9 Search Results for "Wiratma, Lionov"

Document
DOI: 10.4230/LIPIcs.SEA.2026.1
Global Polyline Simplification Under the Fréchet Distance: Theory and Practice

Authors: Christian Abdullahad and Sabine Storandt

Published in: LIPIcs, Volume 371, 24th International Symposium on Experimental Algorithms (SEA 2026)

Abstract
Given an input polyline with n vertices, the global polyline simplification problem seeks a simplified polyline with the minimum number of vertices whose distance to the original polyline does not exceed a given bound. For the vertex-restricted variant, where the simplified polyline is required to be a subsequence of the input vertices, an algorithm with a running time of 𝒪(n³) was presented in previous work, using the Fréchet distance as the polyline similarity measure. A closely related variant is the local polyline simplification problem, in which the distance bound is required to hold for every individual shortcut segment replacing a sub-polyline. This condition implies that any locally valid simplification is also globally valid, whereas the converse does not hold. As a consequence, globally optimal simplifications may use substantially fewer vertices than locally optimal ones. Indeed, in previous work, instances were constructed in which the optimal global simplification is smaller by a constant factor. On the algorithmic side, optimal local simplifications can be computed significantly faster, namely in 𝒪(n² log n) under the Fréchet distance, and efficient heuristics are also available. This raises the question of which problem variant is more suitable for practical application. In this paper, we first show that there exist instances for which the optimal solution sizes of global and local polyline simplification differ by a factor in Θ(n), substantially strengthening the previously known constant-factor separation. We then present the first practical implementations of existing algorithms for global polyline simplification and experimentally evaluate their performance. To this end, we introduce several engineering techniques that considerably accelerate these algorithms. Moreover, we develop an implicit Fréchet framework that allows many Fréchet-related problems to be addressed in a weaker computational model. Within this framework, explicit geometric computations can be reduced to simple comparisons, resulting in significantly more robust implementations. Somewhat surprisingly, our experimental results reveal that, despite the large worst-case gap established by our theoretical result, the difference in solution size between optimal global and local simplifications is negligible in practice. Motivated by this observation, we propose a heuristic for global polyline simplification that is guaranteed to produce solutions of size equal to or smaller than the optimal local simplification. On a benchmark consisting of one million polylines, the heuristic yields suboptimal results on only eight while being significantly faster than the optimal algorithms.
Cite as

Christian Abdullahad and Sabine Storandt. Global Polyline Simplification Under the Fréchet Distance: Theory and Practice. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{abdullahad_et_al:LIPIcs.SEA.2026.1,
  author =	{Abdullahad, Christian and Storandt, Sabine},
  title =	{{Global Polyline Simplification Under the Fr\'{e}chet Distance: Theory and Practice}},
  booktitle =	{24th International Symposium on Experimental Algorithms (SEA 2026)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-422-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{371},
  editor =	{Aum\"{u}ller, Martin and Finocchi, Irene},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2026.1},
  URN =		{urn:nbn:de:0030-drops-260055},
  doi =		{10.4230/LIPIcs.SEA.2026.1},
  annote =	{Keywords: Polyline Simplification, Shortcut Graph, Fr\'{e}chet Distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.17
Fréchet Distance in the Imbalanced Case

Authors: Lotte Blank

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Given two polygonal curves P and Q defined by n and m vertices with m ≤ n, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of 2-ε in 𝒪((nm)^{1-δ}) time for any ε, δ > 0 unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to 1+√2-ε (resp. 3-ε) if the curves lie in the Euclidean space (resp. in the L_∞-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where m = n^α for α ∈ (0,1) and increases the approximation factor of 1.001 by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any L_p space, we present a (3+ε)-approximation algorithm for the continuous and discrete Fréchet distance using 𝒪((n+m²)log n) time, which almost matches the approximation factor of the lower bound for the L_∞ metric.
Cite as

