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Finding Sparser Directed Spanners

Piotr Berman, Sofya Raskhodnikova, Ge RuanPennsylvania State University

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Graph Spanners [Awerbuch85,Peleg Schäffer89]

A subgraph H of G is a k-spanner if for all pairs of vertices u, v in G,

distanceH(u,v) ≤ k distanceG(u,v)

Goal: Given G and k, find a sparsest k-spanner of G

dense graph G sparse subgraph H

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Transitive-Closure Spanners [BhattacharyyaGrigorscu Jung Raskhodnikova Woodruff 09]

Transitive closure TC(G) has an edge from u to v iffG has a path from u to v

k-TC-spanner H of G has distanceH(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)

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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

Computational ResultsProblem Stretch k Known Ratio Authors Our Ratio

DIRECTED SPANNER andTC-SPANNER

k=2 O(log n) [Elkin Peleg]

k=3 Õ(n 2/3 ) [EP, BGJRW] Õ(n1/2 )k=4 Õ(n 3/4 )

[BGJRW]

Õ(n1/2 )

DIRECTED SPANNER

any even k Õ(n1-1/k) Õ(k n 1-2/k )

TC-SPANNERas above Õ(n 1-2/k )

k = (log n/ log log n )

O(n log n / (k2 +k log n))

O(n /k2)

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• 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 n1-1/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(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.

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Counterexample for Spanner Transformation of a general digraph

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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 m2 edges

If we can use any TC-edges, we have a 2-spanner with 2m+3 edges.

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Tight Example for 3- to 2-TC-spanner Transf

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mm

m m m2m2

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.

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:o If possible, a dense nodeo 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.

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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 m2 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).

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