Lecture 11: June 27. Distance Sensitivity Oracles. Succinct Graph Structures and Their Applications Spring Lecturer: Merav Parter

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1 Sccinct Graph Strctres and Their Applications Spring 2018 Lectrer: Merav Parter Lectre 11: Jne 27 Distance Sensitivit Oracles A distance oracle scheme (Lectre 2) consists of two algorithms: a preprocessing algorithm, that given a graph G otpts a (small space) data strctre; and a qer algorithm that given a distance qer s, t, ses onl the data strctre (with no access to G) to report the distance between s and t in G. This setting has three compleit measres: the preprocessing time, the size of the data strctre and the qer time. In this class, we pt emphasis on the latter two measres and the tradeoff between them. In distance sensitivit oracles, the qer consists of three vertices: sorce s, sink t and a failed verte. The otpt of the qer algorithm is dist(s, t, G \ {}). There are two naive schemes. One etreme is to keep in the data strctre, all O(n 3 ) possible distances eplicitl, i.e., for an triplet s, t,, keep dist(s, t, G \ {}). This gives space O(n 3 log n) and constant qer time. On the other etreme, one can keep the entire graph G, and given a qer s, t,, appl Dijkstra in G. This gives an optimal space of O(m) bt qite nsatisfactor qer time of O(m). Toda, we will see a sper elegant constrction that enjos both small space of O(n 2 log 3 n) and constant qer time. Up to logarithmic factors, this constrction is optimal. Theorem 11.1 ([DTCR08]) For an n-verte graph G = (V, E), there eists a constrction of a distance sensitivit oracles in time O(m n 2 + n log n), space of O(n 2 log 3 n) and constant qer time. We will se the following fact abot data strctre for LCA (Lowest Common Ancestor). fact 11.2 [HT84] For an n-verte tree T, there eists an O(n) comptable data strctre which occpies O(n)-space and answers LCA qeries ( what s the LCA of s and t in T? ) in O(1) time. Description of the oracle: A sccinct and an elegant eplanation of the constrction appears in [DP09]. For simplicit, we consider nweighted graphs, althogh the constrction holds for weighted graphs as well. For ever r V, let T r be a BFS tree rooted at r. The oracle O contains the LCA data strctre of T r for ever r V. Let π(, ) be the shortest path in G. We assme niqeness of shortest paths b breaking shortest path ties in a consistent manner. In addition, the oracle O contains the following: (1) A distance matri B 0 where B 0 (, ) = dist(,, G). (2) A collection of distance lists B 1 where B 1 (, ) contains for ever, V : (A) For i {0, 1,..., log dist(,, G) }: Let i (resp., v i ) be the verte at distance 2 i from (resp., ) on π(, ). Keep B 1 (,, w) = dist(,, G \ {w}) for w { i, v i }. (B) For ever i, j {0, 1,..., log dist(,, G) }: Keep B 1 (,, i, v j ) = dist(,, G \ V (π( i, v j ))). It is eas to see that the total space of O is indeed bonded b O(n 2 log 3 n) (mostl de to step (B)). The main challenge is in showing that the stored information is sfficient. In particlar, note that we consider onl O(log n) special vertices on each shortest path π(, ), even thogh there might be Ω(n) vertices that might fail on that path. The magic comes from the strctre of replacement paths that avoids a single failed verte. Last week, in the constrction of FT-BFS, we mainl sed the strctre of new-ending replacement paths, i.e., paths whose last edge is not the edge of the shortest path in G. Here, we will se the general strctre of an replacement path. 11-1