Lotte Blank. Fréchet Distance in the Imbalanced Case. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{blank:LIPIcs.SoCG.2026.17,
  author =	{Blank, Lotte},
  title =	{{Fr\'{e}chet Distance in the Imbalanced Case}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.17},
  URN =		{urn:nbn:de:0030-drops-258232},
  doi =		{10.4230/LIPIcs.SoCG.2026.17},
  annote =	{Keywords: Fr\'{e}chet distance, SETH, Orthogonal Vectors, Lower Bounds, distance oracle, data structures}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.6
A Dimension-Reducing Fréchet Simplification Oracle

Authors: Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh

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

Abstract
Let P be a polygonal curve with n vertices in the plane. We construct a data structure of size O(n log n) suited for simplification queries of the following kind. Given a query line 𝓁 and an integer k ≥ 1, find a curve Q on 𝓁 with at most k vertices that minimizes the discrete Fréchet distance to P, among all such curves. Using our data structure, a query can be handled in O(k² log³ n + k log⁴n) time. More generally, a geometric tree T on n vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair u, v of its vertices, a line 𝓁, and an integer k ≥ 1, one can find a curve Q on 𝓁 with at most k vertices that minimizes the discrete Fréchet distance to the path from u to v in T, in time O(k² polylog n). For the general dimension-reduction problem, where P is a curve in ℝ^d (d ≥ 3), 0 < ε₀ < 1 is a real parameter, and a query specifies a g-flat h (1 ≤ g ≤ d-1) and an integer k ≥ 1, we construct a data structure of size O(nlog n + f(ε₀) n), where f(ε₀) = (1+1/ε₀)^{(d-1)/2}, that allows us to find a curve Q on h with at most k vertices, whose discrete Fréchet distance to P is at most 1+ε₀ times the distance of Q^* to P, where Q^* is such a curve that minimizes the distance to P. The query handling time is O(f(ε₀) k² log² n).
Cite as

Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh. A Dimension-Reducing Fréchet Simplification Oracle. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aronov_et_al:LIPIcs.ISAAC.2025.6,
  author =	{Aronov, Boris and Farhana, Tsuri and Katz, Matthew J. and Ramesh, Indu},
  title =	{{A Dimension-Reducing Fr\'{e}chet Simplification Oracle}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{6:1--6:20},
  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.2025.6},
  URN =		{urn:nbn:de:0030-drops-249149},
  doi =		{10.4230/LIPIcs.ISAAC.2025.6},
  annote =	{Keywords: Computational geometry, discrete Fr\'{e}chet distance, curve simplification oracle, restricted minimum enclosing disk queries}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.36
Faster Fréchet Distance Under Transformations

Authors: Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves π and σ of total complexity n and a threshold δ ≥ 0, we present an 𝒪̃(n^{7 + 1/3}) time algorithm to determine whether there exists a translation t ∈ ℝ² such that the Fréchet distance between π and σ + t is at most δ. This improves on the previous best result, which is an 𝒪(n⁸) time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class T of rationally parametrized transformations with k degrees of freedom, we show that one can determine whether there is a transformation τ ∈ T such that the Fréchet distance between π and τ(σ) is at most δ in 𝒪̃(n^{3k+4/3}) time.
Cite as

Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong. Faster Fréchet Distance Under Transformations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2025.36,
  author =	{Buchin, Kevin and Buchin, Maike and Huang, Zijin and Nusser, Andr\'{e} and Wong, Sampson},
  title =	{{Faster Fr\'{e}chet Distance Under Transformations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.36},
  URN =		{urn:nbn:de:0030-drops-234137},
  doi =		{10.4230/LIPIcs.ICALP.2025.36},
  annote =	{Keywords: Fr\'{e}chet distance, curve similarity, shape matching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.78
Efficient Greedy Discrete Subtrajectory Clustering

