## 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 Gr(S):= (S,E), where E = {(e1,e2) | dist(e1,e2) ≤ r} and dist(e1,e2) = min_{(p,q)}∈e_{1×}e_{2} ∥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(nlog^{2} n)-time algorithm for computing a BFS tree of Gr(S). We apply it to obtain an O^{∗}(n^{6/5})+O(nlog^{2} nlog ∆)-time algorithm for the so-called reverse shortest path problem, in which we want to find the smallest value r^{∗} for which Gr∗(S) contains a path of some specified length between two designated start and target segments (where the O^{∗}(·) notation hides polylogarithmic factors). Here ∆ = 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(log^{5} n) amortized time. The amortized update time is also O(log^{5} 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 Gr(S) can be computed in O^{∗}(n) time. We also obtain an O^{∗}(n)-time algorithm for constructing a minimum spanning tree of Gr(S). Finally, we present an O^{∗}(n^{4/3})-time algorithm for computing a single-source shortest-path tree in Gr(S). This is the only result that does not exploit the disjointness of the input segments.

Original language | English |
---|---|

Title of host publication | 32nd Annual European Symposium on Algorithms, ESA 2024 |

Editors | Timothy Chan, Johannes Fischer, John Iacono, Grzegorz Herman |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773386 |

DOIs | |

State | Published - 1 Sep 2024 |

Event | 32nd Annual European Symposium on Algorithms, ESA 2024 - London, United Kingdom Duration: 2 Sep 2024 → 4 Sep 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 308 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd Annual European Symposium on Algorithms, ESA 2024 |
---|---|

Country/Territory | United Kingdom |

City | London |

Period | 2/09/24 → 4/09/24 |

## Keywords

- BFS
- DFS
- dynamic data structures
- nearest neighbor searching
- segment proximity graphs
- unit-disk graphs

## ASJC Scopus subject areas

- Software