Partially Optimal Edge Fault-Tolerant Spanners
aa r X i v : . [ c s . D S ] F e b Partially Optimal Edge Fault-Tolerant Spanners
Greg BodwinUniversity of Michigan [email protected]
Michael Dinitz ∗ Johns Hopkins University [email protected]
Caleb RobelleUMBC [email protected]
February 24, 2021
Abstract
Recent work has established that, for every positive integer k , every n -node graph has a(2 k − O ( f − /k n /k ) edges that is resilient to f edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood:the best known lower bound for general k is Ω( f − k n /k + f n ). Our main result is to nearlyclose this gap with an improved upper bound, thus separating the cases of edge and vertexfaults. For odd k , our new upper bound is O k ( f − k n /k + f n ), which is tight up to hiddenpoly( k ) factors. For even k , our new upper bound is O k ( f / n /k + f n ), which leaves a gap ofpoly( k ) f / (2 k ) . Our proof is an analysis of the fault-tolerant greedy algorithm, which requiresexponential time, but we also show that there is a polynomial-time algorithm which creates edgefault tolerant spanners that are larger only by factors of k . ∗ Supported in part by NSF award CCF-1909111.
Introduction
Let G = ( V, E ) be a graph, possibly with edge lengths w : E → R ≥ . A t -spanner of G , for t ≥ H = ( V, E ′ ) that preserves all pairwise distances within a factor of t , i.e., d H ( u, v ) ≤ t · d G ( u, v )for all u, v ∈ V (where d X denotes the shortest-path distance in a graph X ). Since H is a subgraphof G it is also true that d G ( u, v ) ≤ d H ( u, v ), and so distances in H are the same as in G up toa factor of t . The distance preservation factor t is called the stretch of the spanner. Spannerswere introduced by Peleg and Ullman [22] and Peleg and Sch¨affer [21], and have a wide rangeof applications in routing [23], synchronizers [3], broadcasting [2, 20], distance oracles [24], graphsparsifiers [15], preconditioning of linear systems [12], etc.The most common objective in spanners research is to achieve the best possible existentialsize-stretch trade-off. Most notably, a landmark result of Alth¨ofer et al. [1] analyzed the followingsimple and natural greedy algorithm: given an n -node graph G and an integer k ≥
1, consider theedges of G in non-decreasing order of their weight and add an edge ( u, v ) to the current spanner H ifand only if d H ( u, v ) > (2 k − w ( u, v ). They proved that this algorithm produces (2 k − existentially optimal size . In particular, the spanner produced has size O ( n /k ), and assumingthe well-known Erd˝os girth conjecture [13] there are graphs in which every (2 k −
1) spanner (andin fact every 2 k -spanner) has at least Ω( n /k ) edges.A crucial aspect of real-life systems that is not captured by the standard notion of spannersis the possibility of failure . If some edges (e.g., communication links) or vertices (e.g., computerprocessors) fail, what remains of the spanner might not still approximate the distances in whatremains of the original graph. This motivates the notion of fault tolerant spanners: Definition 1 (Fault Tolerant Spanners) . A subgraph H is an f -edge fault tolerant ( f -EFT) t -spanner of G = ( V, E ) if d H \ F ( u, v ) ≤ t · d G \ F ( u, v )for all u, v ∈ V and F ⊆ E with | F | ≤ f .In other words, an f -EFT spanner contains a spanner of G \ F for every set of | F | ≤ f edgesthat could fail. The definition for vertex fault tolerance (VFT) is equivalent, with the only changebeing that F ⊆ V \ { u, v } .Fault tolerant spanners were originally introduced in the setting of geometric graphs by Lev-copoulos, Narasimhan, and Smid [16] and have since been studied extensively in that setting [9,17–19]. Chechik, Langberg, Peleg and Roditty [8] were the first to study fault-tolerant spanners in gen-eral graphs, giving a construction of an f -VFT (2 k − O ( f k f +1 · n /k log − /k n )and an f -EFT (2 k − O ( f · n /k ). So they showed that introducing tolerance to f edge faults costs us an extra factor of f in the size of the spanner, while introducing tolerance to f vertex faults costs us a factor of roughly f k f +1 in the size (compared to the size of a non-faulttolerant spanner of the same stretch). Dinitz and Krauthgamer [10] later improved the extra factorpaid in the VFT setting to about f . At this point, it appeared that the EFT setting might besubstantially easier than the VFT setting, in the sense that it allowed for a smaller dependenceon f in spanner size. However, a recent series of papers has developed a set of techniques thatapply equally well to both settings, yielding the same improved bounds for each [5–7, 11]. This hasculminated in the following theorem: Theorem 1.1 (FT upper bounds [6]) . There is a polynomial time algorithm that, given any positiveintegers f, k and any n -node graph G , constructs an f -EFT or VFT (2 k − -spanner of G with O ( f − /k n /k ) edges. panner size Tight? Polytime? Citation O (cid:0) f · n /k (cid:1) X [8] O (cid:0) exp( k ) f − /k · n /k (cid:1) k = 2 only [5] O (cid:0) f − /k · n /k (cid:1) k = 2 only [7] O (cid:0) kf − /k · n /k (cid:1) k = 2 only X [11] O (cid:0) f − /k · n /k (cid:1) k = 2 only X [6] O (cid:0) k f / · n /k + kf n (cid:1) for even k k = 2 only ( ∗ ) this paper O (cid:0) k f / − / (2 k ) · n /k + kf n (cid:1) for odd k all fixed odd k this paper O (cid:0) k / f / · n /k + k f n (cid:1) for even k k = 2 only ( ∗ ) X this paper O (cid:0) k / f / − / (2 k ) · n /k + k f n (cid:1) for odd k all fixed odd k X this paper Table 1: Prior work on the size of f -EFT 2 k − f / − / (2 k ) n /k ), except for k = 2 where thelower bound improves to Ω( f / n /k ). Except for k ∈ { , , } , and for k = 7 and large enough f (see Theorem B.1 to follow), this lower bound and our claims of tightness are conditional on thegirth