TY - JOUR

T1 - All-pairs shortest paths with a sublinear additive error

AU - Roditty, Liam

AU - Shapira, Asaf

PY - 2011/9

Y1 - 2011/9

N2 - We show that, for every 0 ≤ p ≤ 1, there is an O(n 2.575-p/(7.4-2.3p))-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u, v in the graph to within an additive error δp(u, v), where δ(u, v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1. Previously the only way to "beat" the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [2002] that approximates the shortest path distances within a multiplicative error of (1 + ε). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ε) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when ε 1 and the graph has small integer weights.

AB - We show that, for every 0 ≤ p ≤ 1, there is an O(n 2.575-p/(7.4-2.3p))-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u, v in the graph to within an additive error δp(u, v), where δ(u, v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1. Previously the only way to "beat" the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [2002] that approximates the shortest path distances within a multiplicative error of (1 + ε). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ε) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when ε 1 and the graph has small integer weights.

KW - Graph algorithms

KW - Matrix multiplication

KW - Shortest paths

UR - http://www.scopus.com/inward/record.url?scp=80053519825&partnerID=8YFLogxK

U2 - 10.1145/2000807.2000813

DO - 10.1145/2000807.2000813

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AN - SCOPUS:80053519825

SN - 1549-6325

VL - 7

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 4

M1 - 45

ER -