# Two-phase algorithms for the parametric shortest path problem

Sourav Chakraborty, Eldar Fischer, Oded Lachish, Raphael Yuster

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

A parametric weighted graph is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edge-weighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, and arise in various natural scenarios. Parametric weighted graph algorithms consist of two phases. A preprocessing phase whose input is a parametric weighted graph, and whose output is a data structure, the advice, that is later used by the instantiation phase, where a specific value for the variable is given. The instantiation phase outputs the solution to the (standard) weighted graph problem that arises from the instantiation. The goal is to have the running time of the instantiation phase supersede the running time of any algorithm that solves the weighted graph problem from scratch, by taking advantage of the advice. In this paper we construct several parametric algorithms for the shortest path problem. For the case of linear function weights we present an algorithm for the single source shortest path problem. Its preprocessing phase runs in Õ(V4) time, while its instantiation phase runs in only O(E + V log V) time. The fastest standard algorithm for single source shortest path runs in O(VE) time. For the case of weight functions defined by degree d polynomials, we present an algorithm with quasi-polynomial preprocessing time O(V(1+log(d))logV) and instantiation time only Õ(V). In fact, for any pair of vertices u, v, the instantiation phase computes the distance from u to v in only O(log2 V) time. Finally, for linear function weights, we present a randomized algorithm whose preprocessing time is Õ(V3.5) and so that for any pair of vertices u, v and any instantiation variable, the instantiation phase computes, in O(1) time, a length of a path from u to v that is at most (additively) ∈ larger than the length of a shortest path. In particular, an all-pairs shortest path solution, up to an additive constant error, can be computed in O(V2) time.

Original language English STACS 2010 - 27th International Symposium on Theoretical Aspects of Computer Science 167-178 12 https://doi.org/10.4230/LIPIcs.STACS.2010.2452 Published - 2010 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010 - Nancy, FranceDuration: 4 Mar 2010 → 6 Mar 2010

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 5 1868-8969

### Conference

Conference 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010 France Nancy 4/03/10 → 6/03/10

### Bibliographical note

Funding Information:
This study was supported by the National Centre of Scientific Research (CNRS), the JGOFS-France program at the DYFAMED time series station, and “La Société de Secours des Amis des Sciences”. We thank the crew of R/V Tethys II and George-Petit and A. Morel for providing us the lab place and laminar flow hood, allowing a rapid filtration of our samples We are grateful to V. Andersen, M.-D. Pizay, P. Raimbault, J. Chiavérini, P. Statham, and R. Arraes-Mescoff, who provided us useful data for this manuscript. We also thank the reviewers for their thoughtful and helpful suggestions.

## Keywords

• Parametric algorithms
• Shortest path problem

• Software

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