A linear bound on the k-rendezvous time for primitive sets of NZ matrices
Costanza Catalano, Umer Azfar, Ludovic Charlier, Raphaël Jungers
FFundamenta Informaticae XXI (2020) 1–27 DOI 10.3233/FI-2016-0000IOS Press
A linear bound on the k-rendezvous time for primitive sets of NZmatrices
Costanza Catalano C Department of Economics, Statistics and ResearchBanca d’Italia (Central Bank of Italy)Largo Guido Carli 1, 00044 Frascati (Roma), [email protected]
Umer Azfar
ICTEAM, Université Catholique de LouvainAvenue Georges Lemaîtres 4-6, Louvain-la-Neuve,[email protected]
Ludovic Charlier
ICTEAM, Université Catholique de LouvainAvenue Georges Lemaîtres 4-6, Louvain-la-Neuve,[email protected]
Raphaël M. Jungers * ICTEAM, Université Catholique de LouvainAvenue Georges Lemaîtres 4-6, Louvain-la-Neuve,[email protected]
Abstract.
A set of nonnegative matrices is called primitive if there exists a product of thesematrices that is entrywise positive. Motivated by recent results relating synchronizing automataand primitive sets, we study the length of the shortest product of a primitive set having a column ora row with k positive entries, called its k -rendezvous time ( k -RT ), in the case of sets of matriceshaving no zero rows and no zero columns. We prove that the k -RT is at most linear w.r.t. thematrix size n for small k , while the problem is still open for synchronizing automata. We providetwo upper bounds on the k -RT: the second is an improvement of the first one, although the lattercan be written in closed form. We then report numerical results comparing our upper bounds onthe k -RT with heuristic approximation methods. Keywords:
Primitive set of matrices, matrix semigroups, synchronizing automaton, ˇCerný con-jecture. C Corresponding author * R. M. Jungers is a FNRS Research Associate. He is supported by the French Community of Belgium, the Walloon Regionand the Innoviris Foundation. a r X i v : . [ c s . D M ] J a n C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
1. Introduction
Primitive sets of matrices.
A nonnegative matrix M is called primitive if there exists an integer s ∈ N such that M s > entrywise. This notion was introduced by Perron and Frobenius at thebeginning of the 20th century, and it can be extended to sets of matrices: a set of nonnegative matrices M = { M , . . . , M m } is called primitive if there exist some indices i , . . . , i r ∈ { , . . . , m } suchthat the product M i · · · M i r is entrywise positive. A product of this kind is called a positive productand the length of the shortest positive product of a primitive set M is called its exponent and it isdenoted by exp ( M ) . The concept of primitive set has been just recently formalized by Protasovand Voynov [1], but it had appeared before in different fields as in stochastic switching systems [2,3] and time-inhomogeneous Markov chains [4, 5]. It has lately gained more importance due to itsapplications in consensus of discrete-time multi-agent systems [6], cryptography [7] and automatatheory [8, 9, 10, 11]. Deciding whether a set is primitive is a PSPACE-complete problem [9], whilecomputing the exponent of a primitive set is an FP NP [log] -complete problem [9]; for the complexityof other problems related to primitivity and the computation of the exponent, we refer the reader to[9]. For sets of matrices having at least one positive entry in every row and every column (called NZ [9] or allowable matrices [12, 2]), the primitivity problem becomes decidable in polynomial-time [1],although computing the exponent remains NP-hard [9]. Methods for approximating the exponent havebeen proposed [13, 14] as well as cubic upper bounds on the matrix size n [8]. Better upper boundshave been found for some classes of primitive sets (see e.g. [9] and [4], Corollary 2.5). The NZcondition is often met in applications and in particular in the connection with synchronizing automata. Synchronizing automata. A (complete deterministic finite state) automaton is a -tuple A = (cid:104) Q, Σ , δ (cid:105) where Q = { q , . . . , q n } is a finite set of states, Σ = { a , . . . , a m } is a finite set of in-put symbols (the letters of the automaton) and δ : Q × Σ → Q is the transition function . Givensome indices i , i , ..., i l ∈ { , ..., m } , we call w = a i a i ...a i l a word and we define δ ( q, w ) = δ ( δ ( q, a i a i ...a i l − ) , a i l ) . An automaton is synchronizing if it admits a word w , called a synchroniz-ing or a reset word, and a state q such that δ ( q (cid:48) , w ) = q for any state q (cid:48) ∈ Q . In other words, the resetword w brings the automaton from every state to the same fixed state. Remark 1.1.
The automaton A can be equivalently represented by the set of matrices { A , . . . , A m } where, for all i = 1 , . . . , m and l, k = 1 , . . . , n , ( A i ) lk = 1 if δ ( q l , a i ) = q k , ( A i ) lk = 0 otherwise.The action of a letter a i on a state q j is represented by the product e (cid:62) j A i , where e j is the j -th elementof the canonical basis. Notice that the matrices A , . . . , A m are binary and row-stochastic, i.e. eachof them has exactly one entry equal to in every row and zero everywhere else. In this representation,the automaton A is synchronizing if and only if there exists a product of its matrices with a columnwith all entries equal to (also called an all-ones column).The idea of synchronization is quite simple: we want to restore control over a device whose currentstate is unknown. For this reason, synchronizing automata are often used as models of error-resistantsystems [15, 16], but they also find application in other fields such as in symbolic dynamics [17],in robotics [18] or in resilience of data compression [19, 20]. For a recent survey on synchronizing A matrix is binary if it has entries in { , } . . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices automata we refer the reader to [21]. We are usually interested in the length of the shortest reset wordof a synchronizing automaton A , called its reset threshold and denoted by rt ( A ) . Despite the factthat determining whether an automaton is synchronizing can be done in polynomial time (see e.g.[21]), computing its reset threshold is an NP-hard problem [15] . One of the most longstanding openquestions in automata theory concerns the maximal reset threshold of a synchronizing automaton,problem that is traditionally known as The ˇCerný conjecture . Conjecture 1. (The ˇCerný(-Starke) conjecture)
Any synchronizing automaton on n states has a synchronizing word of length at most ( n − .ˇCerný also presented in his pioneering paper [27] (see also its recent english translation [28]) afamily of automata having reset threshold of exactly ( n − , thus demonstrating that the bound in hisconjecture (if true) cannot be improved. Exhaustive search confirmed the ˇCerný conjecture for smallvalues of n [29, 30, 31, 32] and within certain classes of automata (see e.g. [33, 34, 35]), but despite agreat effort has been made to prove (or disprove) it in the last decades, its validity still remains unclear.Indeed on the one hand, the best upper bound known on the reset threshold of any synchronizing n -state automaton is cubic in n [36, 37, 38], while on the other hand automata having quadratic resetthreshold, called extremal automata, are very difficult to find and few of them are known (see e.g.[39, 40, 41, 42]). Some of these families have been found by Ananichev et. al. [43] by coloringthe digraph of primitive matrices having large exponent; this has been probably the first time whereprimitivity has been succesfully used to shed light on synchronization. Connecting primitivity and synchronization.
The following definition and theorem establishthe connection between primitive sets of binary NZ matrices and synchronizing automata. From hereon, we will use the matrix representation of deterministic finite automata as described in Remark 1.1.
Definition 1.2.
