Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs
RRestorable Shortest Path Tiebreaking for Edge-Faulty Graphs
Greg BodwinUniversity of Michigan EECS [email protected]
Merav ParterWeizmann Institute of Science [email protected]
Abstract
The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp.’02] proves that, in an undirected unweighted graph, any replacement shortest path avoidinga failing edge can be expressed as the concatenation of two original shortest paths. However,the lemma is tiebreaking-sensitive : if one selects a particular canonical shortest path for eachnode pair, it is no longer guaranteed that one can build replacement paths by concatenatingtwo selected shortest paths. They left as an open problem whether a method of shortest pathtiebreaking with this desirable property is generally possible.We settle this question affirmatively with the first general construction of restorable tiebreakingschemes . We then show applications to various problems in fault-tolerant network design. Theseinclude a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact)distance labeling schemes, fault-tolerant subset distance preservers and +4 additive spanners withimproved sparsity, and fast distributed algorithms that construct these objects. For example, analmost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributedconstruction of sparse fault-tolerant distance preservers resilient to three faults.
Contents
A Lower bound for f -failures preservers with a consistent and stable tie-breakingscheme 15 a r X i v : . [ c s . D S ] F e b Introduction
This paper builds on a classic work of Afek, Bremler-Barr, Kaplan, Cohen, and Merritt from2002, which initiated a theory of shortest path restoration in graphs [3]. The premise is that one hasa network, represented by a graph G , and one has computed its shortest paths and stored them in arouting table. But then, an edge in the graph breaks, rendering some of the paths unusable. Wewant to efficiently restore these paths, changing the table to reroute them along a new shortest pathbetween the same endpoints in the surviving graph. An ideal solution will both avoid recomputingshortest paths from scratch and only require easy-to-implement changes to the routing table.Motivated by the fact that the multiprotocol label switching (MPLS) allows for efficient concate-nation of paths, Afek et al [3] developed the following elegant structure theorem for the problem,called the restoration lemma . All graphs in this paper are undirected and unweighted. Theorem 1 (Restoration Lemma [3]) . For any graph G = ( V, E ) , vertices s, t ∈ V , and failingedge e ∈ E , there exists a vertex x and a replacement shortest s (cid:32) t path avoiding e that is theconcatenation of two original shortest paths π ( s, x ) , π ( t, x ) in G . We remark that some versions of this lemma were perhaps implicit in prior work, e.g., [23]. Therestoration lemma itself has proved somewhat difficult to apply directly, and most applications ofthis theory use weaker variants instead (e.g., [8, 4, 12]). The issue is that the restoration lemma is tiebreaking-sensitive , in a sense that we next explain.To illustrate, let us try a naive attempt at applying the restoration lemma. One might tryto restore a shortest path π ( s, t ) under a failing edge e by searching over all possible midpoints x , concatenating the existing shortest paths π ( s, x ) , π ( t, x ), and then selecting the replacement s (cid:32) t path to be the shortest among all concatenated paths that avoid e . It might seem that therestoration lemma promises that one such choice of midpoint x will yield a valid replacement shortestpath. But this isn’t quite right: the restoration lemma promises that there exist two shortest pathsof the form π ( s, x ) , π ( t, x ) whose concatenation forms a valid replacement path. But generally therecan be many shortest s (cid:32) x and t (cid:32) x paths, and in designing the initial routing table we implicitlybroke ties to select just one of them. The restoration lemma does not promise that the selected shortest paths can be used to restore the shortest s (cid:32) t path by concatenating two of them.So, is it possible to break shortest path ties in such a way that this restoration-by-concatenationmethod works? Afek et al. [3] discussed this question extensively, and gave a partial negativeresolution: when the input graph is a 4-cycle, one cannot select symmetric shortest paths to enablethe method (see Theorem 22 in the body of this paper for a formal proof). By “symmetric” wemean that, for all nodes s, t , the selected s (cid:32) t and t (cid:32) s shortest paths are the same. However,Afek et al. [3] also point out that the MPLS protocol is inherently asymmetric, and so in principleone can choose different s (cid:32) t and t (cid:32) s shortest paths. They left as a central open questionwhether the restoration lemma can be implemented by an asymmetric tiebreaking scheme (see theirremark at the bottom of page 8). In the meantime, they showed that one can select a larger “baseset” of O ( mn ) paths such that one can restore shortest paths by concatenating two of these paths,and they suggested as an intermediate open question whether their base set size can be improved.This method has found applications in network design (e.g., [3, 8, 4, 12]), but these applicationstend to pay an overhead associated to the larger base set size.The main result of this paper is a positive resolution of the question left by Afek et al. [3]: weprove, perhaps surprisingly, that asymmetry is indeed enough to allow restorable tiebreaking inevery graph. Theorem 2 (Main Result) . In any graph G , one can select a single shortest path for each ordered pair of vertices such that, for any pair of vertices s, t and a failing edge e , there is a vertex x and a eplacement shortest s (cid:32) t path avoiding e that is the concatenation of two selected shortest paths π ( s, x ) , π ( t, x ) . We emphasize again that this theorem is possible only because we select independent shortestpath for each ordered pair of vertices, and thus asymmetry is allowed. If one insists on symmetry,then the theorem can also be viewed as a way to generate a base set with at most two shortestpaths per vertex pair, which thus has size ≤ n − n . The shortest path tiebreaking method usedin this theorem has a few other desirable properties, outlined in Section 2. Most importantly it is consistent , which implies that the selected paths have the right structure to be encoded in a routingtable. It can also be efficiently computed, using a single call to any APSP algorithm that can handledirected weighted input graphs. Replacement Path Algorithms.
Our first applications of our restorable shortest path tiebreak-ing are to computation of replacement paths. The problem has been extensively studied in the single-pair setting, where the input is a graph G = ( V, E ) and a vertex pair s, t , and the goal isto report dist G \{ e } ( s, t ) for every edge e along a shortest s (cid:32) t path. The single-pair setting canbe solved in (cid:101) O ( m + n ) time [23, 25]. Recently, Chechik and Cohen [11] introduced the sourcewise setting, in which one wants to solve the problem for all pairs in { s } × V simultaneously. They gaveconditionally near-optimal algorithms for this problem, which were subsequently generalized to the S × V setting by Gupta, Jain, and Modi [21].We study the natural subsetwise version of the problem, subset-rp , where one is given a graph G and a vertex subset S , and the goal is to solve the replacement path problem simultaneously forall pairs in S × S . We prove: Theorem 3.
Given an n -vertex, m -edge undirected unweighted graph G and | S | = σ source vertices,there is a centralized algorithm that solves solves subset-rp in O ( σm ) + (cid:101) O ( σ n ) time. We remark that, in the case where most pairs s, t ∈ S have dist G ( s, t ) = Ω( n ), the latter termin the runtime σ n is the time required to write down the output. So this term is unimprovable, upto the hidden log factors. The leading term of σm is required for any “combinatorial” algorithm tocompute single-source shortest paths even in the non-faulty setting, and so it will not be improvableeither, except possibly by an algorithm that relies on fast matrix multiplication. Fault-tolerant preservers and additive spanners.