2 11-2 Lectre 11: Jne 27 Wh does it work? some intition. Consider a pair, V and a failed verte. Recall from last class that the replacement path P,, consists of a detor which bpasses the failing verte on the - shortest path π(, ). We need the following definition. Let be the closest verte to on the π(, ) path sch that π(, ) is a power of 2. In the same manner, let r be the closest verte to, sch that π(, r ) is a power of 2. There are three possibilities for the detor of P,,. Option (1) is where the detor goes throgh, option (2) is where the detor passes throgh r, and option (3) is where the detor does not go throgh nor r. In sch a case, b the niqeness of shortest paths, it mst hold that the detor avoids all the vertices on the segment π(, r ). See Fig B the triangle ineqalit, it alwas holds that: dist(,, G \ {}) dist(,, G \ {}) + dist(, r, G \ {}) + dist( r,, G \ {}). B the above possible configrations, we get that: dist(,, G \ {}) = min{ π(, ) + dist(,, G \ {}), dist(, r, G \ {}) + π( r, ), dist(,, G \ π(, r ))}). Note that besides the term dist(,, G \ π(, r )), all other distances in the above eqation are eplicitl kept in or data strctre. That is, since and r are within power of 2 distance from (and not from and respectivel), we did not keep the distance dist(,, G \ π(, r )) eplicitl. We will see how to handle it soon. dist(,, G {}) = dist(,, G) + dist(,, G {}) r dist(,, G {}) = dist(, r, G {}) + dist(,, G) r dist(,, G {}) = dist(,, G π(, r )) r Figre 11.1: The possible configrations of a single falt detor along with the corresponding distances. The qer algorithm: Given a triplet,,, we do as follows. First, to see if π(, ), we appl an LCA qer with, in T. If is the LCA of in T, then we dedce that π(, ). If is not the LCA, then we simpl retrn B 0 (, ) = dist(,, G). From now on, assme that π(, ). Let W = { i, v i i {0, 1,... log(dist(,, G)) }} be all vertices at distance 2 i from either or on π(, ). If W, then retrn B 1 (,, ) = dist(,, G\{}). Note that we keep a lookp table for the B 1 distances with the ke,, w for ever w W, and hence we can check in O(1) time if W and in sch a case retrn the corresponding distance. We finall get to the interesting case where / W and define and r as above. Since we know dist(,, G), we can compte in O(1) time the vertices and r. Observe that since / W (i.e., its distance is not

3 Lectre 11: Jne a power of two from neither nor ), we get that and r. Let dist(,, G) = 2 i and dist(, r, G) = 2 j. Define at the verte at distance 2 i from on π(, ). In the same manner, define as the verte at distance 2 j from on π(, ). See Fig. for an illstration. Since and are in power of 2 distance from and respectivel, we have that, W. Ths, their comptation can be done in O(1) time sing the lookp table of B 1 (,, w) for w W. The qer algorithm retrns the answer: min{b 0 (, ) + B 1 (,, ), B 1 (, r, ) + B 0 (, r ), B 1 (,,, )}. To prove the correctness of the qer algorithm, it is sfficient to show: Claim 11.3 π(, ) and π( r, ) and ths π(, ). Proof: B the definition of, π(, ) 2 i and ths π(, ). Similarl, b the definition of r, it holds that π(, r ) 2 j and ths π(, r ). 2 i 2 j r 2 i 2 j Falt Tolerant Spanners For positive integers f, t 1, a sbgraph H G is an f-edge t-falt-tolerant (FT) spanner if dist(, v, H \ F ) t dist(, v, G \ F ) for ever, v V and for ever F E, F f. The verte variant is defined analogosl onl with F V. Note that this definition does not reqire G to be (f + 1)-verte connected 1. In particlar, if there is a seqence of f falts F which disconnects and v in G, then the -v distance in both G \ F and H \ F is infinit. However, as long as and v are connected in G \ F, it is reqired that the -v distance in H \ F is a factor t approimation to the distance in G \ F. This notion of falt tolerance is known as competitive, see [CP10]. Before diving into the constrctions of sparse falt tolerant spanners, lets tr to compare them to the other falt tolerant strctre we saw last class, namel to falt tolerant BFS strctres. Comparison to FT-BFS: On the one hand, it seems that the reqirement of FT-spanners is stronger than that of FT-BFS: the latter onl concerns the sorcewise distances {s} V while the former concerns with all the pairs V V. On the other hand, FT-BFS insists on preserving the eact distance, while FTspanners settle for an approimation. As we will see toda, the insistence on eact distance plas the ke role here which makes f FT-BFS significantl denser than f-edge FT-spanners for stretch vales t 5. 1 A graph G is k verte ocnnected, in the removal of an k 1 vertices does not disconnect G.

