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DOI: 10.4230/LIPIcs.ISAAC.2025.3

Circle-Segment Intersection Queries in Connected Geometric Graphs

Authors: Peyman Afshani, Yannick Bosch, and Sabine Storandt

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

In this paper, we study the problem of efficiently reporting all intersections between a given set of line segments in the plane and a query circle, focusing on the case where the segments form the edges of a connected geometric graph. While previous data structures for circle-segment intersection queries on general segment sets incur high space or query time costs, we exploit the connectivity of the input to obtain significantly improved performance. In fact, we propose a new circle-segment intersection data structure that can be constructed in 𝒪((n + C) log³ n) time and space on connected graphs with n edges and C edge crossings. It answers intersection queries in 𝒪(k log³ n) time, where k denotes the output size. Our method relies on the construction of efficient circle-graph intersection oracles as well as a novel linear-time algorithm to partition the edges of the graph into balanced, connected components, which might be of independent interest. In a proof-of-concept experimental study on real-world road networks, we show that our novel data structure also performs well in practice. Even on networks with millions of edges, the construction time is within minutes and queries are answered in a few milliseconds.

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Peyman Afshani, Yannick Bosch, and Sabine Storandt. Circle-Segment Intersection Queries in Connected Geometric Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{afshani_et_al:LIPIcs.ISAAC.2025.3,
  author =	{Afshani, Peyman and Bosch, Yannick and Storandt, Sabine},
  title =	{{Circle-Segment Intersection Queries in Connected Geometric Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.3},
  URN =		{urn:nbn:de:0030-drops-249114},
  doi =		{10.4230/LIPIcs.ISAAC.2025.3},
  annote =	{Keywords: Intersection data structure, Graph partitioning, Dobkin-Kirkpatrick hierarchy}
}

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DOI: 10.4230/LIPIcs.SEA.2025.7

Continuous Map Matching to Paths Under Travel Time Constraints

Authors: Yannick Bosch and Sabine Storandt

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract

In this paper, we study the problem of map matching with travel time constraints. Given a sequence of k spatio-temporal measurements and an embedded path graph with travel time costs, the goal is to snap each measurement to a close-by location in the graph, such that consecutive locations can be reached from one another along the path within the timestamp difference of the respective measurements. This problem arises in public transit data processing as well as in map matching of movement trajectories to general graphs. We show that the classical approach for this problem, which relies on selecting a finite set of candidate locations in the graph for each measurement, cannot guarantee to find a consistent solution. We propose a new algorithm that can deal with an infinite set of candidate locations per measurement. We prove that our algorithm always detects a consistent map matching path (if one exists). Despite the enlarged candidate set, we also demonstrate that our algorithm has superior running time in theory and practice. For a path graph with n nodes, we show that our algorithm runs in 𝒪(k² n log {nk}) and under mild assumptions in 𝒪(k n ^λ + n log³ n) for λ ≈ 0.695. This is a significant improvement over the baseline, which runs in 𝒪(k n²) and which might not even identify a correct solution. The performance of our algorithm hinges on an efficient segment-circle intersection data structure. We describe how to design and implement such a data structure for our application. In the experimental evaluation, we demonstrate the usefulness of our novel algorithm on a diverse set of generated measurements as well as GTFS data.

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Yannick Bosch and Sabine Storandt. Continuous Map Matching to Paths Under Travel Time Constraints. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bosch_et_al:LIPIcs.SEA.2025.7,
  author =	{Bosch, Yannick and Storandt, Sabine},
  title =	{{Continuous Map Matching to Paths Under Travel Time Constraints}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.7},
  URN =		{urn:nbn:de:0030-drops-232457},
  doi =		{10.4230/LIPIcs.SEA.2025.7},
  annote =	{Keywords: Map matching, Travel time, Segment-circle intersection data structure}
}

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Documents authored by Bosch, Yannick

Circle-Segment Intersection Queries in Connected Geometric Graphs

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Continuous Map Matching to Paths Under Travel Time Constraints

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