关于几何邻近图中的逆最短路径

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2023-09-11 DOI:10.1016/j.comgeo.2023.102053

Pankaj K. Agarwal , Matthew J. Katz , Micha Sharir

引用次数: 0

摘要

设S是R2中n个恒定复杂度的几何对象（例如，点、线段、圆盘、椭圆）的集合，并且设ϱ：S×S→R≥0是S上的距离函数。对于参数R≥0，我们定义了邻近图G（R）=（S，E），其中E={（e1，e2）∈S×S|e1≠e2，ϱ（e1、e2）≤R}。给定S，S，t∈S，且整数k≥1，反最短路径（RSP）问题要求计算最小值r≥0，使得G（r）包含从S到t的最大长度为k的路径。使用标准的，有时更复杂的半代数范围搜索技术，我们首先给出了决策问题的一个有效算法，即，给定值r≥0，确定G（r）是否包含从s到t的路径，长度至多为k。接下来，我们调整我们的决策算法，并将其与随机抽样方法相结合来计算r，通过在包含r的O（n2）个候选“临界”值的隐式集合上有效地执行二进制搜索。我们通过将其应用于各种几何邻近图来说明我们的通用技术的多功能性。例如，我们得到了（i）一个O（n4/3）期望时间随机化算法（其中O（·）隐藏了polylog（n）因子），其中S是R2中的一组（可能相交）线段，并且ϱ（e1，e2）=minx∈e1，y∈e2⁡‖x−y‖（其中‖是欧几里得距离），以及（ii）当S是位于具有n个顶点的x单调多边形链T上的m个点的集合时的O（n+m4/3）期望时间随机化算法，并且对于p，q∈S，ϱ（p，q）是最小值h，使得点p′：=p+（0，h）和q′：=q+（0、h）彼此可见，即。，线段p′q′上的所有点都位于多边形链T之上或之上。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On reverse shortest paths in geometric proximity graphs

Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in $R^{2}$ , and let $ϱ : S \times S \to R_{\geq 0}$ be a distance function on S. For a parameter $r \geq 0$ , we define the proximity graph $G (r) = (S, E)$ where $E = {(e_{1}, e_{2}) \in S \times S | e_{1} \neq e_{2}, ϱ (e_{1}, e_{2}) \leq r}$ . Given S, $s, t \in S$ , and an integer $k \geq 1$ , the reverse-shortest-path (RSP) problem asks for computing the smallest value $r^{⁎} \geq 0$ such that $G (r^{⁎})$ contains a path from s to t of length at most k.

In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value $r \geq 0$ , determine whether $G (r)$ contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute $r^{⁎}$ , by efficiently performing a binary search over an implicit set of $O (n^{2})$ candidate ‘critical’ values that contains $r^{⁎}$ .

We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an $O^{⁎} (n^{4 / 3})$ expected-time randomized algorithm (where $O^{⁎} (\cdot)$ hides $polylog (n)$ factors) for the case where S is a set of (possibly intersecting) line segments in $R^{2}$ and $ϱ (e_{1}, e_{2}) = \min_{x \in e_{1}, y \in e_{2}} ‖ x - y ‖$ (where $‖ \cdot ‖$ is the Euclidean distance), and (ii) an $O^{⁎} (n + m^{4 / 3})$ expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and $ϱ (p, q)$ , for $p, q \in S$ , is the smallest value h such that the points $p^{'} : = p + (0, h)$ and $q^{'} : = q + (0, h)$ are visible to each other, i.e., all points on the segment $p^{'} q^{'}$ lie above or on the polygonal chain T.

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来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.