Let M be a set of binary NZ matrices. The automaton associated to the set M is theautomaton Aut ( M ) such that A ∈ Aut ( M ) if and only if A is a binary and row-stochastic matrixand there exists M ∈ M such that A ≤ M (entrywise). We denote with Aut ( M (cid:62) ) the automatonassociated to the set M (cid:62) = { M (cid:62) , . . . , M (cid:62) m } .The following example exhibits a primitive set M of NZ matrices and the synchronizing automata Aut ( M ) and Aut ( M (cid:62) ) . Example 1.3.
Here we present a primitive set and its associated automata, see also Figure 1. M = (cid:110)(cid:16) (cid:17) , (cid:16) (cid:17)(cid:111) ,Aut ( M ) = (cid:110) a = (cid:16) (cid:17) , b = (cid:16) (cid:17) , b = (cid:16) (cid:17)(cid:111) Aut ( M (cid:62) ) = (cid:110) a (cid:48) = (cid:16) (cid:17) , b = (cid:16) (cid:17) , b (cid:48) = (cid:16) (cid:17)(cid:111) . Moreover, even approximating the reset threshold of an n -state synchronizing automaton within a factor of n − (cid:15) is knownto be NP-hard for any (cid:15) > , see [22]. ˇCerný, together with Pirická and Rosenaurová, explicitly stated it in 1971 [23], while the first printed version of suchconjecture is attributable to Starke in 1966 [24] (see also its recent english translation [25]). For further details on thepaternity of this conjecture, we refer the reader to [26]. C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices a, b , b b b , b a, b a b , b (cid:48) a (cid:48) , b , b (cid:48) b a (cid:48) , b (cid:48) a (cid:48) Figure 1: The automata
Aut ( M ) (left) and Aut ( M (cid:62) ) (right) of Example 1.3.The following theorem establishes how exp ( M ) , rt (cid:0) Aut ( M ) (cid:1) and rt (cid:0) Aut ( M (cid:62) ) (cid:1) are in relation toeach others. Theorem 1.4. ([8] Theorems 16-17, [9] Theorem 2)
Let M = { M , . . . , M m } be a primitive set of n × n binary NZ matrices. Then Aut ( M ) and Aut ( M (cid:62) ) are synchronizing and it holds that: rt (cid:0) Aut ( M ) (cid:1) ≤ exp ( M ) ≤ rt (cid:0) Aut ( M ) (cid:1) + rt (cid:0) Aut ( M (cid:62) ) (cid:1) + n − . (1) Example 1.5.
Consider the set M and the automata Aut ( M ) and Aut ( M (cid:62) ) of Example 1.3. It holdsthat exp ( M ) = 7 , rt (cid:0) Aut ( M ) (cid:1) = 2 and rt (cid:0) Aut ( M (cid:62) ) (cid:1) = 3 , thus showing that the upper bound ofEq.(1) is tight. A reset word for Aut ( M ) is w = b b , while a reset word for Aut ( M (cid:62) ) is w = b (cid:48) a (cid:48) b (cid:48) .Notice that the requirement in Theorem 1.4 that the set M has to be made of binary matrices is notrestrictive, as the primitivity property does not depend on the magnitude of the positive entries of thematrices of the set. We can thus restrict ourselves to the set of binary matrices by using the Booleanproduct between them ; this means that for any A and B binary matrices, we set ( AB ) ij = 1 any timethat (cid:80) s A is B sj > . In this framework, primitivity can be also rephrased as a membership problem (see e.g. [44, 45]), where we ask whether the all-ones matrix belongs to the semigroup generated bythe matrix set.Equation (1) shows that the behavior of the exponent of a primitive set of NZ matrices is tightlyconnected to the behavior of the reset threshold of its associated automaton. A primitive set M with quadratic exponent implies that one of the automata Aut ( M ) or Aut ( M (cid:62) ) has quadratic resetthreshold; in particular, a primitive set with exponent greater than n − + n − would disprovethe ˇCerný conjecture. This property has been used by the authors in [11] to construct a randomizedprocedure for finding extremal synchronizing automata.The synchronization problem for automata is about finding the length of the shortest word mappingthe whole set of n states onto one single state. We can weaken this request by asking what is the lengthof the shortest word mapping k states onto one single state, for ≤ k ≤ n . In the matrix framework,we are asking what is the length of the shortest product having a column with k positive entries. Thecase k = 2 is trivial, as any synchronizing automaton has a letter mapping two states onto one; for k = 3 Gonze and Jungers [46] presented a quadratic upper bound in the number of the states of the In other words, we work with matrices over the Boolean semiring. . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices automaton while, to the best of our knowledge, the cases k ≥ are still open. Clearly, the case k = n is the problem of computing the reset threshold.In view of the connection between synchronizing automata and primitive sets, we extend the abovedescribed problem to primitive sets by introducing the k-rendezvous time ( k -RT): the k -RT of a prim-itive set M is the length of its shortest product having a row or a column with k positive entries. Thefollowing proposition shows how the k -RT of a primitive set M of NZ matrices (denoted by rt k ( M ) )is linked to the length of the shortest word for which there exists a set of k states mapped by it ontoa single state in the automata Aut ( M ) and Aut ( M (cid:62) ) , where the lengths are denoted respectively by rt k ( Aut ( M )) and rt k ( Aut ( M (cid:62) )) . Proposition 1.6.
Let M be a primitive set of n × n binary NZ matrices and let Aut ( M ) and Aut ( M (cid:62) ) be the automata defined in Definition 1.2. Then for every ≤ k ≤ n , it holds that rt k ( M ) = min (cid:8) rt k (cid:0) Aut ( M ) (cid:1) , rt k (cid:0) Aut ( M (cid:62) ) (cid:1)(cid:9) . The proof of Proposition 1.6 mimics the proof of Theorem 16 in [8]; for the sake of completeness,we provide a self-contained proof.
Proof:
By Definition 1.2, each matrix of
Aut ( M ) and Aut ( M (cid:62) ) is entrywise smaller than a matrix of M .It follows that rt k ( M ) ≤ min (cid:8) rt k (cid:0) Aut ( M ) (cid:1) , rt k (cid:0) Aut ( M (cid:62) ) (cid:1)(cid:9) .Let now M = M i · · · M i u be the product that attains the k -RT, that is a product of length rt k ( M ) having a column or a row with k positive entries. Suppose that M has column with k positive entries:we show that rt k ( M ) ≥ rt k (cid:0) Aut ( M ) (cid:1) . Let j be the index of this column and S be its support. Weclaim that for every r ∈ [ u ] we can safely set to zero some entries of M i r in order to make its rowsbe stochastic while making sure that the final product still has the j -th column with support S . Inother words, we claim that for every r ∈ [ u ] we can select a binary row-stochastic matrix A r ≤ M i r (entrywise) such that the j -th column of the product A · · · A u has support S . If this is true, since byhypothesis the matrices A , . . . , A u belong to Aut ( M ) , it holds that rt k (cid:0) Aut ( M ) (cid:1) ≤ rt k ( M ) .We now prove the claim: let D r be the digraph on n vertices and edge set E r such that p → q ∈ E r ifand only if ( M i r ) pq > . The fact that the j -th column of M i · · · M i u has support S means that forevery s ∈ S there exists a sequence of vertices v s , . . . , v su +1 ∈ [ n ] such that: v s = s , (2) v su +1 = j , (3) v sr → v sr +1 ∈ E r ∀ r = 1 , . . . , u . (4)We can impose an additional property on these sequences: if at step t two sequences share the samevertex, then they have to coincide for all the steps t (cid:48) > t . More formally, if for some t ∈ [ u ] we have that v st = v s (cid:48) t for s (cid:54) = s (cid:48) , then we set v s (cid:48) t (cid:48) = v st (cid:48) for all t (cid:48) > t as the new sequence v s (cid:48) , . . . , v s (cid:48) t , v st +1 , . . . , v su +1 for vertex s (cid:48) fulfills all the requirements (2), (3) and (4). For every r ∈ [ u ] ,we now remove from E r all the edges that are not of type (4). Furthermore, for every r ∈ [ u ] andvertex w / ∈ { v sr } s ∈ S , we remove from E r all the outgoing edges of w but one. We call this new edge C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices set ˜ E r and let ˜ D r be the subgraph of D r with edge set ˜ E r . Then, for every r ∈ [ u ] , we set A r to be theadjacency matrix of ˜ D r . Since M i r is NZ, if A r has some zero-rows we can always add a one in eachof them while preserving the property A r ≤ M i r . We do so in order to make A r row stochastic. Byconstruction, for all r ∈ [ u ] , A r has exactly one positive entry in each row and it is entrywise smallerthan M i r , so A r ∈ Aut ( M ) . Finally, the j -th column of A · · · A u has support S by construction.The case when M has a row with k positive entries can be proved via a similar reasoning by observingthat the product M (cid:62) = M (cid:62) i u · · · M (cid:62) i has a column with k positive entries, and so for every r ∈ [ u ] we can select a binary matrix B r ≤ M (cid:62) i r (entrywise) such that B r ∈ Aut ( M (cid:62) ) and B · · · B u has acolumn with k positive entries. This implies that rt k (cid:0) Aut ( M (cid:62) ) (cid:1) ≤ rt k ( M ) . (cid:117)(cid:116) Our contribution.