We next discuss our applications for theefficient constructions of fault-tolerant distance preservers, defined as follows:
Definition 4 ( S × T f -FT Preserver) . A subgraph H ⊆ G is an S × T f -FT preserver if for every s, t ∈ S × T it holds that dist H \ F ( s, t ) = dist G \ F ( s, t ) , for every F ⊆ E, | F | ≤ f . When T = S the object is called a subset preserver of S , and when T = V (all vertices in theinput graph) the object is sometimes called an FT-BFS structure , since the f = 0 case is then solvedby a collection of BFS trees. The primary objective for all of these objects is to minimize the size ofthe preserver, as measured by its number of edges.For f = 1, it was shown in [8, 7] that one can compute an S × S O ( | S | n )edges, by properly applying the original version of the restoration lemma by Afek et al. [3]. Ourrestorable tie-breaking scheme provides a simple and more general way to convert from S × V ( f − S × S f -FT preservers, which also enjoys better construction time, inthe centralized and distributed settings. For example, for f = 1 we can compute an S × S S , where each BFS tree iscomputed using our tie-breaking scheme. More generally, we get the following bounds: Theorem 5.
Given an n -vertex graph G = ( V, E ) , a set of source vertices S ⊆ V , and a fixednonnegative integer f , there is an ( f + 1) -FT S × S distance preserver of G, S on O (cid:16) n − / f | S | / f (cid:17) edges. This bound is new for all preservers tolerating ≥ f ≥ additive spanners : Definition 6 (FT Additive Spanners) . Given a graph G = ( V, E ) and a set of source vertices S ⊆ V , an f -FT + k additive spanner is a subgraph H satisfying dist H \ F ( s, t ) ≤ dist G \ F ( s, t ) + k for all vertices s, t ∈ S and sets of | F | ≤ f failing edges. Theorem 7.
For any n -vertex graph G = ( V, E ) and nonnegative integer f , there is an ( f + 1) -FT +4 additive spanner on O f (cid:16) n f / (2 f +1) (cid:17) edges. This is also new for ≥ f ≥ f = 0, i.e., thesingle-fault setting, our +4 spanner has O ( n / ) edges which matches the bound originally attainedby Bil´o, Grandoni, Gual´a, Leucci, Proietti [6] and also obtained as corollaries of the previous twopapers on subset preservers [8, 7]. There are many notable constructions of fault-tolerant additivespanners with other error bounds; see for example [6, 28, 9, 8, 10]. Distributed constructions of fault-tolerant preservers.
Distributed constructions of FTpreservers attracted attention recently [15, 20, 16, 29]. In the context of exact distance preservers,Ghaffari and Parter [20] presented the first distributed constructions of fault tolerant distancepreserving structures. For every n -vertex D -diameter graph G = ( V, E ) and a source vertex s ∈ V ,they gave an (cid:101) O ( D )-round randomized algorithm for computing a 1-FT { s } × V preserver with O ( n / ) edges. Recently, Parter [29] extended this construction to 1-FT S × V preservers with (cid:101) O ( (cid:112) | S | n / ) edges and using (cid:101) O ( D + (cid:112) n | S | ) rounds. [29] also presented a distributed constructionof source-wise preservers against two edge -failures, with O ( | S | / · n / ) edges and using (cid:101) O ( D + n / | S | / + | S | / n / ) rounds. These constructions immediately yield +2 additive spanners resilientto two edge failures with subquadratic number of edges, and sublinear round complexity. To thisdate, we are still lacking efficient distributed constructions of f -FT preservers (or additive spanners)for f ≥
3. In addition, no efficient constructions are known for FT spanners with additive stretchlarger than two (which are sparser in terms of number of edges w.r.t the current +2 FT-additivespanners). Finally, there are no efficient constructions of subsetwise FT-preservers, e.g., the onlydistributed construction for 1-FT S × S preserver employs the sourcewise construction of 1-FT S × V preservers, ending with a subgraph of O ( (cid:112) | S | n / ) edges which is quite far from the state-of-the-art(centralized) bound of O ( | S | n ) edges. In this work, we make a progress along all these directions.Combining the restorable tie-breaking scheme with the work of [29] allows us to provide efficientconstructions of f -FT S × S preservers for f ∈ { , , } whose size bounds match the state-of-the-artbounds of the centralized constructions. As a result, we also get the first distributed constructionsof +4 additive spanners resilient to f ∈ { , , } edge faults. In the
CONGEST model of distributed computing [31]. By efficient, we mean with subquadratic number of edges and sublinear round complexity. heorem 8 (Distributed Constructions of Subsetwise FT-Preservers) . For every D -diameter n -vertex graph G = ( V, E ) , there exist randomized distributed CONGEST algorithms for computing: • -FT S × S preservers with O ( | S | n ) edges and (cid:101) O ( D + | S | ) rounds. • -FT S × S preservers with O ( √ Sn / ) edges and (cid:101) O ( D + (cid:112) | S | n ) rounds. • -FT S × S preservers with O ( | S | / · n / ) edges and (cid:101) O ( D + n / | S | / + | S | / n / ) rounds. Using the f -FT S × S preservers for f ∈ { , , } for a subset S of size σ ∈ {√ n, n / , n / } respectively, we get the first distributed constructions of f -FT +4 additive spanners. Corollary 9 (Distributed Constructions of FT-Additive Spanners) . For every D -diameter n -vertexgraph G = ( V, E ) , there exist randomized distributed algorithms for computing: • -FT +4 additive spanners with (cid:101) O ( n / ) edges and (cid:101) O ( D + √ n ) rounds. • -FT +4 additive spanners with (cid:101) O ( n / ) edges and (cid:101) O ( D + n / ) rounds. • -FT +4 additive spanners preservers with (cid:101) O ( n / ) edges and (cid:101) O ( D + n / ) rounds. One can also remove the log factors in these spanner sizes in exchange for an edge bound thatholds in expectation, instead of with high probability.