conjecture [13]. The ( ∗ ) indicates that we conjecture our bound for even k to be tight for allfixed even k .This upper bound on spanner size is notable because it fully matches the known lower boundfor VFT spanners: Theorem 1.2 (VFT lower bounds [5]) . Assuming the girth conjecture [13], for any positive integers n, f, k , there exist n -node graphs in which every f -VFT (2 k − -spanner has Ω( f − /k n /k ) edges. Thus the VFT setting is now essentially completely understood. However, there are still signif-icant gaps in the EFT setting. While the VFT lower bound from [5] also holds for the EFT settingwhen k = 2, for general k it drops off considerably. (We note that the f n term in the followingtheorem statement is not explicitly given in [5], but is a simple folklore result stating that any f -regular graph is the unique f -EFT spanner of itself): Theorem 1.3 (EFT lower bounds [5]) . Assuming the girth conjecture [13], for any positive integers n, f, k , there exist n -node graphs in which every f -EFT (2 k − -spanner H with | E ( H ) | = ( Ω (cid:0) f / n / (cid:1) k = 2Ω (cid:0) f / − / (2 k ) n /k + f n (cid:1) k > f ) gap in the EFT setting when k >
2. So the main openquestion left is how to close this gap: how much extra do we have to pay to achieve edge faulttolerance compared to a non-fault tolerant spanner? Is the right answer f − k , or f − /k , orsomewhere in between? In this paper we nearly resolve this question by improving the upper bound:2 heorem 1.4 (Main Result) . For any positive integers n, f, k , every n -node graph has an f -EFT (2 k − -spanner H with | E ( H ) | = ( O (cid:0) k f / − / (2 k ) n /k + kf n (cid:1) k is odd O (cid:0) k f / n /k + kf n (cid:1) k is even. Hence Theorem 1.4 entirely resolves the question of edge fault tolerance for constant odd k ,and is off from the lower bound (Theorem 1.3) by only quadratic factors of k for nonconstant odd k , and only k f / (2 k ) for even k . See Table 1 for the full context of our results, and see Figure 1 inAppendix A for a visualization.While it is interesting and important to optimize the dependence on k (and close the gapbetween our new upper bounds and the known lower bounds), we remark that spanners are neverused with k larger than O (log n ), since the additional stretch no longer yields additional sparsity.Hence the factor of k in our bounds is at most polylogarithmic in n . On the other hand, the faultparameter f can be significantly larger (for example, polynomial in n ). Hence the more centralquestion in the area is to optimize dependence on f , and we are more concerned with our gap of f / (2 k ) for even k than our gap of k . However, we conjecture that our upper bound is actuallycorrect, and that our current construction and analysis are tight with respect to f : Conjecture 1.5.
For any positive integers n, f and even integer k , there exist n -node graphs inwhich every f -EFT (2 k − -spanner has Ω (cid:0) f / n /k (cid:1) edges. Settling this conjecture, or generally making progress on the gap between the f / dependencein this work and the f − k lower bound on f -dependence from [5], is the main open question leftby this work. We remark that there is some precedent in the literature for fundamentally differentbehavior for (2 k − k . This occurs, for example, in the unbalanced Moorebounds that control the maximum possible density of a bipartite graph with a very different numberof nodes on its two sides [14]. We believe that a similar even/odd effect may occur here, where theimbalance is essentially caused by the structure of edge faults. The fact that the dependence f / is known to be tight for k = 2 partially validates this intuition, and it may be “simpler” for thecorrect bounds to distinguish between even/odd k rather than taking k = 2 as a special case. Algorithmic Efficiency.
The spanner construction algorithm that we use to prove Theorem 1.4is the same greedy algorithm as in [5, 7] (adapted for edge fault tolerance), which requires expo-nential time. However, by combining our new analysis with the ideas used by [11], we can obtainpolynomial time at the price of a slightly worse dependence on k : Theorem 1.6.
There is a polynomial time algorithm that, given positive integers f, k and an n -nodeinput graph, outputs an f -EFT (2 k − -spanner H with | E ( H ) | = ( O (cid:0) k / f / − / (2 k ) n /k + k f n (cid:1) k is odd O (cid:0) k / f / n /k + k f n (cid:1) k is even. Before diving into the details, we first give some intuition and context for our techniques.3 ackground on Non-Faulty Spanners.
In the non-faulty setting, the analysis of the greedyconstruction by Alth¨ofer et al. [1] passes through the girth of the output spanner (recall that thegirth of a graph is the smallest number of edges in a cycle). One proves that the output spanneralways has girth > k , and conversely, any unweighted graph of girth > k has no (2 k − γ ( n, k ) for the maximum number ofedges in any n -node graph of girth > k . Then the greedy algorithm outputs spanners on ≤ γ ( n, k )edges, and this bound is best possible.The only known argument to upper bound the value of γ ( n, k ) is the Moore bounds , which provethat γ ( n, k ) = O ( n /k ) (the girth conjecture [13] is that this is tight, i.e., γ ( n, k ) = Ω( n /k )).These use a counting argument over the simple k -paths of the input graph, where a “simple” pathis one that does not repeat nodes. The Moore bounds are proved in two steps. First, one provesa counting lemma : any n -node graph of average degree d ≫ > k has n · Ω( d ) k totalsimple k -paths. Then, one proves a dispersion lemma : in a graph of girth > k , no two simple k -paths may share endpoints. Together these imply n · Ω( d ) k = O ( n ), and the Moore bound areobtained by rearranging this equation. Background on FT-Spanners.
