Affiliation:
1. Yale University, New Haven, CT
Abstract
We study the <i>s-sources almost shortest paths</i>
(abbreviated <i>s-ASP</i>) problem. Given an unweighted
graph <i>G</i> = (<i>V,E</i>),
and a subset <i>S</i> ⊆ <i>V</i>
of <i>s</i> nodes, the goal is to compute almost
shortest paths between all the pairs of nodes <i>S</i>
× <i>V</i>. We devise an algorithm with
running time
<i>O</i>(∣<i>E</i>∣<i>n</i><sup>ρ</sup>
+ <i>s</i> ·
<i>n</i><sup>1 + ζ)</sup>
for this problem that computes the paths
<i>P</i><sub><i>u,w</i></sub>
for all pairs (<i>u,w</i>) ∈
<i>S</i> × <i>V</i> such that the
length of
<i>P</i><sub><i>u,w</i></sub>
is at most (1 + ε)
<i>d</i><sub><i>G</i></sub>(<i>u,w</i>)
+ β(ζ,ρ,ε), and
β(ζ,ρ,ε) is constant when
ζ, ρ, and ε are arbitrarily small
constants.
We also devise a distributed protocol for the
<i>s</i>-ASP problem that computes the paths
<i>P</i><inf><i>u,w</i></inf>
as above, and has time and communication complexities of
<i>O</i>(<i>s</i> ·
<i>Diam(G)</i> +
<i>n</i><sup>1 +
ζ/2</sup>) (respectively,
<i>O</i>(<i>s</i> ·
<i>Diam(G)</i> log<sup>3</sup>
<i>n</i> + <i>n</i><sup>1
+ ζ/2</sup> log <i>n</i>)) and
<i>O</i>(∣<i>E</i>∣
<i>n</i><sup>ρ</sup> +
<i>s</i> · <i>n</i><sup>1
+ ζ)</sup> (respectively,
<i>O</i>(∣<i>E</i>∣
<i>n</i><sup>ρ</sup> +
<i>s</i> · <i>n</i><sup>1
+ ζ</sup> +
<i>n</i><sup>1 + ρ +
ζ(ρ − ζ/2)/2)) in the
synchronous (respectively asynchronous) setting.
Our sequential algorithm, as well as the distributed protocol,
is based on a novel algorithm for constructing (1 +
ε, β(ζ,ρ, ε))-spanners
of size <i>O</i>(<i>n</i><sup>1
+ ζ</sup>), developed in this article. This
algorithm has running time of
<i>O</i>(∣<i>E</i>∣
<i>n</i><sup>ρ</sup>), which is
significantly faster than the previously known algorithm given in
Elkin and Peleg [2001], whose running time is
<i>Õ</i>(<i>n</i><sup>2
+ ρ</sup>). We also develop the first
distributed protocol for constructing (1 +
ε,β)-spanners. The communication complexity of
this protocol is near optimal.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference38 articles.
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3. A new distributed algorithm to find breadth first search trees
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