This paper comes as an extended version of the one published at the conference
Developments in Language Theory 2019 [47]. With respect to the conference version, the entireSections 4 and 5 have been added; also other small parts and some notation have been changed. Inthis work we provide an analytical upper bound on rt k ( M ) that holds for any primitive set M of n × n NZ matrices. This upper bound is a function of n and k , and it proves in particular that the k -rendezvous time rt k ( M ) is upper bounded by a linear function in n for any fixed k ≤ √ n , problemthat is still open for synchronizing automata. Our result also implies that for any fixed k ≤ √ n , min (cid:8) rt k (cid:0) Aut ( M ) (cid:1) , rt k (cid:0) Aut ( M (cid:62) ) (cid:1)(cid:9) is upper bounded by a linear function in n . This is presentedin Section 3; in particular, in Subection 3.2 we report some numerical experiments and we show thatthis first technique for upper bounding rt k ( M ) cannot be much improved as it is. We then presentin Section 4 a second upper bound for the k -RT that improves the first one for any ≤ k ≤ n . Wereport some numerical experiments in Section 5 comparing our two theoretical upper bounds on the k -RT with the real one (or an approximation when it becomes too hard to compute) for some examplesof primitive sets. Finally, as the second upper bound cannot be written in closed form, in the samesection we present some graphs picturing its behavior, showing that when n is not too big with respectto k the second upper bound significantly improves on the first one.
2. Notation and preliminaries
The set { , . . . , n } is represented by [ n ] . The support of a nonnegative vector v is the set supp ( v ) = { i : v i > } and the weight of a nonnegative vector v is the cardinality of its support.Given a matrix A , we denote by A ∗ j its j -th column and by A i ∗ its i -th row. A permutation matrixis a binary matrix having exactly one positive entry in every row and every column. We remind thatan n × n matrix A is called irreducible if for any i, j ∈ [ n ] , there exists a natural number k such that ( A k ) ij > . A matrix A is called reducible if it is not irreducible.Given M a set of matrices, we denote by M d the set of all the products of at most d matrices from M . A set of matrices M = { M , . . . , M m } is reducible if the matrix (cid:80) i M i is reducible, otherwiseit is called irreducible . Irreducibility is a necessary but not sufficient condition for a matrix set to beprimitive (see [1], Section 1). Given a directed graph D = ( V, E ) , we denote by v → w the directededge leaving v and entering in w and by v → w ∈ E the fact that the edge v → w belongs to thedigraph D . A directed graph is strongly connected if there exists a directed path from any vertex toany other vertex. . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices Lemma 2.1.
Let M be an irreducible set of n × n NZ matrices, A ∈ M and i, j ∈ [ n ] . Then thereexists a matrix B ∈ M n − such that supp ( A ∗ i ) ⊆ supp (( AB ) ∗ j ) . Proof:
We consider the labeled directed graph D M = ( V, E ) where V = [ n ] and i → j ∈ E iff there exists amatrix A ∈ M such that A ij > . We label the edge i → j ∈ E by all the matrices A ∈ M such that A ij > . Notice that a path in D M from vertex k to vertex l having the edges sequentially labeled bythe matrices A s , . . . , A s r ∈ M means that ( A s · · · A s r ) kl > . Since M is irreducible, it followsthat D M is strongly connected and so any pair of vertices in D M is connected by a path of length atmost n − . Consider a path connecting vertex i to vertex j whose edges are sequentially labeled bythe matrices A s , . . . , A s t and let B = A s · · · A s t . Clearly B ∈ M n − ; furthermore it holds that B ij > and so supp ( A ∗ i ) ⊆ supp (cid:0) ( AB ) ∗ j (cid:1) . (cid:117)(cid:116) The following definition will be crucial for the results in the next sections.
Definition 2.2.
Let M be an irreducible set of n × n NZ matrices. We define the pair digraph of theset M as the labeled directed graph PD ( M ) = ( V , E ) where V = { ( i, j ) : 1 ≤ i ≤ j ≤ n } and ( i, j ) → ( i (cid:48) , j (cid:48) ) ∈ E if and only if there exists A ∈ M such that A ii (cid:48) > and A jj (cid:48) > , or A ij (cid:48) > and A ji (cid:48) > . (5)An edge ( i, j ) → ( i (cid:48) , j (cid:48) ) ∈ E is labeled by every matrix A ∈ M for which Eq. (5) holds. A vertex oftype ( s, s ) is called a singleton . Lemma 2.3.
Let M be a finite set of n × n NZ matrices and let PD ( M ) = ( V , E ) be its pair digraph.Let i, j, k ∈ [ n ] and suppose that there exists a path in PD ( M ) from the vertex ( i, j ) to the singleton ( k, k ) having the edges sequentially labeled by the matrices A s , . . . , A s l ∈ M . Then it holds that forevery A ∈ M , supp ( A ∗ i ) ∪ supp ( A ∗ j ) ⊆ supp (( AA s · · · A s l ) ∗ k ) . Suppose now that M is irreducible. Then it holds that M is primitive if and only if for any ( i, j ) ∈ V there exists a path in PD ( M ) from ( i, j ) to some singleton. Proof:
By the definition of the pair digraph PD ( M ) (Definition 2.2), the existence of a path from ver-tex ( i, j ) to vertex ( k, k ) labeled by the matrices A s , . . . , A s l implies that ( A s · · · A s l ) ik > and ( A s · · · A s l ) jk > . By Lemma 2.1, it follows that supp ( A ∗ i ) ∪ supp ( A ∗ j ) ⊆ supp (cid:0) ( AA s · · · A s l ) ∗ k (cid:1) .Suppose now that M is irreducible. If M is primitive, then there exists a product M of matrices from M such that for all i, j , M ij > . By the definition of PD ( M ) , this implies that any vertex in PD ( M ) is connected to any other vertex. On the other hand, if every vertex in PD ( M ) is connectedto some singleton, then for every i, j, k ∈ [ n ] there exists a product A s · · · A s l of matrices from M such that ( A s · · · A s l ) ik > and ( A s · · · A s l ) jk > . Theorem 1 in [48] states that the followingcondition is sufficient for an irreducible matrix set M to be primitive: for all indices i, j , there existsan index k and a product M of matrices from M such that M ik > and M jk > . Therefore, weconclude. (cid:117)(cid:116) C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
3. The k-rendezvous time and a recurrence relation for its upper bound
In this section, we define the k -rendezvous time of a primitive set of n × n NZ matrices, we findan upper bound U k ( n ) on it, and we prove a recurrence relation for U k ( n ) . Definition 3.1.