In the faulty setting, it is natural to consider the “FT-GreedyAlgorithm,” which straightforwardly extends the non-faulty greedy algorithm. That is: considereach edge ( u, v ) in order of nondecreasing weight, and add ( u, v ) to the spanner if and only if thereis a set of | F | ≤ f faults such that d H \ F ( u, v ) > (2 k − · d G \ F ( u, v ). Correctness is again easyto show, but the challenge is to control the number of edges in the output spanner. Informallyspeaking, the output spanner ought to be “close” to a high-girth graph – not in the sense that itsgirth is high (the girth of the output spanner could be 3), but in the sense that the output graphought to be sparse for a similar reason that high-girth graphs must be sparse. There are two basicapproaches to formalizing this intuition:1. The first analysis of the greedy algorithm [5] argued that greedy FT-spanners are similar tohigh-girth graphs in the sense that they are amenable to a generalized version of the Moorebound analysis. This led to a rather complicated analysis that took on exp( k ) factors, butwhich gave optimal dependence on n and f for the size of VFT spanners, assuming the girthconjecture. Indeed, arguments of this type can only hope to achieve conditional optimalityon the girth conjecture, since the Moore bounds themselves are only optimal if the girthconjecture holds.2. The following analyses [6,7,11] took an alternate view that greedy FT-spanners are structurallysimilar to high-girth graphs, in the sense that one can change the spanners into high-girthgraphs by sacrificing only a small amount of their density. This approach has a number ofbenefits over the generalized Moore argument: it is far simpler, it removes all extra factorsof k in the spanner size, and it lets one “black-box” the Moore bounds. That is, it givesbounds that depend directly on γ ( n, k ) rather than the Moore bounds for this function, thusestablishing optimal spanner size in the VFT setting even if the girth conjecture fails. Sothe message of these papers seemed to be that the structural approach, which only uses theMoore bounds as a black-box to plug into γ ( n, k ), dominates the approach that opens theblack box of the Moore bounds in every important way.Since a gap remained for EFT spanners, it was still technically unresolved whether Moore orstructural arguments are to be preferred in this arena. This point was discussed in [7], whichcontained a technical barrier suggesting that it would be difficult to push the EFT upper bounds4ny further using the latter structural approach. Specifically, they suggested the following objectsfor EFT structural analysis: Definition 2 (Edge Blocking Sets [7]) . Given a graph G = ( V, E ), a t -edge blocking set for G is aset B ⊆ (cid:0) E (cid:1) such that, for every cycle C ∈ G with at most t edges, there exists ( e , e ) ∈ B suchthat e , e ∈ C .One proves the previous-best upper bounds on FT-greedy spanners by (1) observing that theoutput spanners H from the EFT-greedy algorithm have 2 k -edge blocking sets of size | B | ≤ f | E ( H ) | , and then (2) proving that any graph with a 2 k -edge blocking set of size | B | ≤ f | E ( H ) | can only have O ( f − /k n /k ) edges. The second lemma is the key to the structural approach,which appeared again in some form in followup work [6, 11]. However, it was further proved in [7]that this second lemma is tight: there exist graphs that have a blocking set of size | B | ≤ f | E ( H ) | and Ω( f − /k n /k ) edges. So, if better EFT upper bounds are to be achieved, they at leastneed to use a different object from blocking sets, and perhaps depart from this style of argumentaltogether. We show, perhaps surprisingly, that the original Moore-based approach seems to dominate thestructural one in the context of EFT spanners. Our analysis is best viewed as a generalization ofthe Moore bounds, and moreover, as an auxiliary result we strengthen the evidence from [7] thatstructural arguments cannot possibly achieve the improved EFT upper bounds obtained in thispaper.First, to sidestep the barrier from [7], we need to change the focus of our analysis. We use strong blocking sets , a natural extension that explicitly takes edge weights into account. While theoriginal definition of blocking sets is just about edges in cycles, we now require one of the edges tobe the heaviest edge. More formally:
Definition 3 (Strong Blocking Sets) . A strong t -blocking set is some B ⊆ (cid:0) E (cid:1) such that, for anycycle C on at most t edges, there is ( e , e ) ∈ B with e , e ∈ C and such that one of e , e is theheaviest edge in C . The elements ( e , e ) ∈ B are called blocks .(We assume that ties between edge weights are broken in some canonical way, and so “theheaviest edge” in C is unambiguous.) It is again easy to prove that the FT-Greedy algorithmproduces spanners that have strong blocking sets of size ≤ f | E ( H ) | . Since the girth of a graphhas nothing to do with its edge weights, we can view this definition as a step away from a purelystructural approach, and towards an approach that cares about the specific interaction between theinput graph and the FT-greedy algorithm (which considers edges in increasing weight order).The main new technical idea goes into our generalized dispersion lemma. The goal is to boundthe total number of simple k -paths in the graph. The core of the argument is the following lemma:up to some technical details that we will not discuss here (see Section 4), due to the existence ofa small strong blocking set, it is nearly true that for any two nodes s, t in the output spanner,there is a set of O ( kf ) edges such that every simple s t k -path uses one of these edges as its heaviest edge. We may then bound the number of simple s t k -paths by considering each edge( u, v ) in this set and recursively bounding the number of s u j -paths and the number of v t ( k − j − j ).This recursion turns out to give the strongest bounds when 1 ≤ j ≤ k −
1; that is, ( u, v ) isa middle edge along a path, not the first or last one. This motivates a narrowing of focus in ourcounting argument from all simple k -paths to only those that are middle-heavy (have their heaviest5dge in a middle position). In fact, we need our paths to stay middle-heavy over the recursion, asthey are repeatedly split over their heaviest edge. So we focus even more narrowly on a class of alternating paths , for which each even-numbered edge is heavier than the two odd-numbered edgeson either side, which have this favorable property. This induces some additional technical work inthe counting lemma: standard arguments guarantee the existence of many simple k -paths, but weneed to rework these arguments to show that a good fraction of these simple k -paths are indeedalternating. Lower Bounds on Structural Analysis.
