Spanner for the $0/1/\infty$ weighted region problem
<|reference_start|>Spanner for the $0/1/\infty$ weighted region problem: 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.<|reference_end|>
@article{gudmundsson2024spanner,
    title={Spanner for the $0/1/\infty$ weighted region problem},
    author={Joachim Gudmundsson, Zijin Huang, Andr'e van Renssen, Sampson Wong},
    journal={arXiv preprint arXiv:2407.01951},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.01951},
    primaryClass={cs.CG cs.DS}
}