Let M be a primitive set of n × n NZ matrices and k an integer such that ≤ k ≤ n .We define the k -rendezvous time ( k -RT) to be the length of the shortest product of matrices from M having a column or a row with k positive entries and we denote it by rt k ( M ) . We indicate with rt k ( n ) the maximal value of rt k ( M ) among all the primitive sets M of n × n NZ matrices.Our goal is to find, for any n ≥ and ≤ k ≤ n , a function U k ( n ) such that rt k ( n ) ≤ U k ( n ) . Definition 3.2.
Let n and k be two integers such that n ≥ and ≤ k ≤ n − . We denote by S kn the set of all the n × n NZ matrices having every row and column of weight at most k and at leastone column of weight exactly k . For any A ∈ S kn , let C A be the set of the indices of the columnsof A having weight equal to k . We define a nk ( A ) = min c ∈C A |{ i : supp ( A ∗ i ) (cid:42) supp ( A ∗ c ) }| and a nk = min A ∈S kn a nk ( A ) .In other words, a nk ( A ) is the minimum over all the indices c ∈ C A of the number of columns of A whose support is not contained in the support of the c -th column of A . Since the matrices are NZ, i.e.zero rows are not allowed, it holds that for any A ∈ S kn , ≤ a nk ≤ a nk ( A ) . Example 3.3. If k = 1 , then S n is the set of n × n permutation matrices. In this case for any A ∈ S n ,it holds that C A = [ n ] and a n ( A ) = n − a n . Consider now the following matrices: A = , B = , C = . It holds that A ∈ S , while B, C / ∈ S because B has a row of weight and C has no column ofweight . Moreover:• C A = { , } ,• |{ i : supp ( A ∗ i ) (cid:42) supp ( A ∗ ) }| = |{ , }| = 2 ,• |{ i : supp ( A ∗ i ) (cid:42) supp ( A ∗ ) }| = |{ , , }| = 3 ,and so a ( A ) = 2 . Therefore it holds that ≤ a ≤ . On the other hand, consider the followingmatrix: ˆ A = . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices We have that ˆ A ∈ S , C ˆ A = { , } and |{ i : supp ( A ∗ i ) (cid:42) supp ( A ∗ ) }| = |{ }| = 1 , so a ( ˆ A ) = 1 .This implies that a = 1 .The following theorem shows that for every n ≥ , we can recursively define a function U k ( n ) ≥ rt k ( n ) on k by using the term a nk . Theorem 3.4.
Let n ≥ be an integer. The following recursive function U k ( n ) is such that for all ≤ k < n , it holds that rt k ( n ) ≤ U k ( n ) : (cid:40) U ( n ) = 1 U k +1 ( n ) = U k ( n ) + n (1 + n − a nk ) / for ≤ k ≤ n − . (6) Proof:
We prove the theorem by induction. Let k = 2 . Any primitive set of NZ matrices must have a matrixwith a row or a column with two positive entries, as otherwise it would be made of just permutationmatrices and hence it would not be primitive. This trivially implies that rt ( n ) = 1 ≤ U ( n ) .Suppose now that rt k ( n ) ≤ U k ( n ) , we show that rt k +1 ( n ) ≤ U k +1 ( n ) . We remind that M d denotesthe set of all the products of matrices from M having length ≤ d . If in M rt k ( M )+ n − there exists aproduct having a column or a row with k + 1 positive entries then rt k +1 ( M ) ≤ rt k ( M ) + n − ≤ U k +1 ( n ) . Suppose now that this is not the case. This means that in M rt k ( M )+ n − every matrix has allthe rows and columns of weight at most k . Let A ∈ M rt k ( M ) be a matrix having a row or a columnof weight k , and suppose it is a column. The case when A has a row of weight k will be studied later.By Lemma 2.1 applied on the matrix A , for every i ∈ [ n ] there exists a matrix W i ∈ M rt k ( M )+ n − having the i -th column of weight k (and all the other columns and rows of weight ≤ k ). Every W i has at least a nk columns (see Definition 3.2) whose support is not contained in the support of the i -thcolumn of W i : we pick a nk indices of these columns and we denote them by c i , c i , . . . , c a nk i . Noticethat any product B of matrices from M of length l such that B is > and B c ji s > for some s ∈ [ n ] and j ∈ [ a nk ] would imply that W i B has the s -th column of weight at least k + 1 and so rt k +1 ( M ) ≤ rt k ( M ) + n − l . We now want to minimize this length l over all i, s ∈ [ n ] and j ∈ [ a nk ] : we will prove that there exists i, s ∈ [ n ] and j ∈ [ a nk ] such that l ≤ n ( n − − a nk ) / . Todo this, we consider the pair digraph PD ( M ) = ( V , E ) (see Definition 2.2) and the vertices (1 , c ) , (1 , c ) , . . . , (1 , c a nk ) , (2 , c ) , . . . , (2 , c a nk ) , . . . , ( n, c n ) , . . . , ( n, c a nk n ) . (7)By Lemma 2.3, for each vertex in Eq.(7) there exists a path in PD ( M ) connecting it to a singleton.By the same lemma, a path of length l from ( i, c ji ) to a singleton ( s, s ) would result in a product B j ofmatrices from M of length l such that W i B j has the s -th column of weight at least k + 1 . We hencewant to estimate the minimal length among the paths connecting the vertices in Eq.(7) to a singleton.Notice that Eq.(7) contains at least (cid:100) na nk / (cid:101) different elements, since each element occurs at mosttwice. It is clear that the shortest path from a vertex in the list (7) to a singleton does not contain anyother element from that list. The vertex set V of PD ( M ) has cardinality n ( n + 1) / and it contains n vertices of type ( s, s ) . It follows that the length of the shortest path connecting some vertex fromthe list (7) to some singleton is at most of n ( n + 1) / − n − (cid:100) na nk / (cid:101) + 1 ≤ n ( n − − a nk ) / . C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
In view of what was said before, we have that there exists a product B of matrices from M of length ≤ n ( n − − a nk ) / and i ∈ [ n ] s.t. W i B j has a column of weight at least k +1 . Since W i B j belongsto M rt k ( M )+ n − n ( n − − a nk ) / , it follows that rt k +1 ( M ) ≤ rt k ( M )+ n ( n +1 − a nk ) / ≤ U k +1 ( n ) .Suppose now A ∈ M rt k ( M ) has a row of weight k . We can use the same argument as above on thematrix set M (cid:62) made of the transpose of all the matrices in M . (cid:117)(cid:116) Notice that the above argument stays true if we replace a nk by a function b ( n, k ) such that for all n ≥ and ≤ k ≤ n − , it holds that ≤ b ( n, k ) ≤ a nk . It follows that Eq.(6) still holds true if we replace a nk by b ( n, k ) . We now find an analytic expression for a lower bound on a nk and we then solve the recurrence (6)in Theorem 3.4 by using this lower bound. We then show that this is the best estimate on a nk we canhope for. Lemma 3.5.
