as its tolerance policy completely in order to continue the reconstructionprocess. Notice that such state imitates the abort condition of RS-R by making v → u = F hold for every u , since:(1) (cid:98) T (cid:101) failed to prevent Reconstruct from degenerating itself while preventing
Recon-struct from imitating RS-E; and(2) (cid:98) F (cid:101) can’t ignore the constraints of SFCM-R completely by Corollary 4.37n addition, as a static v H i can’t have duplicated v H points and the first Reconstruct callcan update J node, | V | − x ∈ φ , is a sufficient conditionto activate every static v H . If Reconstruct makes | V | − x ∈ φ , L e needs to be a hamiltonian sequence in order to not violate any region ordering.Otherwise, Reconstruct is aborted by Strategy 1. In such case, by Corollary 5,
Recon-struct also imitates the stop condition of RS-R, since a valid u must exist for every v foundwhen RS-R is not aborted, and, a distorted error rate curve must exist when RS-R is not aborted.Therefore, Reconstruct is goal-oriented if at most | V | − x ∈ φ , are made by using an optimizable tolerance policy (cid:98) T (cid:101) . In this paper, a novel algorithm to hamiltonian sequence is proposed. Such algorithm tries toreconstruct a potential hamiltonian sequence P by solving a synchronization problem betweenthe forbidden condition of an unknown non-exhaustive hamiltonian sequence characterizationtest, which is a set of unknown sufficient conditions that makes such test fail to produce avalid output, and the forbidden condition of the proposed algorithm, which is a set of sufficientconditions that makes the proposed algorithm fail to produce a valid output. In conclusion,this study suggests that the hamiltonian sequence problem can be treated as a synchronizationproblem involving the two aforementioned forbidden conditions. References [1] Bj¨orklund A (2014) Determining sums for undericted hamiltonicity. SIAM Journal on Com-puting, 43(1):280-299[2] Bj¨orklund A (2016) Bellow all subsets for some permutational counting problems. In 15thScandinavian Symposiym and Workshops on Algorithm Theory (SWAT 2016), volume 53 ofLeibniz International Procedings in Informatics (LIPIcs), pages 1-17[3] Dirac G (1952) Some theorems on abstract graphs. Proceedings of the London MathematicalSociety. 3(1);69. 19[4] Garey MR, Johnson DS (1979) Computers and Intractability: A guide to the Theory ofNP-Completeness[5] Held M, Karp RM (1962) A dynamic programming approach to sequencing problems, J.SIAM, 10(1): 196-210[6] Karp RM (1972) Reducibility among Combinatorial Problems. In: Miller RE, Thatcher JW,Bohlinger JD (eds)
Complexity of Computer Computations . The IBM Research SymposiaSeries. Springer, Boston, MA[7] Mondshein LF (1971) Combinatorial Ordering and the Geometric Embedding of Graphs.PhD thesis, M.I.T. Lincoln Laboratory / Hardvard University[8] Ore O (1960) Note on hamiltonian circuits.