On reverse shortest paths in geometric proximity graphs

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in R2, and let ϱ:S×S→R≥0 be a distance function on S. For a parameter r≥0, we define the proximity graph G(r)=(S,E) where E={(e1,e2)∈S×S|e1≠e2,ϱ(e1,e2)≤r}. Given S, s,t∈S, and an integer k≥1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r≥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≥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(n2) 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(n4/3) expected-time randomized algorithm (where O(⋅) hides polylog(n) factors) for the case where S is a set of (possibly intersecting) line segments in R2 and ϱ(e1,e2)=minx∈e1,y∈e2⁡‖x−y‖ (where ‖⋅‖ is the Euclidean distance), and (ii) an O(n+m4/3) 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∈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 pq lie above or on the polygonal chain T.

Original languageEnglish
Article number102053
JournalComputational Geometry: Theory and Applications
Volume117
DOIs
StatePublished - Feb 2024

Keywords

  • Geometric optimization
  • Proximity graphs
  • Range searching
  • Reverse shortest paths
  • Semi-algebraic sets

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