Let n , k be two integers such that n ≥ and ≤ k ≤ n − , and let a nk as in Definition3.2. It holds that a nk ≥ max { n − k ( k − − , (cid:100) ( n − k ) /k (cid:101) , } . Proof:
We have that a nk ≥ since k ≤ n − and the matrices are NZ. Let now A ∈ S kn (see Definition 3.2)and let a be one of its columns of weight k . Let ζ = supp ( a ) ; by assumption, the rows of A have atmost k positive entries, so there can be at most ( k − k columns of A different from a whose supportis contained in ζ . Therefore, since A is NZ, there must exist at least n − k ( k − − columns of A whose support is not contained in ζ and so a nk ≥ n − k ( k − − .Let again A ∈ S kn and let a be one of its columns of weight k . Let ξ = [ n ] \ supp ( a ) ; ξ has cardinality n − k and since A is NZ, for every s ∈ ξ there exists s (cid:48) ∈ [ n ] such that A ss (cid:48) > . By assumption eachcolumn of A has weight of at most k , so there must exist at least (cid:100) ( n − k ) /k (cid:101) columns of A differentfrom a whose support is not contained in supp ( a ) . It follows that a nk ≥ (cid:100) ( n − k ) /k (cid:101) . (cid:117)(cid:116) In view of the fact that the following inequalities hold:1. (cid:100) ( n − k ) /k (cid:101) ≥ ( n − k ) /k ,2. n − k ( k − − ≥ ( n − k ) /k for k ≤ (cid:98)√ n (cid:99) ,3. ( n − k ) /k ≥ for k ≤ (cid:98) n/ (cid:99) ,the recursion (6) with a nk replaced by max { n − k ( k − − , ( n − k ) /k, } now reads as B k +1 ( n ) = if k = 1 ,B k ( n ) + n (1 + k ( k − / if ≤ k ≤ (cid:98)√ n (cid:99) ,B k ( n ) + n (1 + n ( k − / k ) if (cid:98)√ n (cid:99) + 1 ≤ k ≤ (cid:98) n/ (cid:99) ,B k ( n ) + n / if (cid:98) n/ (cid:99) + 1 ≤ k ≤ n − , (8)where we have denoted by B k ( n ) the function solving (6) with a nk = max { n − k ( k − − , ( n − k ) /k, } . The following proposition shows the solution of the recursion (8). . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices Proposition 3.6.
Equation (8) is fulfilled by the following function: B k ( n ) = n ( k − k + 8 k − if ≤ k ≤ (cid:98)√ n (cid:99) ,B (cid:98)√ n (cid:99) ( n ) + n ( n + 2)( k − (cid:98)√ n (cid:99) )2 − n k − (cid:80) i = (cid:98)√ n (cid:99) i if (cid:98)√ n (cid:99) + 1 ≤ k ≤ (cid:98) n (cid:99) ,B (cid:98) n (cid:99) ( n ) + ( k − (cid:98) n (cid:99) ) n if (cid:98) n (cid:99) + 1 ≤ k ≤ n . (9)Therefore, for any constant k s.t. k ≤ √ n , the k-rendezvous time rt k ( n ) grows at most linearly in n . Proof: If ≤ k ≤ (cid:98)√ n (cid:99) , let C k ( n ) = B k ( n ) /n . By Eq.(8), it holds that C k +1 ( n ) − C k ( n ) = 1 + k ( k − / .By setting C k ( n ) = αk + βk + γk + δ , it follows that αk + (3 α + 2 β ) k + α + β + γ = k / − k/ . Since this must be true for all k , by equating the coefficients we have that C k ( n ) = k / − k / k/ δ . Imposing the initial condition B ( n ) = 1 gives finally the desired result B k ( n ) = n ( k − k + 8 k − / .If (cid:98)√ n (cid:99) + 1 ≤ k ≤ (cid:98) n/ (cid:99) , let again C k ( n ) = B k ( n ) /n . By Eq.(8), it holds that C k +1 ( n ) − C k ( n ) =1+ n ( k − / k and so C k ( n ) = C (cid:98)√ n (cid:99) ( n )+( k − n/ − ( n/ (cid:80) k − i = (cid:98)√ n (cid:99) i − . Since C (cid:98)√ n (cid:99) ( n ) = B (cid:98)√ n (cid:99) ( n ) /n , it follows that B k ( n ) = B (cid:98)√ n (cid:99) ( n ) + ( k − (cid:98)√ n (cid:99) ) n ( n + 2) / − ( n / (cid:80) k − i = (cid:98)√ n (cid:99) i − .If (cid:98) n/ (cid:99) + 1 ≤ k ≤ n − , by Eq.(8) it is easy to see that B k ( n ) = B (cid:98) n/ (cid:99) ( n ) + ( k − (cid:98) n/ (cid:99) ) n / ,which concludes the proof. (cid:117)(cid:116) We now show that a nk = max { n − k ( k − − , (cid:100) ( n − k ) /k (cid:101) , } , and so we cannot improve the upperbound on rt k ( n ) by improving our estimate of a nk . Lemma 3.7.
