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Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-and-Bifurcate Technique

Timothy M. Chan, Zhengcheng Huang

TL;DR

This work addresses a class of geometric optimization problems where parametric search is bottlenecked by non-parallelizable decision steps. It advances the shrink-and-bifurcate framework by introducing a simpler, more scalable interval shrinking method that reduces the number of critical values, enabling a faster bifurcation-based simulation. Across multiple problems including reverse shortest path in unit-disk graphs, segment proximity graphs, and visibility graphs on 1.5D terrains, it achieves a unified improvement to $\tilde{O}(n^{8/7})$ and, in the unweighted unit-disk case, to $\tilde{O}(n^{9/8})$ with Las Vegas randomness. The techniques are anchored in a refined interval shrinking strategy and a careful combination with existing decision algorithms, offering a broadly applicable framework for geometric optimization where decision subproblems resist parallel speedups.

Abstract

In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems -- including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains -- for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in $O^*(n^{6/5})=O^*(n^{1.2})$ time (ignoring subpolynomial factors), and they were obtained using a technique called \emph{shrink-and-bifurcate}. We improve the running time to $\tilde{O}(n^{8/7}) \approx O(n^{1.143})$ for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to $\tilde{O}(n^{9/8})=\tilde{O}(n^{1.125})$.

Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-and-Bifurcate Technique

TL;DR

This work addresses a class of geometric optimization problems where parametric search is bottlenecked by non-parallelizable decision steps. It advances the shrink-and-bifurcate framework by introducing a simpler, more scalable interval shrinking method that reduces the number of critical values, enabling a faster bifurcation-based simulation. Across multiple problems including reverse shortest path in unit-disk graphs, segment proximity graphs, and visibility graphs on 1.5D terrains, it achieves a unified improvement to and, in the unweighted unit-disk case, to with Las Vegas randomness. The techniques are anchored in a refined interval shrinking strategy and a careful combination with existing decision algorithms, offering a broadly applicable framework for geometric optimization where decision subproblems resist parallel speedups.

Abstract

In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems -- including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains -- for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in time (ignoring subpolynomial factors), and they were obtained using a technique called \emph{shrink-and-bifurcate}. We improve the running time to for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to .

Paper Structure

This paper contains 14 sections, 7 theorems, 7 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Shrink-and-Bifurcate Technique
  3. New Results
  4. Interval Shrinking
  5. First Version
  6. Improved Version
  7. Combining with Bifurcation
  8. Reverse Shortest Path in Unweighted Unit-Disk Graphs
  9. Recap of Chan and Skrepetos' Decision Algorithm
  10. Heavy vs. Light Cells
  11. Faster Interval Shrinking for Light Pairs
  12. Computing Distances with $O(n/\Delta)$ Additive Error
  13. Recovering $r^*$
  14. Reverse Shortest Path in Visibility Graphs in 1.5-Dimensional Terrains

Key Result

Theorem 2

Subproblem prb:interval-shrinking (interval shrinking) can be solved in $\tilde{O}(n^{2\alpha}/L^{\alpha} + D(n))$ time w.h.p.

Figures (1)

  • Figure 1: Consider two vertical slabs $\Pi_L$ and $\Pi_R$ divided by a vertical line, two points $p$ and $q$ on $T$ lying in $\Pi_L$ and $\Pi_R$ respectively, and a height $h$. This figure shows the critical rays $\overrightarrow{p(h)v_{p,\Pi_L,h}}$ in blue and $\overrightarrow{q(h)v_{q,\Pi_R,h}}$ in red. In this figure, both critical rays are below the segment $\overline{p(h)q(h)}$, so $p(h)$ and $q(h)$ see each other.

Theorems & Definitions (7)

  • Theorem 2
  • Lemma 3: Bifurcation AvrahamFKKS15
  • Theorem 4
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Theorem 9