New routing techniques and their applications

TL;DR

This paper introduces new routing schemes for unweighted and weighted graphs that achieve near-optimal trade-offs between stretch factor, space complexity, and header size, improving upon previous methods.

Contribution

The paper presents novel routing schemes with improved space and stretch bounds for both unweighted and weighted graphs, nearly matching known lower bounds.

Findings

Achieves $(2+ extepsilon,1)$-stretch routing with $ ilde O(n^{2/3})$ space for unweighted graphs.

Provides $(5+ extepsilon)$-stretch routing with $ ilde O(n^{1/3} extlog D)$ space for weighted graphs.

Introduces a family of schemes with adjustable parameters for different stretch and space trade-offs.

Abstract

Let $G=(V,E)$ be an undirected graph with $n$ vertices and $m$ edges. We obtain the following new routing schemes: - A routing scheme for unweighted graphs that uses $\tilde O(\frac{1}{\epsilon} n^{2/3})$ space at each vertex and $\tilde O(1/\epsilon)$-bit headers, to route a message between any pair of vertices $u,v\in V$ on a $(2 + \epsilon,1)$-stretch path, i.e., a path of length at most $(2+\epsilon)\cdot d(u,v)+1$. This should be compared to the $(2,1)$-stretch and $\tilde O(n^{5/3})$ space distance oracle of Patrascu and Roditty [FOCS'10 and SIAM J. Comput. 2014] and to the $(2,1)$-stretch routing scheme of Abraham and Gavoille [DISC'11] that uses $\tilde O( n^{3/4})$ space at each vertex. - A routing scheme for weighted graphs with normalized diameter $D$, that uses $\tilde O(\frac{1}{\epsilon} n^{1/3}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-bit headers, to route a message between any pair of vertices on a $(5+\epsilon)$-stretch path. This should be compared to the $5$-stretch and $\tilde O(n^{4/3})$ space distance oracle of Thorup and Zwick [STOC'01 and J. ACM. 2005] and to the $7$-stretch routing scheme of Thorup and Zwick [SPAA'01] that uses $\tilde O( n^{1/3})$ space at each vertex. Since a $5$-stretch routing scheme must use tables of $\Omega( n^{1/3})$ space our result is almost tight. - For an integer $\ell>1$, a routing scheme for unweighted graphs that uses $\tilde O(\ell\frac{1}{\epsilon} n^{\ell/(2\ell \pm 1)})$ space at each vertex and $\tilde O(\frac{1}{\epsilon})$-bit headers, to route a message between any pair of vertices on a $(3\pm2/\ell+\epsilon,2)$-stretch path. - A routing scheme for weighted graphs, that uses $\tilde O(\frac{1}{\epsilon}n^{1/k}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-bit headers, to route a message between any pair of vertices on a $(4k-7+\epsilon)$-stretch path.