4 11-4 Lectre 11: Jne 27 FT-spanners for edge failres [CLPR10]: In the setting of FT-spanners, handling edge failres is easier than verte failres. We will present an elegant constrction de to Chechik et al. [CLPR10]. This was the first work which stdied the constrction of sparse falt tolerant sbgraphs for general sbgraph. Theorem 11.4 For ever f 1 and k 1, there eists an f-edge (2k 1) FT-spanner H G with O(f n 1+1/k ). Recall that standard (2k 1) spanners have size of O(n 1+1/k ) edges (First Lectre!), ths the cost of making these strctres robst against f edge falts is linear in f!. The algorithm of [CLPR10] is as follows: Algorithm EFTSpanner(G, f, k) 1. Set G G and H. 2. For i = 1,..., f + 1: H i Spanner(G, 2k 1). H H H i G G \ H i. Figre 11.2: Algorithm for compting an f-edge (2k 1) FT-spanners The size of H is immediate b constrction: Spanner algorithm comptes a (2k 1) spanner with O(n 1+1/k ) edges and H contains O(f) edge disjoint (2k 1) spanners. For the stretch analsis, it is sfficient to show: Claim 11.5 dist(, v, H \ F ) 2k 1 for ever (, v) E and F E, F f and (, v) / F. Proof: If (, v) H then the claim holds triviall. Otherwise, since e = (, v) was not added to an of the f + 1 spanners, it implies that H contains f + 1 edge disjoint -v paths of length at most 2k 1. Since the set F contains at most f edges, one of these paths srvive the failing, i.e., H contains a -v path of length at most 2k 1 that srvives the failing of F. FT-spanners for verte failres: We now trn to discss falt tolerant spanners that are resilient against verte falts. Fiing the nmber of falts to f and the stretch vale to (2k 1), [CLPR10] showed a constrction of f-verte (2k 1) FT-spanners with Õ(f 2 kf+1 n 1+1/k ) edges. This bond becomes Õ(n1+1/k ) when both the nmber of falts and the stretch are constants. Dinitz and krathgamer [DK11] improved this bond to Õ(f 2 1/k n 1+1/k ) edges. Finall, recentl Bodwin et al. [BDPW18] provided a constrction with Õ(f 1 1/k n 1+1/k ) for k = O(1). Assming the girth conjectre, this bond is also tight. We will now describe the algorithm b Dinitz and krathgamer [DK11]. The high level idea is to rn several independent eperiments. In each eperiment (iteration) i, we sample man vertices into the failing set F i and compte a (2k 1) spanner in the remaining graph G \ F i. Note that even thogh the nmber of falts that the spanner shold handle is at most f, in each eperiment we fail (1 1/f) fraction of the vertices! Claim 11.6 H has Õ(f 2 1/k n 1+1/k ) edges. Proof: There are O(f 3 log n) iterations, in each we compte a (2k 1) spanner for a graph that contains O(n/f) vertices. Ths, each H i has O((n/f) 1+1/k ) edges, and the final spanner has O(f 2 1/k log n n 1/k ) edges, as reqired. We now trn to show correctness and as sal consider a neighboring pair, v and a set F of the most f vertices. The end goal is to show that dist(, v, H \ F ) 2k 1 for ever (, v) E and F V. The analsis is based on the following definition. An eperiment (or iteration) i is good for the triplet, v, F if, v / F i and F F i. We claim:

5 Lectre 11: Jne Algorithm VFTSpanner(G, f, k) 1. Set H. 2. For i = 1,..., O(f 3 log n): Sample each v into F i with probabilit p, where { 1 1/f for f 2 p = 1/2 for f = 1 H i = Spanner(G \ F i, 2k 1). H H H i. Figre 11.3: Algorithm for compting an f-verte (2k 1) FT-spanners Claim 11.7 With probabilit of 1 1/n cf for some constant c,, v, F has a good eperiment. Proof: Fi an eperiment i and a triplet, v, F. The probabilit that i is good for this triplet is at least (1 p) 2 p f = 1/f 2 1/e 1/(2f 2 ). Since all eperiments are independent of each