1 Finding Sparser Directed Spanners Piotr Berman, Sofya Raskhodnikova, Ge Ruan Pennsylvania State University

Graph Spanners [Awerbuch85,Peleg Schäffer89] A subgraph H of G is a k-spanner if for all pairs of vertices u, v in G, distance H (u,v) ≤ k distance G (u,v) Goal: Given G and k, find a sparsest k-spanner of G 2 dense graph Gsparse subgraph H

3 Transitive-Closure Spanners [BhattacharyyaGrigorscu Jung Raskhodnikova Woodruff 09] Transitive closure TC(G) has an edge from u to v iff G has a path from u to v k-TC-spanner H of G has distance H (u,v) ≤ k iff G has a path from u to v Alternatively: k-TC-spanner of G is a k-spanner of TC(G) G TC(G)

Applications Directed Spanners: efficient routing protocols in unsynchronized networks parallel /distributed algorithms for approximate shortest paths TC-Spanners: managing keys in access control hierarchies data structures for computing partial products in a semigroup property testing property reconstruction 4

Computational Results Problem Stretch kKnown RatioAuthorsOur Ratio DIRECTED SPANNER and TC-SPANNER k=2O(log n)[Elkin Peleg] k=3 Õ (n 2/3 )[EP, BGJRW] Õ (n 1/2 ) k=4 Õ (n 3/4 ) [BGJRW] Õ (n 1/2 ) DIRECTED SPANNER any even k Õ (n 1-1/k ) Õ (k n 1-2/k ) TC-SPANNER as above Õ (n 1-2/k ) k =  (log n/ log log n ) O(n log n / (k 2 +k log n)) O(n /k 2 ) 5 Õ -notation is hiding at most log n factors Subsequent work on DIRECTED SPANNER [Dinitz Krauthgamer] (independent): Õ (n 1/2 ) for k=3, Õ (n 2/3 ) for all k (better than ours for k>6) [Berman Raskhodnikova Yaroslavtsev]: Õ (n 1/2 ) for all k (better than ours for k>4)

Structural Results Sparsest 2-spanners can be approximated with 2ln n ratio. Question: how good approximation of a k-spanner is provided by the sparsest 2-spanner? We show that a 2k-TC-spanner with m edges can be transformed into a 2-spanner with O(m n 1-1/k ) edges. We show that for every ε>0 there exists digraphs that have O(n 1+1/k ) edges with maximum distance 2k for which minimum number of edges of a 2-spanner is Ω(n 2-ε ) Such a digraph is its own 2k-spanner, so we tightly characterized the quality of approximation for 2k-spanner by an algorithm that finds a 2-spanner. 6

Counterexample for Spanner Transformation of a general digraph 7 mm The upper bound for transformation of a 2k-spanner into a 2-spanner does not hold for arbitrary digraphs. If we can use only the graph edges, the sparsest 2-spanner has more than m 2 edges If we can use any TC-edges, we have a 2-spanner with 2m+3 edges. u

Tight Example for 3- to 2-TC-spanner Transf 8 mm mmm2m2 m2m2 This graph has O(n) edges, longest path length 3, and a sparsest 2- spanner has Ω(m 3 = n 3/2 ) edges. The gap is the same if the longest path is 4. For larger k, to show the largest gap between the size of 2k-spanner and 2-spanner can be obtained adapting a construction by Hesse.

Upper bound for transforming 2k-TC-spanner Def. w is a dense node of D if at least n 1/k edges of D are incident Each TC edge (u,v) has a path of length at most 2k in D and it selects a node on that path using two rules: o If possible, a dense node o If not, the middle node Construction: insert (u,w) and (w,v) to 2-spanner D’ for every (u,v) selecting w. Lemma. If w is incident to a edges of D then it is incident to at most an (k-1)/k edges of D’. Proof. Trivial if w is dense. Otherwise, count (w,v) edges of D’: for each of them there is a path in D with at most k edges; after the first edge e which is incident to w there are at most k-1 branching points and each branching point is sparse, hence at most n (k-1)/k nodes can be reached through edge e. Same argument applies to (u,w) edges. 9

Upper bound for transforming 2k-TC-spanner By the lemma, the degree of each node increases a factor at most n (k-1)/k. Thus |D’ | ≤ |D|n (k-1)/k. We can approximate the sparsest 2-spanner using a greedy algorithm. For k > 4 the newest algorithm delivers a better approximation with a linear program that has m 2 variables. This construction can be generalized for spanners in general graphs, but rather than direct edges, we consider in- and out- arborescences, so we are transforming a 2k-spanner into another 2k-spanner of a restricted type, one that can be efficiently computed within factor O(k log n). 10

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