Constant Workspace Algorithms for Computing Relative Hulls in the Plane

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.