Computing shortest paths amid non-overlapping weighted disks

TL;DR

This paper introduces an approximation algorithm for shortest paths among non-overlapping weighted disks, achieving near-optimal solutions efficiently by discretizing the space and applying Dijkstra's algorithm.

Contribution

It presents a novel approximation algorithm for the Weighted Region Problem with non-overlapping disks, utilizing space discretization and graph algorithms.

Findings

Approximate shortest path length within (1 + ε) factor.

Algorithm runs in (pseudo-)polynomial time.

Effective discretization method for geometric space.

Abstract

In this article, we present an approximation algorithm for solving the Weighted Region Problem amidst a set of $ n $ non-overlapping weighted disks in the plane. For a given parameter $ \varepsilon \in (0,1]$, the length of the approximate path is at most $ (1 +\varepsilon) $ times larger than the length of the actual shortest path. The algorithm is based on the discretization of the space by placing points on the boundary of the disks. Using such a discretization we can use Dijkstra's algorithm for computing a shortest path in the geometric graph obtained in (pseudo-)polynomial time.