Let n and k be two integers such that n ≥ and ≤ k ≤ n − . It holds that: ≤ a nk ≤ u ( n, k ) := (cid:40) n − k ( k − − if n − k ( k − − ≥ (cid:100) ( n − k ) /k (cid:101) , (cid:100) ( n − k ) /k (cid:101) otherwise. Proof:
We need to show that for every n ≥ and ≤ k ≤ n − , there exists a matrix A ∈ S kn such that a nk ( A ) = u ( n, k ) (see Definition 3.2). We define the matrix C m × m i as the m × m matrix havingall the entries of the i -th column equal to and all the other entries equal to , and the matrix R m × m i as the m × m matrix having all the entries of the i -th row equal to and all the other entries equalto . We indicate by m × m the m × m matrix having all its entries equal to zero and by I m × m the m × m identity matrix. Let v nk = (cid:100) ( n − k ) /k (cid:101) + 1 and q = n mod k .Suppose that n − k ( k − − ≥ (cid:100) ( n − k ) /k (cid:101) and set α = n − k ( k − − − (cid:100) ( n − k ) /k (cid:101) . Then C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices the following matrix ˆ A is such that a nk ( ˆ A ) = n − k ( k − − u ( n, k ) : ˆ A = C k × v nk R k × ( k − R k × ( k − · · · R k × ( k − k C k × v nk ... ( n − k ) × [ k ( k − DC k × v nk v nk − C q × v nk v nk , D = k × α I α × α ( n − k − α ) × α . Indeed by construction, the first column of ˆ A has exactly k positive entries. The columns of ˆ A whosesupport is not contained in ˆ A ∗ are the columns ˆ A ∗ i for i = 2 , . . . , v nk and all the columns of D . Intotal we have (cid:100) ( n − k ) /k (cid:101) + α = n − k ( k − − columns, so it holds that a nk ( ˆ A ) = n − k ( k − − .Suppose that n − k ( k − − ≤ (cid:100) ( n − k ) /k (cid:101) . Then the following matrix ˜ A is such that a nk ( ˜ A ) = (cid:100) ( n − k ) /k (cid:101) = u ( n, k ) : ˜ A = C k × v nk R k × ( k − R k × ( k − · · · R k × ( k − k − R k × ( n − v nk − ( k − ) k C k × v nk ... C k × v nk v nk − ( n − k ) × ( n − v nk ) C q × v nk v nk . Indeed by construction, the first column of ˜ A has exactly k positive entries and the columns of ˜ A whose support is not contained in ˜ A ∗ are the columns ˜ A ∗ i for i = 2 , . . . , v nk . Therefore it holds that a nk ( ˜ A ) = v nk − (cid:100) ( n − k ) /k (cid:101) . (cid:117)(cid:116) B k ( n ) We now present some numerical results that compare the theoretical bound B k ( n ) on rt k ( n ) ofEq.(9) with either the exact k -RT or an heuristic approximation of it when the computation of the exactvalue is not computationally feasible for some primitive sets. In Figure 2 we compare our bound with . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices the real k -RT of the primitive sets M CP R and M K reported here below: M CP R = , , M K = , . The sets M K and M CP R are primitive sets of matrices that are based, respectively, on the Kariautomaton [49] and the ˇCerný-Pirická-Rozenaurová automaton [23], which are well known synchro-nizing automata with large (quadratic) reset threshold. The sets M K and M CP R have been createdby adding a in one single column of the matrix corresponding to one particular letter of the Kariand the ˇCerný-Pirická-Rozenaurová automaton respectively, in order to make the matrix set primitive.We can see that for small values of k , the upper bound is fairly close to the actual value of rt k ( M ) .When n is large, computing the k -RT for every ≤ k ≤ n becomes hard, so we compare our upper (a) n = 4 , M = M CPR (b) n = 6 , M = M K Figure 2: Comparison between the bound B k ( n ) , valid for all primitive NZ sets, and rt k ( M ) for (a) M = M CP R and (b) M = M K .bound on the k -RT with a method for approximating it. The Eppstein heuristic is a greedy algorithm C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices developed by Eppstein in [15] for approximating the reset threshold of a synchronizing automaton.Given a primitive set M of binary NZ matrices, we can apply a slightly modified Eppstein heuristicto obtain, for any k , an upper bound on rt k ( M ) . Algorithm 1:
Pseudo-code for the modified Eppstein heuristic
Input:
A primitive matrix set M Output:
A matrix A of elements from M with a positive column A ←− arg max X ∈M max i ∈ [ n ] | supp ( X ∗ i ) | ; i ←− arg max j ∈ [ n ] | supp ( A ∗ j ) | S ←− supp ( A ∗ i ) while S (cid:54) = [ n ] do C ←− { j ∈ [ n ] : supp ( A ∗ j ) (cid:42) S } ; j ∗ ←− arg min j ∈C d PD ( M ) [( i, j ) , ( i, i )] ; /* See Remark 3.7. */ A p , . . . , A p l ←− labels of shortest path from ( i, j ) to ( i, i ) ; A ←− AA p · · · A p l ; S ←− supp ( A ∗ i ) endreturn A ;This modified Eppstein heuristic is formalized in Algorithm 1, where for any nodes ( i, j ) and ( k, l ) in PD ( M ) (see Definition 2.2), we denote by d PD ( M ) [( i, j ) , ( k, l )] the length of the shortestpath from ( i, j ) to ( k, l ) in PD ( M ) . The algorithm looks for the matrix A in the set having the columnwith the maximal number of positive entries; then by making use of the pair graph and Lemma 2.3,it looks for the shortest product B of matrices in the set such that the number of positive entries of ( AB ) ∗ i is strictly greater than the one in A ∗ i . It iterates this procedure until obtaining a matrix witha positive column. Since the algorithm increases the weight of the column i at each iteration of thewhile loop, it provides an upper bound on the k -RT for all ≤ k ≤ n by producing a (reasonablyshort) product whose i -th column is of weight ≥ k . At the same time, we also check the weights ofthe rows of A in case the algorithm happens to produce a larger row weight than the maximal columnweight, thus improving the bound on the k -RT. Remark 3.8.
In our implementation we look for the shortest path to a specific singleton ( i, i ) , whereasit would in general be better to find the shortest path to any singleton. However, one can show thatin the case where among the columns of the matrices of the set M there is only one column withtwo positive entries, the two implementations are equivalent. Since this is the case for all the matrixsets used in the numerical experiments of this paper, the choice of implementation was based onconsiderations of simplicity.We now consider the primitive sets with quadratic exponent presented by Catalano and Jungersin [11], Section 4; here we denote these sets by M C n , where n is the matrix dimension. Figure 3compares our theoretical upper bound B k ( n ) with the results of the Eppstein heuristic on the k -RT of . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices M C n for n = 10 , , , and ≤ k ≤ n . We can see again that for small values of k our genericupper bound is fairly close to the Eppstein heuristic of rt k ( M C n ) .Finally, Figure 4 compares the theoretical upper bound B k ( n ) with the results of the Eppsteinheuristic on the k -RT of the family M C n for fixed k = 4 and ≤ n ≤ . It can be noticed that B k ( n ) does not increase very rapidly as compared to the Eppstein approximation. (a) n = 10 , M = M C (b) n = 15 , M = M C (c) n = 20 , M = M C (d) n = 25 , M = M C Figure 3: Comparison between B k ( n ) and the Eppstein approx. of rt k ( M ) , for (a) M = M C , (b) M = M C , (c) M = M C , (d) M = M C . We recall that B k ( n ) is a generic bound valid for allprimitive NZ sets, while the Eppstein bound is computed on each particular set. C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
Figure 4: Comparison between B k ( n ) and the Eppstein approx. of rt k ( M C n ) for k = 4 . We recallthat B k ( n ) is a generic bound valid for all primitive NZ sets, while the Eppstein bound is computedon each particular set.
4. Improving the upper bound on rt k ( n ) In this section we propose a method for improving the upper bound B k ( n ) in Eq. (9) on rt k ( n ) .The idea behind this improvement is the following: Eq. (9) comes from the recursive formulation (6),where we start from a matrix A with a column (or a row) of weight k and we look for the shortestproduct D that makes AD having a column (or a row) of weight at least k + 1 . This is done by‘ merging ’ two columns of A by post-multiplying with matrix D ; namely, if c is the index of a columnof A of weight k and i is the index of a column of A such that supp ( A ∗ i ) (cid:42) supp ( A ∗ c ) , we look fora (short) product D such that D ij > and D cj > for some j ∈ [ n ] . In this way, since A and D areNZ, the j -th column of AD has at least k + 1 positive entries. What is not exploited in this approachis that if | supp ( A ∗ i ) \ supp ( A ∗ c ) | = l ≥ , then the j -th column of AD has k + l positive entries,thus potentially providing an upper bound on the ( k + l ) -RT as well as on ( k + 1) -RT.In this section we take advantage of this fact to improve the upper bound B k ( n ) . Unfortunately,this refined upper bound comes as a solution of a much more complex recursive equation that isnumerically solvable but that does not seem to be easily expressed in closed form. Therefore, althoughit is possible to prove that this new upper bound is smaller than or equal to B k ( n ) for all values of n and ≤ k ≤ n , we do not provide an analytic formula for it. Extensive numerical experiments showthe behaviour of this new upper bound on rt k ( n ) are presented in Section 5.We start with a technical definition, similar to Definition 3.2, that will be useful for finding thenew upper bound. Definition 4.1.
