Faster and More Robust Mesh-based Algorithms for Obstacle k-Nearest Neighbour

TL;DR

This paper introduces faster, more robust mesh-based algorithms for obstacle-aware k-nearest neighbor searches in the plane, improving efficiency and consistency over previous visibility graph methods.

Contribution

The authors develop new algorithms and heuristics for obstacle k-NN that perform reliably across different neighbor densities, overcoming limitations of prior methods.

Findings

Outperforms previous visibility graph-based algorithms in speed

Maintains robustness across varying neighbor densities

Reduces redundant node expansions in pathfinding

Abstract

We are interested in the problem of finding $k$ nearest neighbours in the plane and in the presence of polygonal obstacles ($\textit{OkNN}$). Widely used algorithms for OkNN are based on incremental visibility graphs, which means they require costly and online visibility checking and have worst-case quadratic running time. Recently $\mathbf{Polyanya}$, a fast point-to-point pathfinding algorithm was proposed which avoids the disadvantages of visibility graphs by searching over an alternative data structure known as a navigation mesh. Previously, we adapted $\mathbf{Polyanya}$ to multi-target scenarios by developing two specialised heuristic functions: the $\mathbf{Interval heuristic}$ $h_v$ and the $\mathbf{Target heuristic}$ $h_t$. Though these methods outperform visibility graph algorithms by orders of magnitude in all our experiments they are not robust: $h_v$ expands many redundant nodes when the set of neighbours is small while $h_t$ performs poorly when the set of neighbours is large. In this paper, we propose new algorithms and heuristics for OkNN which perform well regardless of neighbour density.