other, the probabilit that no eperiment is good is at most (1 1/(2f 2 )) c f 3 log n = 1/n Θ(f). The claim follows. We are now read to complete the stretch argment. Claim 11.8 dist(, v, H \ F ) 2k 1 for ever (, v) E and F V. Proof: B appling the nion bond over all O(n f+2 ) triplets, v, F, b Cl. 11.7, with probabilit 1 1/n c, ever triplet has a good eperiment. Fi a triplet, v, F and let i be sch a good eperiment for this triplet. Then H i either contains (, v) and we are done, or else it contains an -v path P of length at most (2k 1). Since F F i and H i G \ F i, that path P is free from an other failing vertices and ths: dist(, v, H \ F ) dist(, v, H i \ F ) 2k 1. The claim follows. (Conditional) lower bond for f-verte (2k 1) FT-spanners. Lemma 11.9 ([BDPW18]) Assming Erdős girth conjectre, for ever k 1, f 1, there is an n-verte graph G k,f sch that an f-verte (2k 1) FT-spanner H G k,f mst have Ω(f 1 1/k n 1+1/k ) edges. Proof: Erdős girth conjectre (see Lectre 1) states that for ever k 1 and sfficientl large n, there are n-verte graphs G k with Ω(n 1+1/k ) and girth 2 at least 2k + 2. The graph G k,f is obtained from G k b the following operations: Each verte is G k is mltiplied b f = f/2 copies 1,..., f in G k,f. Each edge (, v) G k is replaced b a complete bipartite graph between all copies of and all copies of v. Formall, V (G k,f ) = { 1,..., f V } and E(G k,f ) = {( i, v j ) (, v) E(G k ), i, j [1, f ]}. The nmber of vertices in G k,f is N = O(f n) and the nmber of edges is O(f 2 n 1+1/k ) = O(f 1 1/k N 1+1/k ) edges. We net show that removing an edge e from G k,f reslts in a graph H = G k,f \ {e} which is not a legal f-verte (2k 1) FT-spanner for G k,f. This wold impl that the onl f-verte (2k 1) FT-spanner for G k,f is G k,f, ths contains Ω(f 1 1/k N 1+1/k ) man edges. Let e = ( i, v j ) and let F = ({ 1,..., f } {v 1,..., v f }) \ { i, v j }. Observe that F f. Ths in G k,f \ F, the edge e = ( i, v j ) is the onl edge connecting s copies with v s copies. Since the girth of G k is at least 2k + 2, removing e increases the distance from i to v j in G k,f from 1 to at least 2k + 1. We get that dist( i, v j, G k,f \ F ) = 1 bt dist( i, v j, H \ F ) 2k + 1. The claim follows. 2 Length of shortest ccle in the graph.

6 11-6 Lectre 11: Jne 27 References [BDPW18] Greg Bodwin, Michael Dinitz, Merav Parter, and Virginia Vassilevska Williams. Optimal verte falt tolerant spanners (for fied stretch). In Proceedings of the Twent-Ninth Annal ACM- SIAM Smposim on Discrete Algorithms, pages Societ for Indstrial and Applied Mathematics, [CLPR10] [CP10] Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditt. Falt tolerant spanners for general graphs. SIAM Jornal on Compting, 39(7): , Shiri Chechik and David Peleg. Rigid and competitive falt tolerance for logical information strctres in networks. In Electrical and Electronics Engineers in Israel (IEEEI), 2010 IEEE 26th Convention of, pages IEEE, [DK11] Michael Dinitz and Robert Krathgamer. Falt-tolerant spanners: better and simpler. In Proceedings of the 30th annal ACM SIGACT-SIGOPS smposim on Principles of distribted compting, pages ACM, [DP09] Ran Dan and Seth Pettie. Dal-failre distance and connectivit oracles. In Proceedings of the twentieth annal ACM-SIAM smposim on Discrete algorithms, pages Societ for Indstrial and Applied Mathematics, [DTCR08] Camil Demetresc, Mikkel Thorp, Rezal Alam Chowdhr, and Vijaa Ramachandran. Oracles for distances avoiding a failed node or link. SIAM Jornal on Compting, 37(5): , [HT84] Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. siam Jornal on Compting, 13(2): , 1984.