Fault-Tolerant Exact Distance Labeling A distance labeling scheme is a way to assign shortbitstring labels to each vertex of a graph G such that dist( s, t ) can be recovered by inspectingonly the labels associated with s and t (and no other information about G ) [18, 17, 32]. In an f -FT distance labeling scheme, the labels are assigned to both the vertices and the edges of thegraph, such that for any set of | F | ≤ f failing edges, we can even recover dist G \ F ( s, t ) by inspectingonly the labels of s, t and the edge set F . These are sometimes called forbidden-set labels [14, 13],and they have been extensively studied in specific graph families, especially due to applicationsin routing [1, 2]. The prior work mainly focused on connectivity labels and approximate distancelabels. In those works, the labels were given also to the edges, and one can inspect the labels offailing edges as well.In our setting we consider exact distance labels. Interestingly, our approach will not need touse edge labels; that is, it recovers dist G \ F ( s, t ) only from the labels of s, t and a description ofthe edge set F . Since one can always provide the entire graph description as part of the label, ourmain objective is in providing FT exact distance labels of subquadratic length, To the best of ourknowledge, the only prior labeling scheme for recovering exact distances under faults was given byBil´o et al. [5]. They showed that given a source vertex s , one can recover distances in { s } × V underone failing edge using labels of size O ( n / ). For the all-pairs setting, this would extend to labelsizes of O ( n / ) bits.We prove: Theorem 10 (Subqaudratic FT labels for Exact Distances) . For any fixed nonnegative integer f ≥ , and n -vertex unweighted undirected graph, there is an ( f + 1) -FT distance labeling schemethat assigns each vertex a label of size (cid:101) O (cid:16) n − / f (cid:17) . The hidden log factors are just to store vertex identifiers; in the word memory model, all logfactors disappear. For f = 1, our vertex labels have size (cid:101) O ( n ), improving over O ( n / ) from [5]. Oursize is near-optimal, for f = 1, in the sense that Ω( n ) label sizes are required even for non-faultyexact distance labeling. This is from a simple information-theoretic lower bound: one can recover4he graph from the labeling, and there are 2 Θ( n ) total n -vertex graphs, so Ω( n ) bits are needed intotal.Finally, we remark that FT labels for exact distances are also closely related to distance sensitivityoracles: these are global and centralized data structures that reports s - t distances in G \ F efficiently.For any f = O (log n/ log log n ), Weimann and Yuster [34, 35] provided a construction of distancesensitivity oracles using subcubic space and subquadratic query time. The state-of-the-art boundsfor this setting are given by van den Brand and Saranurak [33]. It is unclear, however, how tobalance the information of these global succinct data-structures among the n vertices, in the formof distributed labels. In this section we formally introduce the framework of tiebreaking schemes in network design,and we extend it into the setting where graph edges can fail. Then we will recap some work onfault-tolerant distance preservers in light of this framework, and we supply a new auxiliary theoremto fill in a gap in the literature highlighted by this framework. The following objects are studied inprior work on non-faulty graphs:
Definition 11 (Shortest Path Tiebreaking Schemes) . In a graph G , a shortest path tiebreakingscheme π is a function from vertex pairs s, t to one particular shortest s (cid:32) t path π ( s, t ) in G (or π ( s, t ) := ∅ if no s (cid:32) t path exists). It is often useful to enforce coordination between the choices of shortest paths. Two basic kindsof coordination are:
Definition 12 (Symmetry) . A tiebreaking scheme π is symmetric if for all vertex pairs s, t , wehave π ( s, t ) = π ( t, s ) . Definition 13 (Consistency) . A tiebreaking scheme π is consistent if, for all vertices s, t, u, v , if u precedes v in π ( s, t ) , then π ( u, v ) is a contiguous subpath of π ( s, t ) . Consistency is important for many reasons; it is worth explicitly pointing out the following two: • As is well known, in any graph G = ( V, E ) one can find a subtree that preserves all { s } × V distances. Consistency gives a natural converse to this statement: if one selects shortest pathsusing a consistent tiebreaking scheme and then overlays the { s } × V shortest paths, one getsa tree. • It is standard to encode the shortest paths of a graph G in a routing table – that is, a matrixindexed by the vertices of G whose ( i, j ) th entry holds the vertex ID of the next hop on an i (cid:32) j shortest path. For example, the standard implementation of the Floyd-Warshall shortestpath algorithm outputs shortest paths via a routing table of this form. Once again, consistencygives a natural converse: if one selects shortest paths in a graph using consistent tiebreaking,then it is possible to encode these paths in a routing table.Since our goal is to study tiebreaking under edge faults, we introduce the following extendeddefinition: Definition 14 (Replacement Path Tiebreaking Schemes) . In a graph G = ( V, E ) , a replacementpath tiebreaking scheme (RPTS) is a function of the form π ( s, t | F ) , where s, t ∈ V and F ⊆ E .The requirement is that, for any fixed set of failing edges F , the two-parameter function π ( · , · | F ) isa shortest path tiebreaking scheme in the graph G \ F .
5e will say that an RPTS π is symmetric or consistent if, for any F , the tiebreaking scheme π ( · , · | F ) is symmetric or consistent over the graph G \ F . Viewing an RPTS π as a single function,rather than a collection of disjoint tiebreaking schemes, lets us discuss coordination between thechoices of shortest paths under different fault sets. For example, we will use the following naturalproperty, which says that selected shortest paths do not change unless this is forced by a new fault: Definition 15 (Stability) . An RPTS π is stable if, for all s, t, F and edge f / ∈ π ( s, t | F ) , we have π ( s, t | F ) = π ( s, t | F ∪ { f } ) . Our main result is expressed formally using an additional coordination property of RPTSesthat we call restorability . We will discuss that in the next section. For the rest of this section, wewill cover some prior work on fault-tolerant S × V preservers (which will be used centrally in ourapplications). These were introduced by Parter and Peleg [30], initially called FT-BFS structures (since the f = 0 case is solved by BFS trees). We mentioned earlier that, for a tiebreaking schemes π , one gets a valid BFS tree by overlaying the { s } × V shortest paths selected by π if π is consistent.To generalize this basic fact, it is natural to ask what properties of an RPTS yield an FT S × V preserver of optimal size, when the structure is formed by overlaying all the replacement pathsselected by π (under fault sets of size | F | ≤ f ). For f = 1, this question was settled by Parter andPeleg [30]: Theorem 16 ([30]) . For any n -vertex graph G = ( V, E ) , set of source vertices S ⊆ V , andconsistent stable RPTS π , the -FT S × V preserver formed by overlaying all S × V replacementpaths selected by π under ≤ failing edge has O (cid:0) n / | S | / (cid:1) edges. This bound is existentially tight. The extension to f = 2 was established by Parter [27] and Gupta and Khan [22]: Theorem 17 ([27, 22]) . For any n -vertex graph G = ( V, E ) and S ⊆ V , there is a -FT S × V preserver with O (cid:16) n / | S | / (cid:17) edges . This bound is also existentially tight.
