A Time-space Trade-off for Computing the Geodesic Center of a Simple Polygon

TL;DR

This paper presents a new algorithm for finding the geodesic center of a simple polygon that balances time and space efficiency, especially under limited workspace conditions.

Contribution

It introduces a novel time-space trade-off algorithm for computing the geodesic center in simple polygons, improving efficiency with limited memory.

Findings

Expected time complexity of O((n^2/s) log^3 n) using current shortest path algorithms.

Uses O(s) space where s is between logarithmic and linear in n.

Applicable to polygons with up to n vertices.

Abstract

In this paper we study the problem of computing the geodesic center of a simple polygon when the available workspace is limited. For an $n$-vertex simple polygon, we give a time-space trade-off algorithm that finds the geodesic center in $O(T(n, s) \log^2 n+ \frac{n^2}{s} \log n)$ expected time and uses $O(s)$ additional words of space where $s\in \Omega(\log n) \cap O(n)$, and $T(n, s)$ is the time needed for constructing the shortest path tree of a given point inside a simple polygon, in depth-first order, with $O(s)$ extra space. Applying the best current known time-space trade-off of Oh and Ahn (Algorithmica 2019) for shortest path tree, our algorithm runs in $O(\frac{n^2}{s} \log^3 n)$ expected time.