NDLI: Additive spanners and (α, β)-spanners

Content Provider	ACM Digital Library
Author	Mehlhorn, Kurt Baswana, Surender Kavitha, Telikepalli Pettie, Seth
Copyright Year	2010
Abstract	An (α, β)-spanner of an unweighted graph $\textit{G}$ is a subgraph $\textit{H}$ that distorts distances in $\textit{G}$ up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) $(2\textit{k}™1,$ 0)-spanner of size $O(n^{1+1/k})$ and an (additive) (1,2)-spanner of size $O(n^{3/2}).$ However no other additive spanners are known to exist. In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1,6)-spanner of size $O(n^{4/3}).$ The construction algorithm can be understood as an economical agent that assigns $\textit{costs}$ and $\textit{values}$ to paths in the graph, $\textit{purchasing}$ affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any $\textit{k},$ a $(\textit{k},\textit{k}™1)-spanner$ with size $O(kn^{1+1/k})$ can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Starting Page	1
Ending Page	26
Page Count	26
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/1868237.1868242
Volume Number	7
Issue Number	1
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2010-12-08
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Spanner Metric embedding
Content Type	Text
Resource Type	Article
Subject	Mathematics

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