Finding Sparser Directed Spanners
Piotr Berman, Sofya Raskhodnikova, Ge Ruan
Pennsylvania 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)
dense graph G
sparse subgraph H
Goal: Given G and k, find a sparsest k-spanner of G
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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
G
TC(G)
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)
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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
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Computational Results
Problem
Stretch k
Known Ratio
Authors
Our Ratio
DIRECTED
SPANNER
and
TC-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 )
Õ(n1/2 )
DIRECTED
SPANNER
any even k
Õ(n1-1/k)
Õ(k n 1-2/k )
[BGJRW]
as above
TC-SPANNER
k = (log n/
log log n )
O(n log n /
(k2 +k log n))
Õ(n 1-2/k )
O(n /k2)
• Õ -notation is hiding at most log n factors
Subsequent work on DIRECTED SPANNER
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)
[Dinitz Krauthgamer] (independent):
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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
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 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
m
m2
m
m
m
m2
This graph has O(n) edges, longest path length 3, and a sparsest 2spanner 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.
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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 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.
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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 outarborescences, 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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