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Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments

Authors Pankaj K. Agarwal , Haim Kaplan , Matthew J. Katz , Micha Sharir

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • segment proximity graphs
  • nearest neighbor searching
  • dynamic data structures
  • BFS
  • DFS
  • unit-disk graphs

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Abstract

In this paper we study a few proximity problems related to a set of pairwise-disjoint segments in {ℝ}². Let S be a set of n pairwise-disjoint segments in {ℝ}², and let r > 0 be a parameter. We define the segment proximity graph of S to be G_r(S) := (S,E), where E = {(e₁,e₂) ∣ dist(e₁,e₂) ≤ r} and dist (e₁,e₂) = min_{(p,q) ∈ e₁× e₂} ‖p-q‖ is the Euclidean distance between e₁ and e₂. We define the weight of an edge (e₁,e₂) ∈ E to be dist(e₁,e₂). 
We first present a simple grid-based O(nlog² n)-time algorithm for computing a BFS tree of G_r(S). We apply it to obtain an O^*(n^{6/5}) + O(nlog²nlogΔ)-time algorithm for the so-called reverse shortest path problem, in which we want to find the smallest value r^* for which G_{r^*}(S) contains a path of some specified length between two designated start and target segments (where the O^*(⋅) notation hides polylogarithmic factors). Here Δ = max_{e ≠ e' ∈ S} dist(e,e')/min_{e ≠ e' ∈ S} dist(e,e') is the spread of S.
Next, we present a dynamic data structure that can maintain a set S of pairwise-disjoint segments in the plane under insertions/deletions, so that, for a query segment e from an unknown set Q of pairwise-disjoint segments, such that e does not intersect any segment in (the current version of) S, the segment of S closest to e can be computed in O(log⁵ n) amortized time. The amortized update time is also O(log⁵ n). We note that if the segments in S∪Q are allowed to intersect then the known lower bounds on halfplane range searching suggest that a sequence of n updates and queries may take at least close to Ω(n^{4/3}) time. One thus has to strongly rely on the non-intersecting property of S and Q to perform updates and queries in O(polylog(n)) (amortized) time each.
Using these results on nearest-neighbor (NN) searching for disjoint segments, we show that a DFS tree (or forest) of G_r(S) can be computed in O^*(n) time. We also obtain an O^*(n)-time algorithm for constructing a minimum spanning tree of G_r(S).
Finally, we present an O^*(n^{4/3})-time algorithm for computing a single-source shortest-path tree in G_r(S). This is the only result that does not exploit the disjointness of the input segments.

Cite As Get BibTex

Pankaj K. Agarwal, Haim Kaplan, Matthew J. Katz, and Micha Sharir. Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.7

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Haim Kaplan
  • School of Computer Science, Tel Aviv University, Israel
Matthew J. Katz
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Agarwal, Pankaj K.: Partially supported by NSF grants IIS-18-14493, CCF-20-07556, and CCF-22-23870.
  • Kaplan, Haim: Partially supported by Israel Science Foundation Grant 1156/23 and the Blavatnik research Foundation.
  • Katz, Matthew J.: Partially supported by Israel Science Foundation Grant 495/23.
  • Sharir, Micha: Partially supported by Israel Science Foundation Grant 495/23.

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