Let n , k and p be three integers s.t. n ≥ , ≤ k < n , and ≤ p ≤ min { k, n − k } .We denote by S kn,p the subset of matrices A in S kn (see Definition 3.2) such that for any column index c ∈ C A and i (cid:54) = c it holds that |{ supp ( A ∗ i ) \ supp ( A ∗ c ) }| ≤ p , and there exists c (cid:48) ∈ C A and j (cid:54) = c (cid:48) such that |{ supp ( A ∗ j ) \ supp ( A ∗ c (cid:48) ) }| = p . For any matrix A ∈ S kn,p , we denote by C pA the subset of . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices columns c in C A for which there exists i (cid:54) = c such that | supp ( A ∗ i ) \ supp ( A ∗ c ) | = p .Finally, for any A ∈ S kn,p we set a nk,p ( A ) = min c ∈C pA |{ i : supp ( A ∗ i ) (cid:42) supp ( A ∗ c ) }| ; a nk,p = min A ∈S kn,p a nk,p ( A ) . In words, a nk,p describes the minimal number of columns that can be summed up to a column of weight k in any matrix in S kn,p in order to increase its weight. Remark 4.2.
Notice that S kn = (cid:83) min { k,n − k } p =1 S kn,p and that for any p such that ≤ p ≤ min { k, n − k } ,it holds that a nk,p ≥ a nk . By mimicking the reasoning used in the proof of Lemma 3.5, it is also easy tosee that a nk,p ≥ (cid:100) ( n − k ) /p (cid:101) . By the same lemma, this implies that a nk,p ≥ ˆ a nk,p := max { n − k ( k − − , (cid:100) ( n − k ) /p (cid:101) , } . (10)Let n ≥ and h , k be two integers such that ≤ h, k ≤ n .We set: O kh ( n ) = max A ∈S hn min { d ∈ N : ∃ D ∈ M d s.t. AD has a column of weight ≥ k } . (11)Notice that O kh ( n ) = 0 any time h ≥ k . In view of Eq. (11), for any h and k s.t. ≤ h ≤ k ≤ n itholds that rt k ( n ) ≤ U h ( n ) + O kh ( n ) ≤ B h ( n ) + O kh ( n ) , where U h ( n ) is defined in Eq. (6) and B h ( n ) is defined in Eq. (8). Since B h ( n ) + O kh ( n ) = B k ( n ) if k = h , it follows that rt k ( n ) ≤ min ≤ h ≤ k { B h ( n ) + O kh ( n ) } ≤ B k ( n ) . (12)Therefore the function min ≤ h ≤ k { B h ( n ) + O kh ( n ) } improves the upper bound B k ( n ) . The rest of thesection is devoted to finding a way to approximate O kh ( n ) for any n , h and k .Let p be an integer such that ≤ p ≤ min { k, n − k } . We set O kh,p ( n ) = max A ∈S hn,p min { d ∈ N : ∃ D ∈ M d s.t. AD has a column of weight ≥ k } . Notice again that O kh,p ( n ) = 0 any time h ≥ k . It follows from Remark 4.2 that O kh ( n ) = max ≤ p ≤ min { h,n − h } O kh,p ( n ) . (13)The following result uses this fact to obtain a recursive upper bound for O kh ( n ) when h < k (for h ≥ k we already know that O kh ( n ) = 0 ). Proposition 4.3.
Let n , k and h integers such that ≤ h < k ≤ n . It holds that O kh ( n ) ≤ max ≤ p ≤ min { h,n − h } min (cid:110) O kh + p ( n ) + n n − O kh +1 ( n ) + n n + 1 − ˆ a nh,p ) (cid:111) . (14) C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
Proof:
By Eq. (13), it suffices to prove that O kh,p ( n ) ≤ O kh + p ( n ) + n ( n − / and O kh,p ( n ) ≤ O kh +1 ( n ) + n ( n + 1 − ˆ a nh,p ) / . To prove the first inequality, we need to show that for any matrix A ∈ S hn,p there exists a product D of length at most n ( n − / such that AD has a column of weight atleast h + p . Let A ∈ S hn,p and c ∈ C pA ; A ∗ c has h positive entries. By the definition of C pA (seeDefinition 4.1), there exists an index i such that | supp ( A ∗ i ) \ supp ( A ∗ c ) | = p . In the pair digraph PD ( M ) (see Definition 2.2) there exists a path from the vertex ( i, c ) to a singleton of length at most ( n ( n + 1) / − n = n ( n − / , in view of Lemma 2.3 and the number of vertices in the pair digraph.By the same lemma, this means that there exists a product D of length at most n ( n − / suchthat AD has a column of weight ≥ h + p . By the definition of O kh + p ( n ) in Eq. (11) it follows that O kh,p ( n ) ≤ O kh + p ( n ) + n ( n − / .To prove the second inequality, we need to show that for any matrix A ∈ S hn,p there exists a product D of length at most n ( n + 1 − ˆ a nh,p ) / such that AD has a column of weight at least h + 1 . This canbe done in view of Eq. (10) and by mimicking the proof of Theorem 3.4. (cid:117)(cid:116) The following theorem provides an upper bound ˜ U kh ( n ) on O kh ( n ) defined by a recurrence relation. Theorem 4.4.
Let n and k be two integers such that ≤ k ≤ n . We define the function ˜ U kh ( n ) for h ≥ in the following way:if h ≥ k , ˜ U kh ( n ) = 0 ; if ≤ h < k , ˜ U kh ( n ) = max ≤ p ≤ min { h,n − h } min (cid:110) ˜ U kh + p ( n ) + n n − U kh +1 ( n ) + n n + 1 − ˆ a nh,p ) (cid:111) . Then it holds that ˜ U kh ( n ) ≥ O kh ( n ) for all h ≥ . In particular it holds that for every ≤ k ≤ n : rt k ( n ) ≤ F k ( n ) := min ≤ h ≤ k { B h ( n ) + ˜ U kh ( n ) } ≤ B k ( n ) . (15) Proof:
The theorem easily follows from Proposition 4.3 and Eq. (12). (cid:117)(cid:116)
We conclude this section by noting that despite the fact that the function ˜ U kh ( n ) defined in Theorem4.4 seems difficult to obtain in closed form, it can be easily implemented and computed by dynamicprogramming. The numerical behavior of the function F k ( n ) in Eq. (15), which represents the im-proved upper bound on rt k ( n ) , is the subject of the next section. These numerical results suggest thetwo following conjectures, for which we do not have a formal proof yet. Conjecture 2.
Let n and k be two integers such that ≤ k ≤ n and consider the function F k ( n ) defined in Eq. (15). Then it holds that F k ( n ) = min ≤ h ≤ k { B h ( n ) + ˜ U kh ( n ) } = B ( n ) + ˜ U k ( n ) ,which implies that rt k ( n ) ≤ U k ( n ) . (16) . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices Figure 5: Comparison between the function k − k + 12 and the values of the smallest n for whichresulted that F k ( n (cid:48) ) = B k ( n (cid:48) ) for all n (cid:48) such that n < n (cid:48) < .Conjecture 2 suggests that the knowledge of the function ˜ U kh ( n ) for h = 2 is enough to establish a(better) upper bound on rt k ( n ) , for every ≤ k ≤ n . Since ˜ U k ( n ) is an upper bound on O k ( n ) (defined in Eq. (11)), Conjecture 2 also suggests that summing up columns (or rows) of weight two ina matrix should be the quickest way to obtain a column (or row) of weight ≥ k , for any k . For further(and future) improvements on the upper bound on rt k ( n ) , we could then think to better exploit theusage of columns of weight two.Numerical simulations also suggest that the two bounds B k ( n ) and F k ( n ) should coincide for n big enough (and fixed k ); this is formalized in the following conjecture. Conjecture 3.