In contrast to Theorem 16 which works with any stable and consistent tiebreaking scheme, theupper bound part of Theorem 17 uses additional properties beyond stability and consistency (in[22], called “preferred” paths). It is a major open question in the area to prove or refute tightness ofthis lower bound for any f ≥
3. The only general upper bound is due to Bodwin, Grandoni, Parter,and Vassilevska-Williams [8], and this one does hold for arbitrary consistent stable RPTSes: Theorem 18 ([8]) . For any n -vertex graph G = ( V, E ) , S ⊆ V , fixed nonnegative integer f , andconsistent stable RPTS π , the f -FT S × V preserver formed by overlaying all S × V replacementpaths selected by π under ≤ f failing edges has O (cid:16) n − / f | S | / f (cid:17) edges. Thus, plugging in f = 1 recovers the result of Parter and Peleg [30], but the bound is non-optimalalready for f = 2. It is natural to ask whether the analysis of consistent stable RPTSes can beimproved, or whether additional tiebreaking properties are necessary. As a new auxiliary result, weshow the latter: consistent and stable tie-breaking schemes cannot improve further on this bound,and in particular these properties alone are non-optimal at least for f = 2. The result in [8] has extra log n factors in size, in exchange for an optimized construction algorithm. These caneasily be removed, with the algorithm still running in polynomial time, by standard techniques. heorem 19. For any nonnegative integers f, σ , there are n -vertex unweighted (directed or undi-rected) graphs G = ( V, E ) , vertex subsets S ⊆ V of size | S | = σ , and RPTSes that are consistent,stable, and (for undirected graphs) symmetric that give rise to S × V preservers on Ω( n − / f σ / f ) edges. The proof of this theorem is given in Appendix A. We also note that the lower bound of Theorem19 holds under a careful selection of one particular “bad” tiebreaking scheme, which happens to beconsistent, stable, and symmetric. This lower bound breaks, for example, for many more specificclasses of tiebreaking: for example, if one uses small perturbations to edge weights to break theties. It is therefore intriguing to ask whether it is possible to provide f -FT S × V preservers withoptimal edge bounds using random edge perturbations to break the replacement paths ties. Thedual failure case of f = 2 should serve here as a convenient starting point. It will be very intriguingto see if one can replace the involved tiebreaking scheme of [27, 22] (i.e., of preferred paths) usingrandom edge perturbations while matching the same asymptotic edge bound (or alternatively, toprove otherwise), especially because these random edge perturbations enable restorable tiebreakingin the sense discussed in the next section. The main result in this paper concerns the following new coordination property:
Definition 20 (Restorability) . An RPTS π is restorable if, for all vertices s, t and nonempty edgefault sets F , there exists a vertex x and a proper fault subset F (cid:48) (cid:40) F such that the concatenationof the paths π ( s, x | F (cid:48) ) , π ( t, x | F (cid:48) ) forms a valid s (cid:32) t replacement path avoiding F . (Note: itis not required that this concatenated path is specifically equal to π ( s, t | F ) , just that it is a validreplacement path.) The motivation for this definition comes from restoration lemma by Afek et al [3], discussedabove. It is not obvious at this point that any or all graphs should admit a restorable RPTS. Thatis the subject of our main result.We will analyze RPTSes generated by the following method, which we call antisymmetric-reweighted tiebreaking . Let G = ( V, E ) be the undirected unweighted input graph whose edgeweights are treated as all 1. Let ε > { u, v } with directed edges ( u, v ) and ( v, u ). Then, for each such pair of directededges, we pick a random number r ( u, v ) uniformly from the interval [ − ε, ε ]. We then add r ( u, v ) tothe weight of ( u, v ) and − r ( u, v ) =: r ( v, u ) to the weight of ( v, u ). This process breaks shortest pathties with probability 1, and we can choose ε > π is defined suchthat π ( s, t | F ) selects the unique shortest directed s (cid:32) t path in the directed reweighted graph G \ F . Theorem 21 (Main Result) . Any RPTS π selected by antisymmetric random reweighting issimultaneously stable, consistent, and restorable. We remark that the detail that reweighting variables r ( u, v ) are selected randomly is just tobreak ties, but otherwise does not play into the following argument. If one selects reweighting We work in the word-RAM model, and so we do not pay much attention to bit complexity added to the edgeweights as a result of this reweighting. However, as is standard in the area, the isolation lemma [26] implies that oneonly needs to add + O (log n ) bit precision to edge weights in order to successfully break ties with high probability(even in the setting where there is a tie between exponentially many shortest paths between a vertex pair). r ( u, v ) = − r ( v, u )) and thatthe reweighting yields unique shortest paths, then we will still have stability, consistency, andrestorability. Proof of Theorem 21.
Consistency and stability follow immediately from the fact that the pathsselected by π are unique shortest paths in some directed weighted graph. The rest of this proofestablishes that π is restorable. We notice that it suffices to verify the restorability definition onlyfor | F | = 1: for larger sets of failing edges, one can select a subset F (cid:48) ⊆ F of all but one failing edge,and then use restorability over the single remaining failing edge with respect to the graph G \ F (cid:48) .So, the following proof assumes | F | = 1. Let s, t be vertices and let f = ( u, v ) be the one failingedge. We may assume without loss of generality that ( u, v ) ∈ π ( s, t ), with that orientation, sinceotherwise by stability we have π ( s, t ) = π ( s, t | f ) and so claim is immediate by choice of (say) x = t . Let x be the last vertex along π ( s, t | f ) such that π ( s, x ) avoids f , and let y be the verteximmediately after x along π ( s, t | f ). Hence ( u, v ) ∈ π ( s, y ). These definitions are all recapped inthe following diagram. s t × fu vx y π ( s , t | f ) π ( s, y ) π ( s , x ) Our goal is now to argue that π ( t, x ) does not use the edge f = ( u, v ), and thus π ( s, x ) ∪ π ( t, x )forms a valid replacement path avoiding f . Since we have assumed that the edge f = ( u, v )is oriented with dist G ( u, t ) > dist G ( v, t ), if ( v, u ) ∈ π ( t, x ) then it appears with that particularorientation, v preceding u .In the following we will write dist ∗ ( · , · ) for the distance function in the antisymmetrically randomreweighted version of the input graph (so dist ∗ is generally non-integral, and it is asymmetric in itstwo parameters). Recall that π ( s, y ) includes the edge ( u, v ), and hence the u (cid:32) y path through v is shorter than the alternate u (cid:32) y path through x . We thus have the inequality:dist ∗ ( u, v ) + dist ∗ ( v, y ) < dist ∗ ( u, x ) + dist ∗ ( x, y ) . Since ( u, v ) , ( x, y ) are single edges we can write(1 + r ( u, v )) + dist ∗ ( v, y ) < dist ∗ ( u, x ) + (1 + r ( x, y )) , where r ( u, v ) , r ( x, y ) are the random reweighting variables selected for these two edges. Rearrangingand using antisymmetry of r , we get(1 + r ( y, x )) + dist ∗ ( v, y ) < dist ∗ ( u, x ) + (1 + r ( v, u )) , and so dist ∗ ( v, y ) + dist ∗ ( y, x ) < dist ∗ ( v, u ) + dist ∗ ( u, x ) . This inequality says that the v (cid:32) x path that passes through y is shorter in the reweighted graphthan the one that passes through u . So ( v, u ) / ∈ π ( v, x ). By consistency and the previously-mentionedfact that dist( v, t ) < dist( u, t ), this also implies that ( v, u ) / ∈ π ( t, x ), as desired.Notice that antisymmetric random reweighting does not give a symmetric RPTS. As observedby Afek et al. [3], one cannot generally have restorability and symmetry:8 heorem 22. There are input graphs that do not admit a tiebreaking scheme that is simultaneouslysymmetric and restorable.Proof.
The simplest example is a C : × s tx y Assume π is symmetric and consider the selected non-faulty shortest paths π ( s, y ) and π ( x, t ) goingbetween the two opposite corners. These paths must intersect on an edge; without loss of generality,suppose this edge is ( s, t ). Then π ( s, t ) is just the single edge ( s, t ). Suppose this edge fails, and sothe unique replacement s (cid:32) t path is q = ( s, x, y, t ). Since both π ( s, y ) and π ( x, t ) use the edge( s, t ), this path does not decompose into two non-faulty shortest paths selected by π . Hence π is notrestorable. In the subset-rp problem, the input is a graph G = ( V, E ) and a set of source vertics S , andthe output is: for every pair of vertics s, t ∈ F and every failing edge e ∈ E , report dist G \{ e } ( s, t ).We next present our algorithm for subset-rp . We will use the following algorithm in prior work,for the single-pair setting, as a black box: Theorem 23 ([25]) . When | S | = 2 , there is an algorithm that solves subset-rp ( G, S ) in time (cid:101) O ( m + n ) . We then use Algorithm 1 to solve subset-rp in the general setting.
Algorithm 1 subset-rp
Algorithm
Input:
Undirected unweighted graph G = ( V, E ) on n vertics, vertex subset S ⊆ V of size | S | = σ .Let G (cid:48) be a directed weighted graph obtained from G by antisymmetric random reweighting. for all s ∈ S do Compute an outgoing shortest path tree T s , rooted at s , in the graph G (cid:48) . for all s , s ∈ S do Using Theorem 23, solve subset-rp on vertics { s , s } in the graph T s ∪ T s . Theorem 24.