Although strong blocking sets move away from struc-tural analyses a bit, it is still reasonable to ask whether our argument can be replaced with a struc-tural one somehow. Next, we explain our new evidence that the answer to this question is probably“no”: we show that a structural analysis would by necessity involve solving a long-standing openproblem in extremal graph theory. As mentioned earlier, a key advantage of structural analyses isthat they give spanner upper bounds directly in terms of the function γ . For example, [7] showsthat all graphs have EFT or VFT (2 k −
1) spanners on O ( f · γ ( n/f, k )) edges. The Moorebounds are applied as a secondary step to bound γ , giving the upper bound of O ( f − /k n /k )from Theorem 1.1. But if the girth conjecture fails, and there is an even better upper bound for γ than the Moore bounds, then the upper bounds from these arguments automatically improveas well. Conversely, if one can prove unconditionally that f -VFT spanners need Ω( f − /k n /k )edges, this would imply a lower bound on γ that would be enough to prove the girth conjecture,which is currently unknown and is a famous and longstanding open problem.Let us now entertain the hypothetical that one could prove our upper bounds for EFT spannerswith a structural argument, arguing that greedy EFT-spanners can be reduced to high-girth graphswhile sacrificing only a small amount of density. For example, analogous to [7], let us imagine thatwe have a bound of the form O k ( f · γ ( n/ √ f , k ) + nf ) edges for odd k , which would in turn implyour upper bound of O k ( f / − / (2 k ) n /k + nf ) for odd k by applying the Moore bounds to γ . Thenif one could prove unconditionally that f -EFT spanners need Ω( f / − / (2 k ) n /k + nf ) edges forfixed odd k , it would imply a lower bound on γ that would be enough to prove the girth conjecturefor fixed odd k , which would again be a major breakthrough.In fact, we can prove a new unconditional bound of this type – not for all fixed odd k , butspecifically for k = 7. A proof of the following theorem appears in Appendix B. Theorem 2.1.
For k = 7 and f = Ω( n / ) , the lower bound on EFT spanners in Theorem 1.3holds unconditionally. Thus, a structural argument as explained above would be enough to prove the girth conjecturefor k = 7, which is currently unknown. Since no new setting of the girth conjecture has been settledin over 50 years, this is likely very hard, and it would still constitute a breakthrough in the area. We begin our technical work by proving that the greedy FT-spanner algorithm analyzed in [5, 7]gives spanners with small strong blocking sets.This algorithm implicitly breaks ties between edges of equal weight, to consider one before theother in the main for-loop. This tiebreaking is arbitrary, but in the sequel it will be convenient tounambiguously refer to the heaviest edge among a set of edges, which means the heaviest edge underthis tiebreaking (equivalently, the edge considered last by the greedy algorithm). In particular, in6 lgorithm 1
Greedy f -EFT (2 k − function FT-GREEDY( G = ( V, E, w ) , k, f ) H ← ( V, ∅ , w ) for all ( u, v ) ∈ E in nondecreasing weight order doif there exists a set F of at most f edges such that d H \ F ( u, v ) > (2 k − w ( u, v ) then add ( u, v ) to H return Hthe definition of strong blocking sets, it is required that for each cycle C , there is a block ( e , e )with e , e ∈ C and such that either e , e is the heaviest edge in C under this tiebreaking .The following lemma is a straightforward adaptation of Lemma 3 of [7], where the same state-ment was proved for blocking sets rather than strong blocking sets. Lemma 3.1.
Any graph H returned by Algorithm 1 is an f -EFT (2 k − -spanner that has a strong k -blocking set of size at most f | E ( H ) | .Proof. See Appendix C.One major downside of this algorithm is its running time. In each iteration of this algorithm,the algorithm needs to decide the following question: given an edge ( u, v ), is there a fault set F ⊆ E where | F | ≤ f and d H \ F ( u, v ) ≤ (2 k − · d G \ F ( u, v )? Doing this in the obvious way (checking allsets of f or fewer edges) takes time exponential in f . And, unfortunately, it turns out that thisquestion is equivalent to the Length Bounded Cut problem (LBC), which is known to be NP-hard[4]. In order to develop a polynomial time algorithm for constructing FT-spanners, [11] replacedthis exponential time subroutine with one that ran in polynomial time. They designed a simple(2 k − unweighted setting which is essentially the standardfrequency approximation of Set Cover. In particular, when used for the decision variant it has theproperty that if there is a fault set of size at most f that intersects all u − v paths with at most2 k − k − f then it will return NO (if neither is true then the algorithm can return either answer).We call this algorithm A ( G, u, v, f, k )A key insight is of [11] was that, even though A is designed for the unweighted setting, it canstill be used in the weighted setting. In particular, if we use the weights to set the initial orderingof edges (as in Algorithm 1) but from then on pretend that the graph is unweighted, this modifiedgreedy algorithm still returns a valid f -EFT (2 k − Algorithm 2
Modified Greedy EFT Spanner Algorithm function
FT-GREEDY( G = ( V, E ) , k, f ) H ← ( V, ∅ ) for all ( u, v ) ∈ E in nondecreasing weight order doif A ( H, u, v, f, k ) returns YES then
Add ( u, v ) to H return HIn order to analyze the sparsity of spanners constructed by this algorithm, [11] showed thatthe blocking set analysis of [7] can be applied with an extra loss of O ( k ) in the size (due to theapproximation). We show the same, but for our new notion of strong blocking sets.7 emma 3.2. The graph H returned by Algorithm 2 is an f -EFT (2 k − -spanner that has a strong k -blocking set of size at most O ( kf ) · | E ( H ) | .Proof. See Appendix C
In this section, we will prove:
Theorem 4.1.
