A Polynomial Time Algorithm to Compute an Approximate Weighted Shortest Path

TL;DR

This paper presents a polynomial-time approximation scheme for finding near-optimal weighted shortest paths in triangulated regions, improving previous algorithms significantly and enabling efficient query answering.

Contribution

The authors develop a new polynomial-time approximation algorithm for weighted shortest paths and introduce data structures for fast query responses, surpassing prior methods.

Findings

Achieves a cubic factor improvement over previous algorithms.

Provides a polynomial-time preprocessing method for efficient queries.

Offers an approximation within (1+epsilon) of the optimal path.

Abstract

We devise a polynomial-time approximation scheme for the classical geometric problem of finding an approximate short path amid weighted regions. In this problem, a triangulated region P comprising of n vertices, a positive weight associated with each triangle, and two points s and t that belong to P are given as the input. The objective is to find a path whose cost is at most (1+epsilon)OPT where OPT is the cost of an optimal path between s and t. Our algorithm initiates a discretized-Dijkstra wavefront from source s and progresses the wavefront till it strikes t. This result is about a cubic factor (in n) improvement over the Mitchell and Papadimitriou '91 result, which is the only known polynomial time algorithm for this problem to date. Further, with polynomial time preprocessing of P, a set of data structures are computed which allow answering approximate weighted shortest path queries in polynomial time.