Showing papers by "Liam Roditty published in 2005"

Deterministic constructions of approximate distance oracles and spanners

Abstract: Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn$^{\rm 1/{\it k}}$) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn$^{\rm 1+1/{\it k}}$) capable of returning in O(k) time an approximation $\hat{\delta}(u,v)$ of the distance δ(u,v) from u to v in G that satisfies $\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)$, for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn$^{\rm 1+1/{\it k}}$log n). We present here a deterministic O(kmn$^{\rm 1/{\it k}}$) time algorithm for constructing oracles of size O(kn$^{\rm 1+1/{\it k}}$). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn$^{\rm 1+1/{\it k}}$) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

Replacement paths and k simple shortest paths in unweighted directed graphs

Abstract: Let G=(V,E) be a directed graph and let P be a shortest path from s to t in G. In the replacement paths problem we are required to find, for every edge e on P, a shortest path from s to t in G that avoids e. We present the first non-trivial algorithm for computing replacement paths in unweighted directed graphs (and in graphs with small integer weights). Our algorithm is Monte-Carlo and its running time is ${\tilde O}(m\sqrt{n})$. Using the improved algorithm for the replacement paths problem we get an improved algorithm for finding the ksimple shortest paths between two given vertices.