Memory-Constrained Algorithms for Simple Polygons

TL;DR

This paper presents algorithms for triangulating simple polygons and preprocessing for shortest path queries under strict memory constraints, achieving quadratic time complexity with minimal workspace.

Contribution

It introduces memory-constrained algorithms for polygon triangulation and shortest path preprocessing, optimizing performance within strict space limitations.

Findings

Triangulation achieved in O(n^2) time with O(1) extra space.

Preprocessing for shortest paths takes O(n^2) time, enabling queries in O(n^2/s) time.

Algorithms operate under read-only input and minimal workspace constraints.

Abstract

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of $O(\log n)$ bits, where $n$ is the size of the input. We assume that a simple $n$-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in $O(n^2)$ time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow $s$ words of working space. After a preprocessing of $O(n^2)$ time, we are able to solve shortest path queries between any two points inside the polygon in $O(n^2/s)$ time.