A Fast Algorithm for Source-wise Round-trip Spanners
release_bj2go2mo2fdybbrvkndf7su6pu
by
Chun Jiang Zhu, Song Han, Kam-Yiu Lam
2021
Abstract
In this paper, we study the problem of fast constructions of source-wise
round-trip spanners in weighted directed graphs. For a source vertex set
S⊆ V in a graph G(V,E), an S-sourcewise round-trip spanner of G
of stretch k is a subgraph H of G such that for every pair of vertices
u,v∈ S× V, their round-trip distance in H is at most k times of
their round-trip distance in G. We show that for a graph G(V,E) with n
vertices and m edges, an s-sized source vertex set S⊆ V and an
integer k>1, there exists an algorithm that in time O(ms^1/klog^5n)
constructs an S-sourcewise round-trip spanner of stretch O(klog n) and
O(ns^1/klog^2n) edges with high probability. Compared to the fast
algorithms for constructing all-pairs round-trip spanners <cit.>,
our algorithm improve the running time and the number of edges in the spanner
when k is super-constant. Compared with the existing algorithm for
constructing source-wise round-trip spanners <cit.>, our algorithm
significantly improves their construction time Ω(min{ms,n^ω})
(where ω∈ [2,2.373) and 2.373 is the matrix multiplication exponent)
to nearly linear O(ms^1/klog^5n), at the expense of paying an extra
O(log n) in the stretch. As an important building block of the algorithm, we
develop a graph partitioning algorithm to partition G into clusters of
bounded radius and prove that for every u,v∈ S× V at small round-trip
distance, the probability of separating them in different clusters is small.
The algorithm takes the size of S as input and does not need the knowledge of
S. With the algorithm and a reachability vertex size estimation algorithm, we
show that the recursive algorithm for constructing standard round-trip spanners
<cit.> can be adapted to the source-wise setting.
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