Spanner for the $0/1/\infty$ weighted region problem

TL;DR

This paper introduces a data structure for efficiently approximating shortest paths in a plane with regions of zero, infinite, or unit weights, improving computation times for complex weighted subdivision problems.

Contribution

The authors present a novel data structure that efficiently approximates shortest paths in weighted regions with weights in {0, 1, ∞}, including a $(1 + ext{ε})$-spanner for polygonal regions.

Findings

Expected construction time is $O(N + (n/ε^3)( ext{log}(n/ε) + ext{log} N))$.

Approximate shortest path can be computed in $O(N + n/ε^3 + (n/ε^2) ext{log}(n/ε) + ( ext{log} N)/ε)$ time.

Contains a $(1 + ε)$-spanner for polygonal regions, enabling efficient path approximation.

Abstract

We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$. We present a data structure $B$, which stores a set of convex, non-overlapping regions. These include zero-cost regions (0-regions) with a weight of $0$ and obstacles with a weight of $\infty$, all embedded in a plane with a weight of $1$. The data structure $B$ can be constructed in expected time $O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$, where $n$ is the total number of regions, $N$ represents the total complexity of the regions, and $1 + \varepsilon$ is the approximation factor, for any $0 < \varepsilon < 1$. Using $B$, one can compute an approximate weighted shortest path from any point $s$ to any point $t$ in $O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)$ time. In the special case where the 0-regions and obstacles are polygons (not necessarily convex), $B$ contains a $(1 + \varepsilon)$-spanner of the input vertices.