Given an n -vertex, m -edge graph G and | S | = σ source vertics, Algorithm 1 solves subset-rp in O ( σm ) + (cid:101) O ( σ n ) time.Proof. First we show correctness. Let π be the RPTS associated to the antisymmetric reweightedgraph G (cid:48) . By Theorem 21, π is restorable. Thus, for any s , s ∈ S and failing edge e , there exists avertex x such that π ( s , x ) ∪ π ( s , x ) form a shortest s (cid:32) s path in G \ { e } . Since π ( s , x ) , π ( s , x )are both contained in the graph T s ∪ T s , the algorithm correctly outputs a shortest s (cid:32) s replacement path avoiding e . 9e now analyze runtime. Computing G (cid:48) from G takes O ( m ) time, due to the selection of randomedge weights. Using Dijkstra’s algorithm, it takes O ( m + n log n ) time to compute an outgoingshortest path tree T s in G (cid:48) for each of the | S | = σ source vertics, which costs O ( σ ( m + n log n ))time in total. Finally, we solve subset-rp for each pair of vertics s , s ∈ S , on the graph T s ∪ T s which has only O ( n ) edges. By Theorem 23 each pair requires (cid:101) O ( n ) time, for a total of (cid:101) O ( σ n )time. Here we prove:
Theorem 25.
For any fixed nonnegative integer f ≥ , there is an ( f + 1) -FT distance labelingscheme that assigns each vertex a label of size (cid:101) O (cid:16) n − / f (cid:17) . Proof.
Let π be a consistent stable restorable RPTS from Theorem 21 on the input graph G = ( V, E ).For each vertex s ∈ V , compute an f -FT { s } × V preserver by overlaying the { s } × V replacementpaths selected by π with respect to ≤ f edge failures. The label of s just explicitly stores the edgesof this preserver; the bound on label size comes from the number of edges in Theorem 18 and thefact that O (log n ) bits are required to describe each edge.To compute dist G \ F ( s, t ), we simply read the labels of s, t to determine their associated preserversand we union these together. By the restorability of π , there exists a valid s (cid:32) t replacement pathavoiding F formed by concatenating a path included in the preserver of s and a path included inthe preserver of t . So, it suffices to remove the edges of F from the union of these two preserversand then compute dist( s, t ) in the remaining graph. We next show applications in fault-tolerant network design. The following theorem extendsTheorem 2 in [8] which was shown (by a very different proof) for f = 1, to any f ≥ Theorem 26.
Given an n -vertex graph G = ( V, E ) , a set of source vertics S ⊆ V , and a fixednonnegative integer f , there is an ( f +1) -EFT S × S distance preserver of G, S on O (cid:16) n − / f | S | / f (cid:17) edges.Proof. Using Theorem 21, let π be an RPTS that is simultaneously stable, consistent, and restorable.The construction is to build an f -FT S × V preserver by overlaying all S × V replacement pathsselected by π with respect to ≤ f edge faults. By Theorem 18, this has the claimed number of edges.To prove correctness of the construction, we invoke restorability. Consider some vertics s, t ∈ S and set of | F | ≤ f + 1 edge faults. Since π is restorable, there is a valid s (cid:32) t replacement pathavoiding F that is the concatenation of two shortest paths of the form π ( s, x | F (cid:48) ) , π ( t, x | F (cid:48) ),where | F (cid:48) | ≤ f and x ∈ V . These replacement paths are added as part of the f -FT preservers, andhence the union of the preservers includes a valid s (cid:32) t replacement path avoiding F .We can then plug these subset distance preservers into a standard application to additivespanners. We will black-box the relationship between subset preservers and additive spanners, sothat it may be applied again when we give distributed constructions below. The following lemma isstandard in the literature on spanners. 10 emma 27. Suppose that we can construct an f -FT S × S distance preserver on g ( n, σ, f ) edges,for any set of | S | = σ vertics in an n -vertex graph G . Then G has an f -FT +4 additive spanner on O ( g ( n, σ, f ) + n f /σ ) edges.Proof. For simplicity, we will give a randomized construction where the error bound holds deter-ministically and the edge bound holds in expectation. Naturally, one can repeat the construction O (log n ) times and select the sparsest output spanner to boost the edge bound to a high-probabilityguarantee.Let C ⊆ V be a subset of σ vertics, selected uniformly at random. Call the vertics in C clustercenters . For each vertex v ∈ V , if it has at least f + 1 neighbors in C , then add an arbitrary f + 1edges connecting v to vertics in C to the spanner H , and we will say that v is clustered . Otherwise,if v has ≤ f neighbors in C , then add all edges incident to v to the spanner, and we will say that v is unclustered . By a standard probabilistic analysis (omitted), each vertex contributes O ( f n/σ ) edgesin expectation in this step, which gives the second term in our claimed edge bound. The secondand final step in the construction is to add an f -FT subset distance preserver over the vertex set C to the spanner H . This costs g ( n, σ, f ) edges, which gives the first term in our claimed edge bound.Now we prove that H is an f -EFT +4 spanner of G (deterministically). Consider any two vertics s, t and a set of | F | ≤ f edge faults, and let q = q ( s, t | F ) be any replacement path between them.Let x be the first clustered vertex and y the last clustered vertex in q . Let c x , c y be cluster centersadjacent to x, y , respectively, in the graph G \ F (since x, y are each adjacent to f + 1 cluster centersinitially, at least one adjacency still holds after F is removed). We then have:dist H \ F ( s, t ) ≤ dist H \ F ( s, x ) + dist H \ F ( x, y ) + dist H \ F ( y, t )= dist G \ F ( s, x ) + dist H \ F ( x, y ) + dist G \ F ( y, t ) all unclustered edges in H ≤ dist G \ F ( s, x ) + (cid:0) H \ F ( c x , c y ) (cid:1) + dist G \ F ( y, t ) triangle inequality= dist G \ F ( s, x ) + (cid:0) G \ F ( c x , c y ) (cid:1) + dist G \ F ( y, t ) C × C preserver ≤ dist G \ F ( s, x ) + (cid:0) G \ F ( x, y ) (cid:1) + dist G \ F ( y, t ) triangle inequality= dist G \ F ( s, t ) + 4 . where the last equality follows since x, y lie on a valid s (cid:32) t replacement path.Using this, we get: Theorem 28.
For any n -vertex graph G = ( V, E ) and nonnegative integer f , there is an ( f + 1) -FT +4 additive spanner on O f (cid:16) n f / (2 f +1) (cid:17) edges.Proof. This follows immediately by applying Lemma 27 to the subset distance preservers fromTheorem 26.