Let k ∈ N and n k = 2 k − k + 12 . Then:1. if ≤ k ≤ , for all n > k it holds that F k ( n ) = B k ( n ) ;2. if k > , for all n > n k it holds that F k ( n ) = B k ( n ) .Notice that n k > k for all k > . Conjecture 3, if true, states that our two upper bounds on the k -RT coincide when k is fixed and n is big enough. Equivalently, our two upper bounds on the k -RTcoincide when n is fixed and k is small enough. This would imply that the linear upper bound on rt k ( n ) for fixed k ≤ √ n in Eq. (8) is not improved by our new techinque. On the other hand, forvalues of n and k not fulfilling requirements 1. and 2. of Conjecture 3, numerical results seem tosuggest that F k ( n ) < B k ( n ) , that is our new techinque strictly improves the bound B k ( n ) . This willbe shown in detail in the next section, in particular in Figures 8 and 9. Finally, Figure 5 shows foreach fixed < k ≤ , the smallest value of n < found such that F k ( n (cid:48) ) = B k ( n (cid:48) ) for all n (cid:48) s.t. n < n (cid:48) < ; it clearly supports Conjecture 3. C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices (a) n = 4 , M = M CPR (b) n = 6 , M = M K Figure 6: Comparison between the upper bounds F k ( n ) and B k ( n ) , valid for all primitive NZ sets,and rt k ( M ) for (a) M = M CP R and (b) M = M K .
5. Numerical results for the upper bound F h ( n ) In this section, we first present the analogue of Figures 2, 3 and 4 for the new upper bound F k ( n ) ,and then we show some graphs picturing the behavior of F k ( n ) , first for fixed k and then for fixed n .Figure 6 compares F k ( n ) with B k ( n ) and with the real k -RT of the primitive sets M CP R and M K (see Section 3.2). We can see that for k ≤ √ n , the upper bound F k ( n ) coincides with B k ( n ) and itis fairly close to the actual value of rt k ( M ) , while for k > √ n , F k ( n ) strictly improves on B k ( n ) in almost all cases. Figure 7 compares the upper bounds F k ( n ) and B k ( n ) , and the results of theEppstein heuristic on the k -RT of the primitive sets M C , M C , M C and M C (see Section 3.2).It can be noticed that for k big enough, F k ( n ) substantially improves on B k ( n ) . Regarding Figure4, since k is fixed equal to and n ranges from to , in view of Conjecture 3 (and confirmed bynumerical experiments) the values of F k ( n ) coincide with the values of B k ( n ) .Figure 8 compares the new upper bound F k ( n ) on the k -RT with the bound B k ( n ) obtained inSection 3.1 for fixed values of k ( k = 10 , , , , , ). The pictures show the values of n forwhich F k ( n ) is a real improvement of B k ( n ) , as for n big enough the two bounds seem to coincide(see also Conjecture 3). Furthermore for every k and n we obtained that the minimum of (15) isreached at h = 2 , that is F k ( n ) = B ( n ) + ˜ U k ( n ) , thus supporting Conjecture 2.Figure 9 compares F k ( n ) with B k ( n ) for fixed values of n ( n = 10 , , , , , ) and k ranging from to n . We again observe that for small values of k the two bounds coincide (seealso Conjecture 3), while for bigger values of k the bound F k ( n ) is a substantial improvement of B k ( n ) . Again, for every k and n we obtained that the minimum of (15) is reached at h = 2 , that is F k ( n ) = B ( n ) + ˜ U k ( n ) , supporting Conjecture 2.Finally, Figure 10 compares the upper bounds F k ( n ) and B k ( n ) for k = n , that is the upper . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices M = M C (b) M = M C (c) M = M C (d) M = M C Figure 7: Comparison between the bounds F k ( n ) and B k ( n ) , and the Eppstein approximation of rt k ( M ) , for (a) M = M C , (b) M = M C , (c) M = M C , (d) M = M C .bound on the length of the smallest product having a column or a row with all positive entries. Sinceby Proposition 1.6 any generic upper bound on the reset threshold of a synchronizing automaton is anupper bound on rt n ( n ) , in Figure 10 we also represent the function (15617 n + 7500 n + 9375 n − / , which is the upper bound on the reset threshold of a synchronizing automaton on n states found by Szykuła in [38]. Szykuła’s upper bound has been recently improved by Shitov[50], who proved that we can upper bound the reset threshold of a synchronizing automaton by afunction αn + o ( n ) with α ≤ . (while in Szykula’s upper bound α ≈ . ). We decidedto picture Szykuła’s bound because it has a precise analytical expression, while Shitov’s one does not,and because the difference between the two bounds is negligible with respect to our purposes. IndeedFigure 10 shows that up to now, our techniques fall short of improving the upper bound on the n -RT,as more efficient bounds are already known. C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices (a) k = 10 (b) k = 20 (c) k = 30 (d) k = 40 (e) k = 50 (f) k = 100 Figure 8: Comparison between the bounds F k ( n ) and B k ( n ) for fixed values of k . . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices n = 10 (b) n = 50 (c) n = 100 (d) n = 200 (e) n = 500 (f) n = 1000 Figure 9: Comparison between the bounds F k ( n ) and B k ( n ) for fixed values of n . C. Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices
Figure 10: Comparison between the two upper bounds F n ( n ) , B n ( n ) , and Szykuła’s upper bound.The function n / has been pictured for reference.
6. Conclusions
In this paper we have shown that we can upper bound the length of the shortest product of a primi-tive NZ set M having a column or a row with k positive entries by a linear function of the matrix size n ,for any constant k ≤ √ n . We have called this length the k -rendezvous time ( k -RT) of the set M , andwe have shown that the same linear upper bound holds for min (cid:8) rt k (cid:0) Aut ( M ) (cid:1) , rt k (cid:0) Aut ( M (cid:62) ) (cid:1)(cid:9) ,where Aut ( M ) and Aut ( M (cid:62) ) are the synchronizing automata defined in Definition 1.2. We havealso showed that our technique cannot be improved as it is, because it already takes into account theworst cases. We have then presented a new strategy to obtain a better upper bound on rt k ( n ) , whichtakes into account the weights of the columns (or rows) that we are summing up to obtain a column(or row) of higher weight; numerical results show that this new upper bound significantly improvesthe previous one when n is not too large with respect to k . The notion of k -RT comes as an extensionof a similar notion for synchronizing automata introduced in [46]. For automata, the problem whetherthere exists a linear upper bound on the k -RT for small k is still open, as the only nontrivial result onthe k -RT that appears in the literature proves a quadratic upper bound on the -RT [46]. We believethat our result, as well as the new technique developed in Section 4, could help in shedding light tothis problem and possibly to the ˇCerný conjecture, in view of the connection between synchronizingautomata and primitive NZ sets established by Theorem 1.4. . Catalano, U. Azfar, L. Charlier, R.M. Jungers / A linear bound on the k-rt for primitive sets of NZ matrices References [1] Protasov VY, Voynov AS. Sets of nonnegative matrices without positive products.
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