An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains

TL;DR

This paper introduces a new algorithm that efficiently finds minimum-link rectilinear paths in polygonal domains with obstacles, improving preprocessing time and enabling quick path queries.

Contribution

The authors present an algorithm that constructs a data structure in O(n+h log h) time, allowing fast minimum-link path queries in rectilinear polygonal domains.

Findings

Data structure construction in O(n+h log h) time

Query time for path edges is O(log n)

Path output time is linear in path length

Abstract

We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $\mathcal{P}$ of $h$ pairwise-disjoint rectilinear polygonal obstacles with a total of $n$ vertices in the plane, a minimum-link rectilinear path between two points is a rectilinear path that avoids all obstacles with the minimum number of edges. In this paper, we present a new algorithm for finding minimum-link rectilinear paths among $\mathcal{P}$. After the plane is triangulated, with respect to any source point $s$, our algorithm builds an $O(n)$-size data structure in $O(n+h\log h)$ time, such that given any query point $t$, the number of edges of a minimum-link rectilinear path from $s$ to $t$ can be computed in $O(\log n)$ time and the actual path can be output in additional time linear in the number of the edges of the path. The previously best algorithm computes such a data structure in $O(n\log n)$ time.