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Improved Algorithms for Distance Selection and Related Problems

Haitao Wang, Yiming Zhao

TL;DR

The paper develops deterministic and randomized algorithmic frameworks for geometric optimization problems whose optimal value equals an interpoint distance. Central to the approach is a partial batched range searching (BRS) technique built on hierarchical cuttings, enabling subquadratic time improvements, notably a deterministic $O(n^{4/3} \log n)$ solution for distance selection. These frameworks yield broad improvements: faster two-sided and one-sided discrete Fréchet distance with shortcuts, and faster reverse shortest paths in unit-disk graphs, plus a generalizable template for distance-based optimization problems. The results combine careful geometric partitioning with expanders and bifurcation-tree strategies to minimize calls to decision procedures and to manage uncertain pairs efficiently, offering practical impact for a wide class of distance-related geometric tasks.

Abstract

In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the distance selection problem is to find the $k$-th smallest interpoint distance among all pairs of points of $P$. The previously best deterministic algorithm solves the problem in $O(n^{4/3} \log^2 n)$ time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to $O(n^{4/3} \log n)$ time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fréchet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly $\log^2(m+n)$ (resp., $(m+n)^ε$), where $m$ and $n$ are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.

Improved Algorithms for Distance Selection and Related Problems

TL;DR

The paper develops deterministic and randomized algorithmic frameworks for geometric optimization problems whose optimal value equals an interpoint distance. Central to the approach is a partial batched range searching (BRS) technique built on hierarchical cuttings, enabling subquadratic time improvements, notably a deterministic solution for distance selection. These frameworks yield broad improvements: faster two-sided and one-sided discrete Fréchet distance with shortcuts, and faster reverse shortest paths in unit-disk graphs, plus a generalizable template for distance-based optimization problems. The results combine careful geometric partitioning with expanders and bifurcation-tree strategies to minimize calls to decision procedures and to manage uncertain pairs efficiently, offering practical impact for a wide class of distance-related geometric tasks.

Abstract

In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set of points in the plane and an integer , the distance selection problem is to find the -th smallest interpoint distance among all pairs of points of . The previously best deterministic algorithm solves the problem in time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fréchet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly (resp., ), where and are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.
Paper Structure (21 sections, 12 theorems, 16 equations, 3 figures)

This paper contains 21 sections, 12 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two-sided DFD.
  3. One-sided DFD.
  4. Reverse shortest paths in unit-disk graphs.
  5. Recap.
  6. Outline.
  7. Partial batched range searching
  8. Distance selection
  9. Proof of the claim.
  10. The set $\mathcal{G}_1$.
  11. The set $\mathcal{G}_2$.
  12. Summary.
  13. A bipartite version.
  14. Two-sided discrete Fréchet distance with shortcuts
  15. Improving the decision algorithm.
  16. ...and 6 more sections

Key Result

Lemma 1

For any $r$ with $1\leq r\leq \min\{m^{1/3},n^{1/3}\}$, we can compute in $O(mr \log r + nr)$ time two collections $\Gamma(A, B, \alpha, \beta)= \{A_t \times B_t\ |\ A_t \subseteq A, B_t \subseteq B\}$ and $\Pi(A, B, \alpha, \beta) = \{A'_s \times B'_s\ |\ A'_s \subseteq A, B'_s \subseteq B\}$ of ed

Figures (3)

  • Figure 1: An example of Problem \ref{['prob:BRS']} with $\Gamma = \{ \{a_1, a_2\} \times \{b_1\}, \{a_4, a_5\} \times \{b_1, b_2\}\}$ and $\Pi = \{\{a_3, a_6, a_7\} \times \{b_2\}\}$. Note that the uncertain pair $(a_3, b_2)$ has a distance $\lVert a_3 b_2 \rVert \notin (\alpha, \beta]$.
  • Figure 2: Illustrating an annulus $D_p$ (the grey region).
  • Figure 3: Illustrating a pseudo-trapezoid.

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Lemma 5
  • ...and 2 more