Throughout this section, we consider the standard
CONGEST model of distributed computing[31]. In this model, the network is abstracted as an n -vertex graph G = ( V, E ), with one processoron each vertex. Initially, these processors only know their incident edges in the graph, and thealgorithm proceeds in synchronous communication rounds over the graph G = ( V, E ). In each round,vertices are allowed to exchange O (log n ) bits with their neighbors and perform local computation.Throughout, the diameter of the graph G = ( V, E ) is denoted by D .11 emma 29 (Distributed Tie-Breaking SPT) . For every unweighted and undirected n -vertex graph G = ( V, E ) and a randomized tie-breaking (possibly anti-symmetric) weight function ω : E → [1 − (cid:15), (cid:15) ] , and every source vertex s , there is a deterministic algorithm that computes a shortest-path tree rooted at s (based on the weights of ω ) within O ( D ) rounds. The total number of messagessent through each edge is bounded O (1) .Proof. Let dist( u, v ) denote the unweighted u - v distance in G , and dist ∗ ( u, v ) denote the (directed) u (cid:32) v distance under the weights of ω . Since ω is only a tie-breaking weight function, any shortestpath tree of s under ω is also a legit BFS tree for s . In other words, all vertices at unweighteddistance d from the source s must appear on level d in the SPT of s under ω . The constructiontherefore is almost analogous to the standard distributed BFS construction, and the only differenceis that each vertex uses the weights of ω in order to pick its parent in the tree.The algorithm works in O ( D ) steps, each step is implemented within O (1) rounds. The invariantat the beginning of phase i ∈ { , . . . , D } is as follows: all vertices in the first i layers L , . . . , L i − of the SPT are marked, and each of these vertices v know their distance dist ∗ ( s, v ). This clearlyholds for i = 1 as L = { s } and dist ∗ ( s, s ) = 0. In phase i , all vertices u of layer L i broadcast thedistance dist ∗ ( s, u ) to their neighbors. Every vertex v / ∈ (cid:83) i − j =0 L j that receives messages from itsneighbors in layer L i picks its parent u to be the vertex that minimizes its dist ∗ ( s, v ) distance. Thatis, u = arg min w ∈ L i ∩ N ( v ) dist ∗ ( s, w ) + ω ( w, v ). It is easy to see by induction on i , that the invariantis now satisfied at the beginning of phase i + 1. After D rounds, the construction of the tree iscompleted. As in the standard BFS construction, only O (1) number of messages are sent througheach edge in the graph.We make an extensive use of the random delay approach of [24, 19]. Specifically, we use thefollowing theorem: Theorem 30 ([19, Theorem 1.3]) . Let G be a graph and let A , . . . , A m be m distributed algorithmsin the CONGEST model, where each algorithm takes at most d rounds, and where for each edgeof G , at most c messages need to go through it, in total over all these algorithms. Then, thereis a randomized distributed algorithm (using only private randomness) that, with high probability,produces a schedule that runs all the algorithms in O ( c + d · log n ) rounds, after O ( d log n ) roundsof pre-computation. Using the random delay approach for computing SPT with respect to our restorable te-breakingscheme, provides an efficient distributed construction of 1-FT S × S preserves, and consequentlyalso FT +4-additive spanners. Lemma 31 (Dist. 1-FT S × S Preserver) . For every unweighted and undirected n -vertex graph G = ( V, E ) and subset of sources S , there is a randomized algorithm that computes a -FT S × S preserver with O ( | S | n ) edges within (cid:101) O ( D + | S | ) rounds, with high probability.Proof. First, the vertices locally compute the restorable tie-breaking weight function ω , by lettingeach vertex u sample the weights for its incident edges, and sending it to the second edge endpoints.This is done in a single communication round. Then we apply the SPT construction of Lemma29 under ω , for every source s ∈ S . We run all these algorithms, A s , . . . , A s σ , simultaneously inparallel using the random delay approach. By sharing a shared seed of O (log n ) bits, each vertexcan compute the starting time of each algorithm A s i . As the total congestion of these algorithms is O ( | S | ), by Thm. 30, the round complexity is bounded by (cid:101) O ( D + | S | ) rounds, w.h.p.By applying the constructions of 1-FT S × V preservers and 2-FT S × V preservers of [29]and using the restorable weight function to break the BFS ties, we immediately get 2-FT S × S S × S preservers. Specifically, Theorem 8 (2) (resp., (3)) follows by using therestorable weight function with Theorem 1 (resp., Theorem 2) of [29]. E.g., to compute the 2-FT S × S preserver, we apply the construction of 1-FT S × V preservers of Theorem 1 in [29] with theonly difference being that the shortest path ties are decided based on the restorable weight functioninstead of breaking it arbitrarily. This can be easily done by augmenting the BFS tokens with thelength of the path from the root.Finally, the construction of FT S × S preservers can naturally yield FT +4-additive spanners.No prior constructions of such spanners have been known before (not even for f = 1). Proof of Cor. 9.
Let S be a sample of σ = Θ( √ n log n ) sources sampled independently in V . Item(1) of the corollary follows by applying the construction of Theorem 8(1) and the correctnessfollows by Lemma 27. In the same manner, item (2) of the corollary follows by applying the aboveconstruction with σ = n / surces, and using Theorem 8(2). Finally, item (3) of the corollary followsby applying Theorem 8 (3) with σ = n / . References [1]
