TY - GEN

T1 - Faster approximation of distances in graphs

AU - Berman, Piotr

AU - Kasiviswanathan, Shiva Prasad

PY - 2007

Y1 - 2007

N2 - Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ε V reports distance no greater than 2dG(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)dG(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1)ε-3log(nε-1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

AB - Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ε V reports distance no greater than 2dG(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)dG(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1)ε-3log(nε-1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

UR - http://www.scopus.com/inward/record.url?scp=38149066832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149066832&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-73951-7_47

DO - 10.1007/978-3-540-73951-7_47

M3 - Conference contribution

AN - SCOPUS:38149066832

SN - 3540739483

SN - 9783540739487

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 541

EP - 552

BT - Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings

PB - Springer Verlag

T2 - 10th International Workshop on Algorithms and Data Structures, WADS 2007

Y2 - 15 August 2007 through 17 August 2007

ER -