A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-off Algorithms

TL;DR

This paper introduces a new method for subdividing simple polygons that optimizes time and space, leading to improved algorithms for shortest path, shortest path tree, and triangulation problems within polygons.

Contribution

It presents an $O(n^2/s)$-time $s$-workspace algorithm for polygon subdivision, enhancing existing time-space trade-offs for key geometric problems.

Findings

Improved time-space trade-offs for shortest path and triangulation.

New subdivision algorithm reduces complexity and resource usage.

Enhanced algorithms for shortest path tree computation.

Abstract

We are given a read-only memory for input and a write-only stream for output. For a positive integer parameter s, an s-workspace algorithm is an algorithm using only $O(s)$ words of workspace in addition to the memory for input. In this paper, we present an $O(n^2/s)$-time $s$-workspace algorithm for subdividing a simple polygon into $O(\min\{n/s,s\})$ subpolygons of complexity $O(\max\{n/s,s\})$. As applications of the subdivision, the previously best known time-space trade-offs for the following three geometric problems are improved immediately: (1) computing the shortest path between two points inside a simple $n$-gon, (2) computing the shortest path tree from a point inside a simple $n$-gon, (3) computing a triangulation of a simple $n$-gon. In addition, we improve the algorithm for the second problem even further.