[Paper Review] Minimum-Link Paths Revisited

Joseph S. B. Mitchell, Valentin Polishchuk+1 more|arXiv (Cornell University)|Feb 13, 2013

Computational Geometry and Mesh Generation25 references4 citations

TL;DR

This paper presents a subquadratic-time O(√h)-approximation algorithm for minimum-link paths in general polygonal domains with h holes, and introduces an optimal O(C²n log n)-time, O(Cn)-space algorithm for computing C-oriented minimum-link paths in C-oriented domains. It also establishes 3SUM-hardness of exact minimum-link path computation and shows that C-oriented paths approximate robust paths within an additive error of 1.

ABSTRACT

A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case (C = 2). We study exact and approximation algorithms for minimum-link C-oriented paths and paths with unrestricted orientations, both in C-oriented and in general domains. Our two main algorithms are as follows: A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains. An algorithm to find a minimum-link C-oriented path in a C-oriented domain. Our algorithm is simpler and more time-space efficient than the prior algorithm. We also obtain several related results: - 3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain). - An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones. - An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain. - A more efficient algorithm to compute a 2-approximate C-oriented minimum-link path. - A notion of "robust" paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1.

Motivation & Objective

• To develop faster and more space-efficient algorithms for computing minimum-link paths in polygonal domains with holes.
• To address the long-standing open question of whether a subquadratic algorithm exists for general minimum-link path computation.
• To investigate the approximation quality of C-oriented paths for paths with unrestricted orientations.
• To construct efficient data structures for approximate link distance queries with bounded additive error.
• To resolve the complexity of the minimum-link path problem by proving 3SUM-hardness for exact computation.

Proposed method

• Proposes a subquadratic-time O(√h)-approximation algorithm for general minimum-link paths in domains with h holes, using visibility and geometric partitioning techniques.
• Introduces a novel O(C²n log n)-time, O(Cn)-space algorithm for C-oriented minimum-link paths in C-oriented domains, improving upon prior work in both time and space complexity.
• Employs a geometric transformation to convert φ-robust paths into Cφ-oriented paths with at most one additional link, enabling robust path approximation.
• Uses elementary data structures (e.g., range trees, visibility graphs) to simplify and optimize rectilinear path computation, achieving optimal bounds.
• Constructs a data structure of size O(n/φ² log n) and query time O(log n / φ) to answer φ-robust link distance queries with additive error 1.
• Reduces the problem of finding minimum-link paths to visibility and geometric sweep operations, enabling efficient preprocessing and query support.

Experimental results

Research questions

• RQ1 Can a subquadratic-time algorithm be designed for the general minimum-link path problem in polygonal domains with holes?
• RQ2 Is the exact computation of minimum-link paths 3SUM-hard, even in rectilinear domains?
• RQ3 How well can C-oriented paths approximate minimum-link paths with unrestricted orientations?
• RQ4 Can efficient data structures be built to support approximate link distance queries with bounded additive error?
• RQ5 What is the optimal time and space complexity for computing minimum-link rectilinear paths in rectilinear domains?

Key findings

• The exact computation of minimum-link paths with unrestricted orientations is 3SUM-hard, even in rectilinear domains, resolving an open problem in computational geometry.
• A subquadratic O(√h)-approximation algorithm is presented for general minimum-link paths in polygonal domains with h holes, improving upon the previous O(n²) bound.
• An optimal O(C²n log n)-time, O(Cn)-space algorithm is developed for computing C-oriented minimum-link paths in C-oriented domains, simplifying and improving prior approaches.
• The algorithm constructs exact or approximate link distance maps in the same time and space bounds, enabling efficient query processing.
• C-oriented paths approximate φ-robust paths within an additive error of 1, enabling the construction of a data structure for φ-robust link distance queries with O(log n / φ) query time.
• A 2-approximation algorithm for C-oriented paths runs in O(Cn log n) time and O(n) space, improving efficiency over prior methods.