A parallel algorithm for the constrained shortest path problem on lattice graphs

TL;DR

This paper introduces a parallel algorithm for the constrained shortest path problem on lattice graphs, focusing on active vertices related to first passage percolation, with GPU implementation analysis.

Contribution

It presents a novel parallel algorithm that efficiently finds constrained shortest paths on lattice graphs and analyzes its GPU implementation.

Findings

Active vertices can form large fractals, challenging assumptions of small active sets.

The algorithm effectively utilizes GPU parallelism for lattice graphs.

OpenCL implementation demonstrates practical performance on high-dimensional cubes.

Abstract

The edges of a graph are assigned weights and passage times which are assumed to be positive integers. We present a parallel algorithm for finding the shortest path whose total weight is smaller than a pre-determined value. In each step the processing elements are not analyzing the entire graph. Instead they are focusing on a subset of vertices called {\em active vertices}. The set of active vertices at time $t$ is related to the boundary of the ball $B_t$ of radius $t$ in the first passage percolation metric. Although it is believed that the number of active vertices is an order of magnitude smaller than the size of the graph, we prove that this need not be the case with an example of a graph for which the active vertices form a large fractal. We analyze an OpenCL implementation of the algorithm on GPU for cubes in $\mathbb Z^d$.