# Improved Approximation for the Directed Spanner Problem

Ontology type: schema:Chapter

### Chapter Info

DATE

2011

AUTHORS ABSTRACT

We give an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\sqrt{n}\log n)$\end{document}-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V,E) with nonnegative edge lengths d: E → ℝ ≥ 0 and a stretchk ≥ 1, a subgraph H = (V,EH) is a k-spanner of G if for every edge (u,v) ∈ E, the graph H contains a path from u to v of length at most k ·d(u,v). The previous best approximation ratio was \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{O}(n^{2/3})$\end{document}, due to Dinitz and Krauthgamer (STOC ’11).We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{O}(n^{1/3})$\end{document} which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer, had approximation ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{O}(\sqrt{n})$\end{document}. More... »

PAGES

1-12

### Book

TITLE

Automata, Languages and Programming

ISBN

978-3-642-22005-0
978-3-642-22006-7

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-22006-7_1

DOI

http://dx.doi.org/10.1007/978-3-642-22006-7_1

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1036034221

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