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A Lower bound for f -failures preservers with a consistent andstable tie-breaking scheme In this section we prove Theorem 19 by giving a lower bound constructions for S × V distancepreservers using a consistent and stable tie-breaking scheme.We begin by showing the construction for the single source case (i.e., σ = 1) and then extend itto the case of multiple sources. Our construction is based on the graph G f ( d ) = ( V f , E f ), definedinductively. For f = 1, G ( d ) = ( V , E ) consists of three components:15. a set of vertices U = { u , . . . , u d } connected by a path P = [ u , . . . , u d ],2. a set of terminal vertices Z = { z , . . . , z d } (viewed by convention as ordered from left to right),3. a collection of d vertex disjoint paths { Q i } , where each path Q i connects u i and z i and haslength of d − i + 1 edges, for every i ∈ { , . . . , d } .The vertex r ( G ( d )) = u is fixed as the root of G ( d ), hence the edges of the paths Q i are viewedas directed away from u i , and the terminal vertices of Z are viewed as the leaves of the graph,denoted Leaf ( G ( d )) = Z . Overall, the vertex and edge sets of G ( d ) are V = U ∪ Z ∪ (cid:83) di =1 V ( Q i )and E = E ( P ) ∪ (cid:83) di =1 E ( Q i ).For ease of future analysis, we assign labels to the leaves z i ∈ Leaf ( G ( d )). Let Label f : Leaf ( G f ( d )) → E ( G f ( d )) f . The label of each leaf corresponds to a set of edge faults under which thepath from root to leaf is still maintained (as will be proved later on). Specifically, Label ( z i , G ( d )) =( u i , u i +1 ) for i ∈ [1 , d − P ( z i , G ( d )) = P [ r ( G ( d )) , u i ] ◦ Q i to be the pathfrom the root u to the leaf z i .To complete the inductive construction, let us describe the construction of the graph G f ( d ) =( V f , E f ), for f ≥
2, given the graph G f − ( √ d ) = ( V f − , E f − ). The graph G f ( d ) = ( V f , E f )consists of the following components. First, it contains a path P f = [ u f , . . . , u fd ], where the vertex r ( G f ( d )) = u f is fixed to be the root. In addition, it contains d disjoint copies of the graph G (cid:48) = G f − ( √ d ), denoted by G (cid:48) , . . . , G (cid:48) d (viewed by convention as ordered from left to right), whereeach G (cid:48) i is connected to u fi by a collection of d vertex disjoint paths Q fi , for i ∈ { , . . . , d } , connectingthe vertices u fi with r ( G (cid:48) i ). The length of Q fi is d − i + 1, and the leaf set of the graph G f ( d ) is theunion of the leaf sets of G (cid:48) j ’s, Leaf ( G f ( d )) = (cid:83) dj =1 Leaf ( G (cid:48) j ).Next, define the labels Label f ( z ) for each z ∈ Leaf ( G f ( d )). For every j ∈ { , . . . , d } and anyleaf z j,i ∈ Leaf ( G (cid:48) j ), let Label f ( z j,i , G f ( d )) = ( u fj , u fj +1 ) ◦ Label f − ( z j,i , G (cid:48) j ).Denote the size (number of vertices) of G f ( d ) by N ( f, d ), its depth (maximum distance betweenthe root vertex r ( G f ( d )) to a leaf vertex in Leaf ( G f ( d ))) by depth ( f, d ), and its number of leavesby nLeaf ( f, d ) = | Leaf ( G f ( d )) | . Note that for f = 1, N (1 , d ) = 2 d + d ≤ d , depth (1 , d ) = d and nLeaf (1 , d ) = d . We now observe that the following inductive relations hold. Observation 1. (a) depth ( f, d ) = O ( d ) , (b) nLeaf ( f, d ) = d − / f − and (c) N ( f, d ) = 2 f · d .Proof. (a) follows by the length of Q fi , which implies that depth ( f, d ) = d + depth ( f − , √ d ) ≤ d .(b) follows by the fact that the terminals of the paths starting with u f , . . . , u fd are the terminals ofthe graphs G (cid:48) , . . . , G (cid:48) d which are disjoint copies of G f − ( √ d ), so nLeaf ( f, d ) = d · nLeaf ( f − , √ d ).(c) follows by summing the vertices in the d copies of G (cid:48) i (yielding d · N ( f, d )) and the vertices in d vertex disjoint paths, namely Q f , . . . , Q fd of total d vertices, yielding N ( f, d ) = d · N ( f − , √ d )+ d ≤ f d .Consider the set of leaves in G f ( d ), namely, Leaf ( G f ( d )) = (cid:83) di =1 Leaf ( G (cid:48) i ) = { z , . . . , z λ } ,ordered from left to right according to their appearance in G f ( d ).For every leaf vertex z ∈ Leaf ( G f ( d )), we define inductively a path P ( z, G f ( d )) connecting theroot r ( G f ( d )) = u f with the leaf z . As described above for f = 1, P ( z i , G ( d )) = P [ r ( G ( d )) , u i ] ◦ Q i . Consider a leaf z ∈ Leaf ( G f ( d )) such that z is the i th leaf in the graph G (cid:48) j . We thereforedenote z as z i,j , and define P ( z j,i , G f ( d )) = P f [ r ( G f ( d )) , u j ] ◦ Q fj ◦ P ( z j,i , G (cid:48) j ). We next claim thefollowing on these paths. Lemma 32.
For every leaf z j,i ∈ Leaf ( G f ( d )) it holds that:(1) The path P ( z j,i , G f ( d )) is the only u f − z j,i path in G f ( d ) . P ( z j,i , G f ( d )) ⊆ G \ (cid:83) i ≥ j Label f ( z j,i , G f ( d )) ∪ (cid:83) k ≥ j,(cid:96) ∈ [1 , nLeaf ( f − , √ d )] Label f ( z k,(cid:96) , G f ( d )) .(3) P ( z j,i , G f ( d )) (cid:54)⊆ G \ Label f ( z k,(cid:96) , G f ( d )) for k < j and every (cid:96) ∈ [1 , nLeaf ( f − , √ d )] , as wellas for k = j and every (cid:96) ∈ [1 , i − . (4) | P ( z, G f ( d )) | = | P ( z (cid:48) , G f ( d )) | for every z, z (cid:48) ∈ Leaf ( G f ( d )) .Proof. We prove the claims by induction on f . For f = 1, the lemma holds by construction. Assumethis holds for every f (cid:48) ≤ f − G f ( d ). Recall that P f = [ u f , . . . , u fd ], and let G (cid:48) , . . . , G (cid:48) d be d copies of the graph G f − ( √ d ), viewed as ordered from left to right, where G (cid:48) j is connected to u fj . That is, there are disjoint paths Q fj connecting u fj and r ( G (cid:48) j ), for every j ∈ { , . . . , d } .Consider a leaf vertex z j,i , the i th leaf vertex in G (cid:48) j . By the inductive assumption, there exists asingle path P ( z j,i , G (cid:48) j ) between the root r ( G (cid:48) j ) and the leaf z j,i , for every j ∈ { , . . . , d } . We nowshow that there is a single path between r ( G f ( d )) = u f and z j,i for every j ∈ { , . . . , d } . Sincethere is a single path P (cid:48) connecting r ( G f ( d )) and r ( G (cid:48) j ) given by P (cid:48) = P f [ u f , u fj ] ◦ Q fj , it followsthat P ( z j,i , G f ( d )) = P (cid:48) ◦ P ( z j,i , G (cid:48) j ) is a unique path in G f ( d ).We now show (2). We first show that P ( z j,i , G f ( d )) ⊆ G \ (cid:83) (cid:96) ≥ i | z j,(cid:96) ∈ Leaf ( G (cid:48) j ) LAB f ( z j,(cid:96) , G f ( d )).By the inductive assumption, P ( z j,i , G (cid:48) j ) ∈ G \ (cid:83) (cid:96) ≥ i Label f − ( z j,(cid:96) , G (cid:48) j ). Since Label f ( z j,i , G f ( d )) =( u fj , u fj +1 ) ◦ Label f − ( z j,i , G (cid:48) j ), it remains to show that e (cid:96) = ( u f(cid:96) , u f(cid:96) +1 ) / ∈ P (cid:48) for (cid:96) ≥ i . Since P (cid:48) diverges from P f at the vertex u fj , it holds that e j , . . . , e d − / ∈ P ( z j,i , G f ( d )). We next complete theproof for every leaf vertex z k,(cid:96) for z k,(cid:96) ∈ Leaf ( G (cid:48) q ) for k > j and every (cid:96) ∈ nLeaf ( f − , √ d ). Theclaim holds as the edges of G (cid:48) j and G (cid:48) k are edge-disjoint, and e j , . . . , e d − / ∈ P ( z j,i , G f ( d )).Consider claim (3) and a leaf vertex z j,i ∈ Leaf ( G (cid:48) j ) for some j ∈ { , . . . , d } and i ∈ nLeaf ( f − , √ d ). Let Z = { z j,(cid:96) ∈ Leaf ( G (cid:48) j ) | (cid:96) < i } be the set of leaves to the left of z j,i