Let f be a positive integer, let k be a positive integer, and let H be an n -node graphthat has a strong k -blocking set B of size | B | ≤ | E | f . Then | E ( H ) | = ( O (cid:0) k f / − / (2 k ) n /k + nf k (cid:1) k is odd O (cid:0) k f / n /k + nf k (cid:1) k is even . This theorem together with Lemma 3.1 directly implies Theorem 1.4. For the polynomial-time version, note that Lemma 3.2 gives us a polytime algorithm with | B | = O ( kf | E ( H ) | ), so wecan reparameterize Theorem 4.1 by setting the f in the theorem equal to O ( kf ). This impliesTheorem 1.6. Our proof will be a counting argument over a special type of path in the graph H : Definition 4 (Alternating Paths) . An alternating k -path in H is a path π with edge sequence( e , e , . . . , e k ) with the following property: every even-numbered edge is heavier than the two odd-numbered edges adjacent to it. (When k is even, the last edge e k is only required to be heavierthan e k − .)More specifically, we will count simple unblocked alternating paths. As usual, a simple path isone that does not repeat nodes. An unblocked path is defined as follows: Definition 5 (Blocked Paths) . A path π is blocked by a strong blocking set B if there is ( e , e ) ∈ B with e , e ∈ π . Otherwise, π is unblocked . (Note: unlike for cycles, π is blocked even if neither e nor e is the heaviest edge in π .)In the remainder of this section we follow the outline discussed in Section 2, giving a countinglemma and a dispersion lemma for simple unblocked alternating k -paths. But we first start bymaking some useful simplifications, which are all without loss of generality:1. We will use the stronger hypothesis that in the strong blocking set B , each edge appearsin ≤ f blocks. This is without loss of generality for the following reason. Since every blockcontains two edges, the average edge participates in ≤ f blocks, and hence a simple countingargument (or Markov’s inequality) implies that at most half of the edges participate in atleast 4 f blocks. Suppose that, while there is an edge that participates in ≥ f blocks, wedelete that edge from the graph and the corresponding blocks from the blocking set. We thusdelete at most half of the graph edges in this way, and we arrive at a graph with a blockingset in which every block participates in < f blocks. We can then reparametrize f ← f while changing our claimed upper bound on spanner size by only a constant factor.8. We assume that H has average degree d ≥ f k . In the regime where this does not hold, H has nd/ O ( nf k ) total edges and so the claimed bounds hold already due to the trailingadditive term.3. Additionally, letting d be the average degree in H , we will assume that the maximum degreeis O ( d ). This is justified by the following operation: while there is a node v of degree ≥ d ,split it into two nodes v , v , equitably partitioning the edges that use v between its twocopies. Notice that this splitting operation does not create cycles or edges (though it maydestroy some cycles), and thus B is still a strong blocking set for the modified graph. A quickcounting argument, standard in prior work (for example, [5]), shows that one introduces only O ( n ) new nodes in this way, and hence this again changes our claimed upper bounds onspanner size only by a constant factor.4. We assume that all edges in H have distinct weights, so that we may unambiguously refer to the heaviest or lightest edge among a set of edges. This is justified by simply breaking weightties in some arbitrary but consistent way, as in our discussion of the FT-greedy algorithm. First, we prove a counting lemma for alternating paths. We will start with some weaker versions ofthe counting lemma we need, and then gradually bootstrap them into a full one. In the followinglemma statements, an edge-simple alternating path is an alternating path that does not repeatedges (but which might repeat nodes).
