[Paper Review] Shortest Path in a Polygon using Sublinear Space

Sariel Har-Peled|arXiv (Cornell University)|Dec 2, 2014

Computational Geometry and Mesh Generation13 references3 citations

TL;DR

This paper presents a novel algorithm for computing the shortest path in a simple polygon using sublinear space and subquadratic time. By decomposing the polygon into O(n/m) corridors of size O(m), and using a modified, constant-space variant of Seidel's linear programming algorithm on violator spaces, the method achieves O(n²/m) expected time with O(m) working memory, resolving an open problem posed by Asano on avoiding quadratic preprocessing.

ABSTRACT

$ enewcommand{\Re}{{ m I\!\hspace{-0.025em} R}} ewcommand{\SetX}{\mathsf{X}} ewcommand{\VorX}[1]{\mathcal{V} \pth{#1}} ewcommand{\Polygon}{\mathsf{P}} ewcommand{\Space}{\overline{\mathsf{m}}} ewcommand{\pth}[2][\!]{#1\left({#2} ight)}$ We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon $\Polygon$ with $n$ vertices in a read only memory, and additional working memory of size $\Space$, the new algorithm computes the shortest path (in $\Polygon$) in $O( n^2 /\, \Space )$ expected time. This requires several new tools, which we believe to be of independent interest.

Motivation & Objective

• To resolve Tetsuo Asano's open problem of computing shortest paths in polygons without quadratic preprocessing time.
• To design an algorithm that uses O(m) working memory and subquadratic time, where m < n/log²n.
• To enable efficient shortest path computation in constrained memory settings, such as embedded systems or streaming environments.
• To develop a space-efficient solution for violator space problems, applicable to geometric optimization.

Proposed method

• Decomposes the polygon into O(n/m) corridors, each of size O(m), using a corridor decomposition based on the medial axis.
• Uses a modified version of Seidel's randomized linear programming algorithm to solve violator space problems in expected linear time with O(1) space.
• Employs pseudo-random sequences in place of true random permutations to maintain constant space while preserving expected performance.
• Applies backward analysis to the modified algorithm, proving correctness under pseudo-random sequences.
• Walks through corridors iteratively, maintaining a funnel structure and using funnel reduction to track the shortest path.
• Uses a point location query in canonical decomposition to determine the next corridor, leveraging the violator space framework.

Experimental results

Research questions

• RQ1 Can the shortest path in a polygon be computed in subquadratic time using only O(m) working memory, without quadratic preprocessing?
• RQ2 Can violator space problems be solved in expected linear time using only O(1) space?
• RQ3 Does backward analysis remain valid when using pseudo-random sequences instead of true random permutations?
• RQ4 Can corridor decomposition be used to reduce the space-time trade-off in geometric shortest path computation?

Key findings

• The algorithm computes the shortest path in O(n²/m) expected time using O(m) working memory, for m = O(n/log²n).
• Violator space problems can be solved in expected linear time using only O(1) space by modifying Seidel's algorithm and using pseudo-random sequences.
• Backward analysis remains valid when using pseudo-random sequences, ensuring correctness and expected performance.
• The funnel reduction step requires O(log m) queries per corridor, but can be optimized to constant queries per corridor by stopping at size O(m/4).
• The overall running time is O(n²/m + n log m log⁴(n/m)) for general m, improving to O(n²/m) when m = O(n/log²n).
• The approach resolves Asano's open problem by eliminating the need for quadratic preprocessing while maintaining subquadratic time complexity.