Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

TL;DR

This work presents the first substantial space reductions for two-point geodesic shortest-path queries in polygonal domains while preserving fast query times. The core approach combines augmented shortest-path maps, a low-envelopes framework for the minimum over region-pair functions, and a multilevel cutting-tree construction built on $1/r$-cuttings of Tarski cells. The main results deliver $O(n^{10+\varepsilon})$ space and $O(\log n)$ query time (also $O(n^{9+\varepsilon})$ with $O(\log^2 n)$), plus a flexible space-time trade-off and improved boundary-case bounds, with further reductions in special boundary-restricted scenarios. These advances enhance feasibility for geodesic-diameter and geodesic-center computations and suggest directions for tighter lower bounds and further space reductions in polygonal-domain query structures.

Abstract

We devise a data structure that can answer shortest path queries for two query points in a polygonal domain $P$ on $n$ vertices. For any $\varepsilon > 0$, the space complexity of the data structure is $O(n^{10+\varepsilon })$ and queries can be answered in $O(\log n)$ time. Alternatively, we can achieve a space complexity of $O(n^{9+\varepsilon })$ by relaxing the query time to $O(\log^2 n)$. This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They present a data structure with $O(n^{11})$ space complexity and $O(\log n)$ query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with $O(n^{9+\varepsilon}/\hspace{1pt} \ell^{4 + O(\varepsilon )})$ space complexity and $O(\ell \log^2 n )$ query time, for any integer $1 \leq \ell \leq n$. Furthermore, we present improved data structures with $O(\log n)$ query time for the special case where we restrict one (or both) of the query points to lie on the boundary of $P$. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes $O(n^{6+\varepsilon})$. When both query points are on the boundary, the space complexity is decreased further to $O(n^{4+\varepsilon })$, thereby improving an earlier result of Bae and Okamoto.

Figures (7)

• Figure 1: A tangible example of a two-point shortest path problem: finding the shortest path among islands for a boat to an emergency.
• Figure 2: Given $P$ and the query points $s,t$ we want to compute the shortest path efficiently.
• Figure 3: The augmented shortest path map of a vertex $v$. The shortest path map edges are solid, and the additional edges in the augmented shortest path map are dotted. Each region is bounded by three curves, of which at least two are line segments. Two regions and their apices are highlighted.
• Figure 4: Two pairs of relevant regions in red and blue with the path whose length is $f_{ST}(s,t)$.
• Figure 5: Overview of our data structure. The first level cutting tree (red) is built by recursively constructing a cutting $\Xi\xspace$ on the (orange) regions that intersect a cell $\Delta$ (purple). For each cell $\Delta$, we store a second level cutting tree (blue). For each cell $\Delta'$ in $\Xi_\Delta\xspace$, we build a Lower Envelope data structure on all regions that fully contain $\Delta$ (dark red) and $\Delta'$ (dark blue).

Theorems & Definitions (9)

• Theorem 1: Main Theorem
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10