Lemma 4.2 (Weak Counting Lemma) . For any positive integer k , any n -node graph Γ with ≥ kn edges has an edge-simple alternating k -paths.Proof. The proof is by induction on k . For the base case, when k = 0, any individual node may beviewed as an edge-simple alternating 0-path.Now we prove the inductive step. We will first assume that k is odd. Assume without loss ofgenerality that H has no isolated nodes, and preprocess Γ by removing the lightest edge incidentto each node. We remove at most n edges in this way, so at least ( k − n edges remain. By theinductive hypothesis, there exists an edge-simple alternating k − π in the surviving graph.Let ( u, v ) be the last edge of π . Since k − π into an alternating k -pathby appending any edge incident to v that is lighter than ( u, v ). Letting ( v, x ) be the edge removedin the preprocessing, we know that ( v, x ) is lighter than ( u, v ), since it was selected to be removedinstead of ( u, v ). Thus we can extend π into an alternating k -path by appending ( v, x ), and since π was generated in a graph that did not contain ( v, x ), it remains edge-simple under this extension.The case where k is even is similar, except that we remove the heaviest edge incident to eachnode in the preprocessing.We next bootstrap into a stronger counting lemma over the same kinds of paths: Lemma 4.3 (Intermediate Counting Lemma) . For any positive integer k , any n -node graph Γ with ≥ kn edges has ≥ kn total edge-simple alternating k -paths.Proof. Consider the following process: pick an edge that is contained in at least one edge-simplealternating k -path, remove it, and repeat until there are no more edge-simple alternating k -paths.Since we remove the edge we find in each iteration, there are at least as many edge-simple alternating k -paths as there are iterations. And there are at least kn iterations, since as long as ≥ kn edgesremain in the graph, Lemma 4.2 implies that there will still be an edge-simple alternating k -path.Hence there are at least kn edge-simple alternating k -paths in total.9ow we bootstrap this once again into the counting lemma we will actually use in our mainargument. While the previous two lemmas hold for arbitrary graphs Γ, the following one holdsspecifically for the graph H under consideration, which has a small strong blocking set. Lemma 4.4 (Full Counting Lemma) . The graph H has n · Ω( d/k ) k total unblocked simple alter-nating k -paths, where d is its average degree.Proof. In this proof, we will actually count the unblocked edge-simple alternating k -paths in H .We notice that, if an edge-simple k -path repeats a node, then it must have a cycle of length ≤ k as a subpath, and this cycle must be blocked by B . Thus every path π that is edge-simple, butnot simple, is already blocked, and hence the number of unblocked edge-simple alternating k -pathsis equal to the number of unblocked simple alternating k -paths. In the following, let α ( H ) denotethe number of unblocked (edge-)simple alternating k -paths in H (and we use similar notation forother graphs that will arise in the argument).Let H ′ be a random subgraph of H obtained by including each edge with probability 8( d/k ) − .So we have E (cid:2) | E ( H ′ ) | (cid:3) = | E ( H ) | · d/k ) − = nd · d/k ) − = 4 nk. Since each edge is sampled independently, a standard Chernoff bound implies that with highprobability | E ( H ′ ) | ≥ (15 / nk .Let B ′ ⊆ B be the strong blocking set for H ′ obtained by keeping the blocks ( e , e ) ∈ B whereboth e , e survive in H ′ . We have E (cid:2) | B ′ | (cid:3) = | B | · d/k ) − ≤ | E ( H ) | f · d/k ) − = nd · f · d/k ) − = 32 nf k /d ≤ nk where the last inequality uses our assumption that d ≥ f k . Hence another standard Chernoffbound implies that | B ′ | ≤ (5 / nk with high probability. Now, build a subgraph H ′′ from H ′ byconsidering each block ( e , e ) ∈ B ′ and deleting e . We thus have that with high probability, | E ( H ′′ ) = | E ( H ′ ) | − | B ′ | ≥ (15 / nk − (5 / nk = (5 / nk. By Lemma 4.3, a graph on ≥ nk edges has ≥ kn total edge-simple alternating k -paths.By construction no k -paths in H ′′ are blocked (since we have removed at least one edge fromevery block originally in B ). Thus, we have that α ( H ′′ ) ≥ kn with high probability, and hence E [ α ( H ′′ )] = Ω( kn ). Since H ′′ ⊆ H , this implies that E [ α ( H ′ )] = Ω( kn ).Let us now compute E [ α ( H ′ )] in a different way. Recall that H ′ was obtained from H byincluding each edge with probability O ( d/k ) − . So each individual edge-simple k -path from H survives in H ′ with probability O ( d/k ) − k . So we have E [ α ( H ′ )] = α ( H ) · O ( d/k ) − k . Putting together these two bounds on E [ α ( H ′ )], we get α ( H ) = Ω( kn ) · Ω( d/k ) k = n · Ω( d/k ) k , as claimed. 10 .3 Dispersion Lemma for Alternating Paths In this part, we bound the maximum number of alternating paths that can go between two nodes.First we prove a useful intermediate lemma which applies to all unblocked k -paths (not just alter-nating): Lemma 4.5.
For any nodes s, t , there is a set F s,t containing | F s,t | ≤ kf + 1 edges such that everysimple unblocked s t path of length ≤ k in H uses an edge in F s,t as its heaviest edge.Proof. We build F s,t iteratively as follows. Initially F s,t := ∅ . In each round, pick the heaviest edgecontained in any simple unblocked s t path of length ≤ k whose heaviest edge is not already in F s,t , and add this edge to F s,t . Halt once no such path exists.To get the size bound on F s,t , suppose for contradiction that after kf + 1 rounds there remainsa simple unblocked s t path π of length ≤ k whose heaviest edge is not already in F s,t . For each f ∈ F s,t , by construction there is a simple unblocked s t path π ′ of length ≤ k that uses f as itsheaviest edge, and f is heavier than any edge in π . Thus π ∪ π ′ contains a cycle C of length ≤ k ,which has f as its heaviest edge. This cycle C must be blocked by B , so there is ( f, e ) ∈ B with f, e ∈ C . Since the path