[Paper Review] A linear-time algorithm for the geodesic center of a simple polygon

Hee-Kap Ahn, Luis Barba+4 more|arXiv (Cornell University)|Jan 3, 2015

Computational Geometry and Mesh Generation21 references4 citations

TL;DR

This paper presents the first linear-time algorithm for computing the geodesic center of a simple polygon with n vertices, achieving O(n) time complexity—resolving a longstanding open problem since Pollack et al.'s O(n log n) algorithm in 1989. The approach uses a novel combination of planar cuttings, ε-nets, and upper envelope computation over O(n) apexed triangular functions to recursively narrow down the search space while maintaining linear work per iteration.

ABSTRACT

Given two points in a simple polygon $P$ of $n$ vertices, its geodesic distance is the length of the shortest path that connects them among all paths that stay within $P$. The geodesic center of $P$ is the unique point in $P$ that minimizes the largest geodesic distance to all other points of $P$. In 1989, Pollack, Sharir and Rote [Disc. \& Comput. Geom. 89] showed an $O(n\log n)$-time algorithm that computes the geodesic center of $P$. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

Motivation & Objective

• To resolve the long-standing open problem of whether the geodesic center of a simple polygon can be computed in linear time.
• To improve upon the O(n log n) time bound established by Pollack, Sharir, and Rote in 1989.
• To develop a method that avoids iterative refinement on triangulations and instead uses geometric cuttings and function envelopes for efficient search.
• To prove that the geodesic center can be computed in O(n) time by reducing the problem size by a constant fraction in each recursive step.

Proposed method

• The algorithm constructs O(n) apexed triangular functions whose upper envelope matches the farthest-point geodesic distance function F_P(x) in the polygon.
• It uses planar cuttings of O(n) planes derived from chord boundaries and separating planes between apexed triangles to partition the search space.
• An ε-net-based technique ensures that each recursive step reduces both the number of active functions and the size of the candidate region by a constant factor.
• A side-decision algorithm based on restricted optimization on planes (inspired by Megiddo) determines which side of a plane contains the geodesic center.
• The recursion terminates when the problem reduces to a constant-size region or a single triangle, where convex optimization is applied to find the minimum.
• The overall recurrence T(m) = T(3m/4) + O(m) solves to O(m), and since m = O(n), the total time is O(n).

Experimental results

Research questions

• RQ1 Can the geodesic center of a simple polygon be computed in linear time, improving upon the O(n log n) bound from 1989?
• RQ2 Is it possible to avoid reliance on triangulation-based search structures and instead use geometric cuttings to reduce problem size?
• RQ3 Can the farthest-point geodesic distance function be represented as the upper envelope of O(n) triangular functions to enable efficient minimization?
• RQ4 How can a side-decision procedure be implemented efficiently to guide recursive search in 3D space using planar cuttings?

Key findings

• The geodesic center of any simple polygon with n vertices can be computed in O(n) time, achieving the first linear-time solution to this problem.
• The algorithm uses a novel approach based on planar cuttings and ε-nets to reduce the number of active functions and candidate regions by a constant factor per iteration.
• By constructing O(n) apexed triangular functions whose upper envelope equals F_P(x), the problem is reduced to minimizing a piecewise-defined convex function in a bounded region.
• The side-decision algorithm allows efficient determination of which side of a plane contains the geodesic center, enabling recursive refinement in 3D space.
• The recursive reduction leads to a recurrence T(m) = T(3m/4) + O(m), which solves to O(m), proving the overall linear time complexity.
• The method successfully avoids the O(n log n) bottleneck of prior approaches by replacing chord-oracle search on triangulations with a more efficient geometric decomposition.