[Paper Review] Approximate Point-to-Face Shortest Paths in R^3

Yam Ki Cheung, Ovidiu Daescu|arXiv (Cornell University)|Apr 9, 2010

Computational Geometry and Mesh Generation20 references3 citations

TL;DR

This paper presents three approximation algorithms for computing (1+ε)-shortest paths from a point to a face in 3D space amid polyhedral obstacles. By extending point-to-point shortest path techniques and using visibility computation via plane sweeping and Steiner point insertion, the authors achieve an O(n²λ(n)log(n/ε)/ε⁴ + n²log(nρ)log(nlogρ)) time complexity, significantly improving upon prior methods that required repeated point-to-point queries.

ABSTRACT

We address the point-to-face approximate shortest path problem in R: Given a set of polyhedral obstacles with a total of n vertices, a source point s, an obstacle face f, and a real positive parameter epsilon, compute a path from s to f that avoids the interior of the obstacles and has length at most (1+epsilon) times the length of the shortest obstacle avoiding path from s to f. We present three approximation algorithms that take by extending three well-known "point-to-point" shortest path algorithms.

Motivation & Objective

• To address the point-to-face shortest path problem in R³ with polyhedral obstacles, where the target is a face rather than a point.
• To reduce the computational cost of approximate shortest path computation by avoiding repeated point-to-point shortest path queries.
• To develop a method that adds Steiner points directly to the visibility graph to maintain asymptotic efficiency.
• To achieve a time complexity that depends on ε⁻⁴ but remains polynomial, improving upon prior approaches requiring O(1/ε²) point-to-point computations.

Proposed method

• Extends Clarkson’s point-to-point shortest path algorithm by adding Steiner points on obstacle edges to capture visibility to the target face.
• Uses plane sweeping in 2D cross-sections (perpendicular to the target face) to compute visibility between Steiner points and their projections on the target face.
• Transforms the 3D visibility problem into O(n) 2D subproblems by considering each obstacle edge and its orthogonal projection onto the target plane.
• Applies a sweep line data structure with O(log n) update time per event to partition each edge into O(n) segments with uniform visibility properties.
• Introduces a visibility graph refinement step that adds new Steiner points only where visibility is blocked, ensuring minimal overhead.
• Employs ray shooting with fixed perpendicular direction (to the target face) to optimize visibility checks, reducing the cost below naive O(n) per pair.

Experimental results

Research questions

• RQ1 Can we compute an (1+ε)-approximate shortest path from a source point to a face in 3D polyhedral environments more efficiently than by repeated point-to-point queries?
• RQ2 What is the minimal set of Steiner points on obstacle edges that ensures visibility to the target face is correctly captured for path approximation?
• RQ3 Can visibility computation between Steiner points and their projections be optimized to sublinear time per pair using geometric decomposition?
• RQ4 How does the time complexity scale with ε⁻⁴ in the context of 3D point-to-face path approximation, and can it be bounded independently of the number of obstacle vertices?
• RQ5 Is it possible to avoid the O(1/ε²) factor from prior methods by modifying the visibility graph directly instead of re-running point-to-point algorithms?

Key findings

• The proposed algorithm achieves an O(n²λ(n)log(n/ε)/ε⁴ + n²log(nρ)log(nlogρ)) time complexity, which is independent of the number of point-to-point queries.
• The method avoids the O(1/ε²) factor inherent in prior approaches by directly extending Clarkson’s algorithm with targeted Steiner point insertion.
• Visibility computation is optimized via plane sweeping in 2D cross-sections, reducing the cost per visibility check to O(log n) per event.
• The number of Steiner points added is bounded by O(n²) per edge, and the total time for visibility and Steiner point computation is O(n²log n + N), where N is the number of Steiner points.
• The algorithm maintains the same asymptotic complexity as the underlying point-to-point method, making it the first to achieve this for the point-to-face problem.
• The approach is robust to the ratio ρ of the longest edge to the source-to-face distance, with logarithmic dependence in the final time bound.