Constant Workspace Algorithms for Computing Relative Hulls in the Plane

TL;DR

This paper introduces the first constant workspace algorithms for efficiently computing various types of relative hulls in the plane, addressing problems involving polygons and point sets with minimal memory use.

Contribution

It presents novel algorithms for computing relative hulls in the plane using only constant workspace, a significant advancement over prior methods.

Findings

Algorithms successfully compute minimal perimeter relative hulls.

First known constant workspace algorithms for these problems.

Applicable to polygons and point sets within polygons.

Abstract

Constant workspace algorithms use a constant number of words in addition to the read-only input to the algorithm. In this paper, we devise algorithms to efficiently compute relative hulls in the plane using a constant workspace. Specifically, we devise algorithms for the following three problems: (i) Given two simple polygons P and Q with P \subset Q, compute a simple polygon P' with a perimeter of minimum length such that P \subseteq P' \subseteq Q. (ii) Given two simple polygons P and Q such that Q does not intersect the relative interior of P but it does intersect the relative interior of the convex hull of P, compute a weakly simple polygon P' with a perimeter of minimum length such that P \subseteq P', the convex hull of P contains P', and P' does not intersect the relative interior of Q. (iii) Given a set S of points located in a simple polygon P, compute a weakly simple polygon P' with a perimeter of minimum length such that P' \subseteq P and P' contains all the points in S. To our knowledge, no prior work devised algorithms to compute relative hulls using a constant workspace, and this work is the first such attempt.