π ′ is unblocked, and f ∈ π ′ , it must be that e ∈ π .But there are ≤ k edges in π , and there are kf + 1 edges in F s,t , so by the pigeonhole principleone edge in π appears in ≥ f + 1 blocks. This contradicts our initial assumption from Section 4.1that each edge appears in ≤ f total blocks in B . Lemma 4.6 (Dispersion Lemma) . For any nodes s, t and positive integer j ≤ k , the number ofsimple unblocked alternating s t paths of length j in H is ( O (cid:0) k f (cid:1) ( j − / j is odd O (cid:0) k f (cid:1) j/ j is evenProof. The proof is by strong induction over j . The base cases are when j = 0, in which case thereis at most one 0-path (when s = t ), and also when j = 1, in which case only a single ( s, t ) edge canbe a 1-path.For the inductive step, we apply Lemma 4.5 to find a set of | F s,t | = O ( kf ) edges for which everysimple unblocked alternating s t path of length k has its heaviest edge in F s,t . The alternating s t paths may then be partitioned into O ( k f ) equivalence classes, by (1) which edge in F s,t istheir heaviest, and (2) in which of the j ≤ k possible positions in the path the heaviest edge occurs.We now bound the size of each individual equivalence class using the inductive hypothesis.Consider the equivalence class of simple unblocked alternating paths whose heaviest edge is ( u, v ) ∈ F s,t , and which use ( u, v ) in the i th position. Notice that each such path may be viewed as theconcatenation of a simple unblocked s u alternating path of length i −
1, and then the edge( u, v ), and then a simple unblocked v t alternating path of length j − i . So the number of suchpaths may be upper bounded by the product of the number of simple unblocked s u alternatingpaths of length i − v t paths of length j − i .We have that i is always even, since the heaviest edge in any alternating path is even-numbered.So i − s u simple unblockedalternating paths of length i − O ( k f ) ( i − / . The parity j − i depends on the parity of j , sowe need to split into two cases: • if j is even, and so j − i is even, by the strong inductive hypothesis the number of simpleunblocked alternating v t paths is O ( k f ) ( j − i ) / . In this case the product is bounded by O ( k f ) ( i − / · O ( k f ) ( j − i ) / = O (cid:0) k f (cid:1) ( j − / . O ( k f ) total equivalence classes, the total number of simple unblocked alter-nating s t j -paths is O ( k f ) · O (cid:0) k f (cid:1) ( j − / = O (cid:0) k f (cid:1) j/ , as claimed. • if j is odd, and so j − i is odd, by the strong inductive hypothesis the number of simpleunblocked alternating v t paths is O ( k f ) ( j − i − / . In this case the product is bounded by O ( k f ) ( i − / · O ( k f ) ( j − i − / = O (cid:0) k f (cid:1) ( j − / . Since there are O ( k f ) total equivalence classes, the total number of simple unblocked alter-nating s t j -paths is O ( k f ) · O (cid:0) k f (cid:1) ( j − / = O (cid:0) k f (cid:1) ( j − / , also as claimed.We have stated this dispersion lemma for general j , for the sake of its inductive proof, but wewill only need to use the setting j = k in the following. Now we prove Theorem 4.1. Let d be the average degree of H . By our counting lemma, H hasat least n · Ω( d/k ) k total simple unblocked alternating k -paths. To apply our dispersion lemma,let us first consider the case where k is odd. Then there are O (cid:0) k f (cid:1) ( k − / total simple unblockedalternating k -paths between any given pair of nodes in H , for an upper bound of n · O (cid:0) k f (cid:1) ( k − / on the total number of simple unblocked alternating k -paths in the entire graph. We can put thisupper and lower bound together and rearrange as follows: n · Ω( d/k ) k = n · O (cid:0) k f (cid:1) ( k − / = ⇒ d/k = n /k O (cid:0) k f (cid:1) / − / (2 k ) = ⇒ | E ( H ) | = nd O (cid:16) k n /k f / − / (2 k ) (cid:17) . In the case where k is even, we proceed similarly: the total number of simple unblocked al-ternating k -paths in the entire graph is at most n · O (cid:0) k f (cid:1) k/ , and so we put this upper boundtogether with the same lower bound as before to get: n · Ω( d/k ) k = n · O (cid:0) k f (cid:1) k/ = ⇒ d/k = n /k O (cid:0) k f (cid:1) / = ⇒ | E ( H ) | = nd O (cid:16) k n /k f / (cid:17) . The additional additive + nf k term in Theorem 4.1 is needed to justify the assumption that d = Ω( f k ), which was used to prove our counting lemma.12 eferences [1] Ingo Alth¨ofer, Gautam Das, David P. Dobkin, Deborah Joseph, and Jos´e Soares. On sparsespanners of weighted graphs. Discrete & Computational Geometry , 9:81–100, 1993.[2] Baruch Awerbuch, Alan Baratz, and David Peleg. Efficient broadcast and light-weight span-ners.
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A FiguresB Unconditional Lower Bounds for EFT Spanners
In this section, we point out an extension to the lower bound of [5] for EFT spanners:
Theorem B.1.
For k = 7 and f = Ω( n / ) , there is an infinite family of n -node graphs for whichany f -EFT (2 k − -spanner has Ω (cid:0) f / n / (cid:1) edges. This is the same bound as in Theorem 1.3, but without the need to condition on the girthconjecture despite the fact that the girth conjecture is still unproved for the k = 7 case. Ourconstruction is based on an object in incidence geometry called a Ree-Tits Octagon (see [25] fordiscussion and a relatively simple construction). This is a kind of incidence structure, meaning it isa collection of subsets (called “lines”) of the universe [ n ] (called “points”). For integers k ≥
