Elsevier

Theoretical Computer Science

Volume 873, 10 June 2021, Pages 87-113
Theoretical Computer Science

Almost linear time algorithms for minsum k-sink problems on dynamic flow path networks

https://doi.org/10.1016/j.tcs.2021.05.003Get rights and content
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Highlights

  • Study the minsum k-sink problems on dynamic flow path networks for the confluent/non-confluent flow model.

  • Develop algorithms which run in almost linear time regardless of the number of sinks k.

  • Improve upon the previous results for the same problem with the confluent flow model.

  • Provide the first polynomial time algorithm for the problem with the non-confluent flow model.

  • The main theoretical contribution is to construct novel data structures for solving subproblems in polylogarithmic time.

Abstract

We address the facility location problems on dynamic flow path networks. A dynamic flow path network consists of an undirected path with positive edge lengths, positive edge capacities, and positive vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. In the dynamic flow network, given particular points on edges or vertices, called sinks, all the people evacuate from the vertices to the sinks as quickly as possible. The problem is to find the location of sinks on a dynamic flow path network in such a way that the aggregate evacuation time (i.e., the sum of evacuation times for all the people) to sinks is minimized. We consider two models of the problem: the confluent flow model and the non-confluent flow model. In the former model, the way of evacuation is restricted so that all the people at a vertex have to evacuate to the same sink, and in the latter model, there is no such restriction. In this paper, for both the models, we develop algorithms which run in almost linear time regardless of the number of sinks. It should be stressed that for the confluent flow model, our algorithm improves upon the previous result by Benkoczi et al. [Theoretical Computer Science, 2020], and one for the non-confluent flow model is the first polynomial time algorithm.

Keywords

Dynamic flow networks
Facility location problems
Minimum k-link path problem
Persistent data structures

Cited by (0)

A preliminary version of this paper appeared in the proceedings of COCOA 2020 [17].

1

Supported by JSPS KAKENHI Grant Number 20K19746.

2

Supported by JSPS KAKENHI Grant Number 19H04068.

3

Supported by JST CREST Grant Number JPMJCR1402.