that belongto G (cid:48) j , and let Z = { z k,(cid:96) / ∈ Leaf ( G (cid:48) j ) | j > k } be the complementary set of leaves to the leftof z j,i . By the inductive assumption, P ( z j,i , G (cid:48) j ) (cid:42) G \ Label f − ( z j,(cid:96) , G (cid:48) j ) for every z j,(cid:96) ∈ Z .The claim holds for Z as the order of the leaves in G (cid:48) j agrees with their order in G f ( d ), and Label f − ( z k,(cid:96) , G (cid:48) j ) ⊂ Label f ( z k,(cid:96) , G f ( d )).Next, consider the complementary leaf set Z to the left of z j,i . Since for every z k,(cid:96) ∈ Z , thedivergence point of P ( z k,(cid:96) , G f ( d )) and P f is at u fk for k < j , it follows that e k = ( u fk , u fk +1 ) ∈ P ( z j,i , G f ( d )), and thus P ( z j,i , G f ( d )) (cid:42) G \ Label f ( z k,(cid:96) , G f ( d )) for every z k,(cid:96) ∈ Z . Finally,consider (4). By setting the length of the paths Q fj to d − j + 1 for every j ∈ { , . . . , d } , we havethat dist( u f , r ( G (cid:48) j )) = d for every j ∈ [1 , d ]. The proof then follows by induction as well, since | P ( z j,i , G (cid:48) j ) | = | P ( z k,(cid:96) , G (cid:48) k ) | for every k, j ∈ [1 , d ] and i, (cid:96) ∈ [1 , nLeaf ( f − , √ d ].Finally, we turn to describe the graph G ∗ f ( V, E, W ) which establishes our lower bound, where W is a particular bad edge weight function that determines the consistent tie-breaking schemewhich provides the lower bound. The graph G ∗ f ( V, E, W ) consists of three components. The firstis the graph G f ( d ) for d = (cid:98) (cid:112) n/ (4 f ) (cid:99) . By Obs. 1, N ( f, d ) = | V ( G f ( d )) | ≤ n/
2. The secondcomponent of G ∗ f ( V, E, W ) is a set of vertices X = { x , . . . , x χ } , where the last vertex of P f , namely, u fd is connected to all the vertices of X . The cardinality of X is χ = n − N ( f, d ) −
1. The thirdcomponent of G ∗ f ( V, E, W ) is a complete bipartite graph B connecting the vertices of X with theleaf set Leaf ( G f ( d )), i.e., the disjoint leaf sets Leaf ( G (cid:48) ) , . . . , Leaf ( G (cid:48) d ). We finally define theweight function W : E → (1 , /n ). Let W ( e ) = 1 for every e ∈ E \ E ( B ). The weights of thebipartite graph edges B are defined as follows. Consider all leaf vertices Leaf ( G f ( d )) from leftto right given by { z , . . . , z λ } . Then, W ( z j , x i ) = ( λ − j ) /n for every z j and every x i ∈ X . Thevertex set of the resulting graph is thus V = V ( G f ( d )) ∪ { v ∗ } ∪ X and hence | V | = n . By Prop. (b)of Obs. 1, nLeaf ( G f ( d )) = d − / f − = Θ(( n/f ) − / f ) , hence | E ( B ) | = Θ(( n/f ) − / f ).17e now complete the proof of Thm. 19 for the single source case. Thm. 19 for | S | = 1 . Let s = u f be the chosen source in the graph G ∗ f ( V, E, W ). We first claimthat under the weights W , there is a unique shortest path, denoted by π ( s, x i | F ) for every x i ∈ X and every fault set F ∈ { Label f ( z , G f ( d )) , . . . , Label f ( z (cid:96) , G f ( d )) } . By Lemma 32(1), there is aunique shortest path from each s to each z j ∈ Leaf ( G f ( d )) denoted by P ( z j , G f ( d )).In addition, by Lemma 32(4), the unweighted length of all the s - z j paths are the same for every z j . Since each x i is connected to each z j with a distinct edge weight in (1 , /n ), we get thateach x i has a unique shortest path from s in each subgraph G \ Label f ( z j , G f ( d )). Note that sincethe uniqueness of π is provided by the edge weights it is both consistent and stable. Also notethat the weights of W are sufficiently small so that they only use to break the ties between equallylength paths.We now claim that a collection of { s } × X replacement paths (chosen based on the weights of W ) contains all edges of the bipartite graph B . Formally, letting P = (cid:91) x i ∈ X (cid:91) z j ∈ Leaf ( G f ( d )) π ( s, x i | Label f ( z j , G f ( d ))) , we will show that E ( B ) ⊆ (cid:83) P ∈P P which will complete the proof. To see this we show that π ( s, x i | Label f ( z j , G f ( d ))) = P ( z j , G f ( d )) ◦ ( z j , x i ). Indeed, by Lemma 32(2), we have that P ( z j , G f ( d )) ⊆ G \ Label f ( z j , G f ( d )). It remains to show that the shortest s - x i path (based onedge weights) in G \ Label f ( z j , G f ( d )) goes through z j . By Lemma 32(2,3), the only z k vertices in Leaf ( G f ( d )) that are connected to s in G \ Label f ( z j , G f ( d )) are { z , . . . , z j } . Since W ( z , x i ) >W ( z , x i ) > . . . > W ( z j , x i ), we have that ( z j , x i ) is the last edge of π ( s, x i | Label f ( z j , G f ( d ))).As this holds for every x i ∈ X and every z j ∈ Leaf ( G f ( d )), the claim follows. Extension to multiple sources.
Given a parameter σ representing the number of sources,the lower bound graph G includes σ copies, G (cid:48) , . . . , G (cid:48) σ , of G f ( d ), where d = O ( (cid:112) ( n/ f σ )). ByObs. 1, each copy consists of at most n/ σ vertices. Let y i be the vertex u fd and s i = r ( G (cid:48) i )in the i th copy G (cid:48) i . Add a vertex v ∗ connected to a set X of O ( n ) vertices and connect v ∗ to each of the vertices y i , for i ∈ { , . . . , σ } . Finally, connect the set X to the σ leaf sets Leaf ( G (cid:48) ) , . . . , Leaf ( G (cid:48) σ ) by a complete bipartite graph B (cid:48) , adjusting the size of the set X in theconstruction so that | V ( G ) | = n . Since nLeaf ( G (cid:48) i ) = Ω(( n/ ( f σ )) − / f ) (see Obs. 1), overall | E ( G ) | = Ω( n · σ · nLeaf ( G f ( d ))) = Ω( σ / f · ( n/f ) − / f ). The weights of all graph edges not in B (cid:48) are set to 1. For every i ∈ { , . . . , σ } , the edge weights of the bipartite graph B j = ( Leaf ( G (cid:48) ) , X )are set in the same manner as for the single source case. Since the path from each source s i to X cannot aid the vertices of G (cid:48) j for j (cid:54) = i , the analysis of the single-source case can be applied to showthat each of the bipartite graph edges in necessary upon a certain sequence of at most f -edge faults.This completes the proof of Thm. 19. 18 𝑧 𝑧 𝑧 𝑧 𝑢 𝑢 𝑢 𝑢 𝑃 𝑄 𝑄 𝑄 𝑄 𝐺 𝑑 𝐺 ′ 𝑢 𝑢 𝑢 𝑑1 𝑃 𝑓 𝑄 𝑄 𝑑 𝑓 𝐺 ′ 𝑄 𝐺 𝑑′ … … 𝐺 𝑓 𝑑 Figure 1:
Illustration of the graphs G ( d ) and G f ( d ) . Each graph G (cid:48) i is a graph of the form G f ( √ d ) . 𝑛 𝑠 … 𝑛 𝑍 = 𝑛
𝑋 = 𝑛 𝐺 (𝑉, 𝐸, 𝑊) 𝐵 Figure 2:
Illustration of the lower bound graph G ∗ f ( V, E, W ) for f = 2 . The edge weights of thebipartite graph are monotone increasing as a function of the leaf index from left to right.. The edge weights of thebipartite graph are monotone increasing as a function of the leaf index from left to right.