2, let ussay that a k -gon is a circularly-ordered collection of k pairwise-distinct points ( v , v , . . . , v k = v )such that, for any two adjacent points v i , v i +1 , there exists a line that contains both points. Theproperties are: 14 dependence on f (log f scale) f / f / f / f f : f / for even k and f / − / (2 k ) for odd k . Blue points are where our f dependence is tight, and black/red points represent non-matchingupper/lower bounds. The shaded regions represent knowledge gaps, and the white points are theprevious best f dependence ( f − /k ) that was known before this paper, in all of [5–7, 11]. Theorem B.2 (Ree-Tits Octagon, see e.g. [25]) . For infinitely many values of n , there is anincidence structure consisting of Θ( n / ) lines over the points [ n ] such that: • The structure has no k -gon for any ≤ k ≤ , • Every point is contained in Θ( n / ) lines, and • Every line contains Θ( n / ) points. The structure has some other interesting properties as well, but these are the properties relevantto our proof. We can associate this structure to an incidence graph, which has the followingproperties:
Corollary B.3 (Ree-Tits Octagon Incidence Graph) . For infinitely many values of n , there is abipartite graph with n nodes on one side, Θ( n / ) nodes on the other side, Θ( n / ) total edges,and girth > .Proof. Build the incidence graph associated to the Ree-Tits Octagon. This is a bipartite graphwith n nodes on the left side, corresponding to the points [ n ], and Θ( n / ) nodes on the rightside corresponding to the lines. We put an edge from a point x on the left to a line ℓ on the rightiff x ∈ ℓ . The number of edges is immediate from the properties of the Ree-Tits Octagon, andwe notice that a 2 k -cycle in the incidence graph corresponds to a k -gon in the original incidencestructure. Thus the graph has no cycles of length 4 , , , ,
12, or 14, and since it is bipartite ithas no odd-length cycles either. So its girth is > G = ( V, E ) from Corollary B.3, copy each node v on the smaller sideof the bipartite graph n / · ( f /n / ) / times, and replace each edge ( u, v ) with an edge ( u, v i ) forevery copy v i . We similarly copy each node on the larger side of the bipartite graph ( f /n / ) / times in the same way. (We assume for convenience that these quantities are integers, which affectsour bounds only by lower-order terms.) The number of nodes in the modified graph isΘ n · (cid:18) fn / (cid:19) / ! = Θ (cid:16) n / · f / (cid:17) on the left side, and Θ n / · n / · (cid:18) fn / (cid:19) / ! = Θ (cid:16) n / · f / (cid:17) on the right side as well. The number of edges in the modified graph is | E ( G ) | = Θ (cid:18) n / · n / · fn / (cid:19) = Θ (cid:16) f · n / (cid:17) . Writing N := n / · f / , so that there are Θ( N ) total nodes, we can then reparametrize thenumber of edges as | E ( G ) | = Θ (cid:16) f / · N / (cid:17) . Notice that this is exactly our claimed lower bound. Next, we argue that G is the only f -EFTspanner of itself, and thus no edges can be removed. To see this, let us consider an edge ( u i , v j ),which is a copy of an edge ( u, v ) from the original graph. The total number of copies of ( u, v ) isthe product of the number of copies of u by the number of copies of v , which is n / · (cid:18) fn / (cid:19) / · (cid:18) fn / (cid:19) / = f. So we may let F be all other copies of ( u, v ), besides the edge ( u i , v j ) under consideration, and wehave | F | ≤ f . With these definitions of G, F , we have:
Lemma B.4. In G \ F , there is no cycle C of length ≤ that includes the edge ( u i , v j ) .Proof. Suppose for contradiction that such a cycle C exists. Let φ be the homomorphism from G \ F to the original Ree-Tits octagon incidence graph R , which maps each node x i back to theoriginal node x from which it was copied. Let φ ( C ) be the nodewise image of C , which is a closedwalk in R . Notice that φ ( C ) only contains the edge ( u, v ) once, since we removed all copies of ( u, v )in G except for ( u i , v j ).We then modify φ ( C ) as follows: while there exists a node x that occurs at least twice in C , replace φ ( C ) by its x x subwalk that contains the single copy of ( u, v ). Once this processhalts, we have a simple circularly-ordered walk that contains the edge ( u, v ) only once, which musttherefore be a cycle in R of length ≥ ≤
15. This contradicts that R has girth >
15, completingthe proof.It follows from this lemma that the shortest u i v j path in G \ ( F ∪ { ( u i , v j ) } ) has length >
14. Thus, we must keep ( u i , v j ) in any f -EFT 14-spanner. Since ( u i , v j ) was an arbitrary edge,this means G is the only f -EFT 14-spanner of itself, completing the proof.16 Missing Proofs from Section 3
Lemma C.1.
Any graph H returned by Algorithm 1 is an f -EFT (2 k − -spanner that has astrong k -blocking set of size at most f | E ( H ) | .Proof. The fact that H is an f -EFT (2 k − H has a small strong 2 k -blocking set. For each edge e = ( u, v ) ∈ E ( H ), let F e denote the set of edges that forced the algorithm to add e ; i.e., the setwith | F e | ≤ f such that d H \ F e ( u, v ) > (2 k − · w ( e ) when e is added to H . Let B := { ( x, e ) | e ∈ E ( H ) , x ∈ F e } . Since | F e | ≤ f for all e , we have that | B | ≤ f | E ( H ) | . We can now show that B is a strong 2 k -blocking set for H . Let C be any cycle on ≤ k edges in the final graph H and let e = ( u, v ) be thelast edge in C considered by the greedy algorithm. By construction there is a u v path (through C ) of total weight ≤ (2 k − · w ( e ) when e is added to H , and so some edge x ∈ C \ { ( u, v ) } must be included in F e . Thus ( x, e ) ∈ B . Moreover, since e is the last edge in C added to H bythe algorithm it is the heaviest edge in C (under tiebreaking), as the algorithm considers edges innon-decreasing weight order. Hence B is a strong 2 k -blocking set. Lemma C.2.
The graph H returned by Algorithm 2 is an f -EFT (2 k − -spanner that has astrong k -blocking set of size at most O ( kf ) · | E ( H ) | .Proof. The fact that H is an f -EFT (2 k − H has a small strong 2 k -blocking set. For each edge e = ( u, v ) ∈ E ( H ), let H e ⊆ H denote the subgraph of H containing only the edges consideredbefore e by the algorithm. Since e was added to H by Algorithm 2, A must have returned YES.Thus there is some set F e ⊆ E \ ( u, v ) with | F e | ≤ f (2 k −
1) such that ˆ d H e \ F e ( u, v ) > k − d to denote the unweighted distance). Let B := { ( x, e ) | e ∈ E ( H ) , x ∈ F e } . Since | F e | ≤ f (2 k −