Finding Sparser Directed Spanners

1. Finding Sparser Directed Spanners Piotr Berman, SofyaRaskhodnikova, GeRuan Pennsylvania State University

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

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

4. 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

5. Computational Results • Õ -notation is hiding at most log n factors Subsequent work on DIRECTED SPANNER [DinitzKrauthgamer] (independent): Õ(n 1/2 ) for k=3, Õ(n 2/3 ) for all k (better than ours for k>6) [Berman RaskhodnikovaYaroslavtsev]: Õ(n 1/2 ) for allk (better than ours for k>4)

6. 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 n1-1/k) edges. • We show that for every ε>0 there exists digraphs that have O(n1+1/k) edges with maximum distance 2k for which minimum number of edges of a 2-spanner is Ω(n2-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.

7. Counterexample for Spanner Transformation of a general digraph The upper bound for transformation of a 2k-spanner into a 2-spanner does not hold for arbitrary digraphs. u m m If we can use only the graph edges, the sparsest 2-spanner has more than m2edges If we can use any TC-edges, we have a 2-spanner with 2m+3 edges.

8. Tight Example for 3- to 2-TC-spanner Transf m m m2 m m m2 This graph has O(n) edges, longest path length 3, and a sparsest 2-spanner has Ω(m3 = n3/2) 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.

9. Upper bound for transforming 2k-TC-spanner Def. w is a dense node of D if at least n1/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: • If possible, a dense node • 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)/kedges 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 ewhich is incident to w there are at most k-1 branching points and each branching point is sparse, hence at most n(k-1)/knodes can be reached through edge e. Same argument applies to (u,w) edges.

10. 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 m2variables. • 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).