Authors: Ivor van der Hoog, Lara Ost, Eva Rotenberg, and Daniel Rutschmann

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We cluster a set of trajectories 𝒯 using subtrajectories of 𝒯. We require for a clustering C that any two subtrajectories (𝒯[a, b], 𝒯[c, d]) in a cluster have disjoint intervals [a,b] and [c, d]. Clustering quality may be measured by the number of clusters, the number of vertices of 𝒯 that are absent from the clustering, and by the Fréchet distance between subtrajectories in a cluster. A Δ-cluster of 𝒯 is a cluster 𝒫 of subtrajectories of 𝒯 with a centre P ∈ 𝒫, where all subtrajectories in 𝒫 have Fréchet distance at most Δ to P. Buchin, Buchin, Gudmundsson, Löffler and Luo present two O(n² + n m 𝓁)-time algorithms: SC(max, 𝓁, Δ, 𝒯) computes a single Δ-cluster where P has at least 𝓁 vertices and maximises the cardinality m of 𝒫. SC(m, max, Δ, 𝒯) computes a single Δ-cluster where 𝒫 has cardinality m and maximises the complexity 𝓁 of P. In this paper, which is a mixture of algorithms engineering and theoretical insights, we use such maximum-cardinality clusters in a greedy clustering algorithm. We first provide an efficient implementation of SC(max, 𝓁, Δ, 𝒯) and SC(m, max, Δ, 𝒯) that significantly outperforms previous implementations. Next, we use these functions as a subroutine in a greedy clustering algorithm, which performs well when compared to existing subtrajectory clustering algorithms on real-world data. Finally, we observe that, for fixed Δ and 𝒯, these two functions always output a point on the Pareto front of some bivariate function θ(𝓁, m). We design a new algorithm PSC(Δ, 𝒯) that in O(n² log⁴ n) time computes a 2-approximation of this Pareto front. This yields a broader set of candidate clusters, with comparable quality to the output of the previous functions. We show that using PSC(Δ, 𝒯) as a subroutine improves the clustering quality and performance even further.
Cite as

Ivor van der Hoog, Lara Ost, Eva Rotenberg, and Daniel Rutschmann. Efficient Greedy Discrete Subtrajectory Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.SoCG.2025.78,
  author =	{van der Hoog, Ivor and Ost, Lara and Rotenberg, Eva and Rutschmann, Daniel},
  title =	{{Efficient Greedy Discrete Subtrajectory Clustering}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{78:1--78:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.78},
  URN =		{urn:nbn:de:0030-drops-232308},
  doi =		{10.4230/LIPIcs.SoCG.2025.78},
  annote =	{Keywords: Algorithms engineering, Fr\'{e}chet distance, subtrajectory clustering}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.52
Convex Partial Transversals of Planar Regions

Authors: Vahideh Keikha, Mees van de Kerkhof, Marc van Kreveld, Irina Kostitsyna, Maarten Löffler, Frank Staals, Jérôme Urhausen, Jordi L. Vermeulen, and Lionov Wiratma

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
We consider the problem of testing, for a given set of planar regions R and an integer k, whether there exists a convex shape whose boundary intersects at least k regions of R. We provide polynomial-time algorithms for the case where the regions are disjoint axis-aligned rectangles or disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.
Cite as

Vahideh Keikha, Mees van de Kerkhof, Marc van Kreveld, Irina Kostitsyna, Maarten Löffler, Frank Staals, Jérôme Urhausen, Jordi L. Vermeulen, and Lionov Wiratma. Convex Partial Transversals of Planar Regions. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{keikha_et_al:LIPIcs.ISAAC.2018.52,
  author =	{Keikha, Vahideh and van de Kerkhof, Mees and van Kreveld, Marc and Kostitsyna, Irina and L\"{o}ffler, Maarten and Staals, Frank and Urhausen, J\'{e}r\^{o}me and Vermeulen, Jordi L. and Wiratma, Lionov},
  title =	{{Convex Partial Transversals of Planar Regions}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{52:1--52:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.52},
  URN =		{urn:nbn:de:0030-drops-100003},
  doi =		{10.4230/LIPIcs.ISAAC.2018.52},
  annote =	{Keywords: computational geometry, algorithms, NP-hardness, convex transversals}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.GISCIENCE.2018.64
An Experimental Comparison of Two Definitions for Groups of Moving Entities (Short Paper)

Authors: Lionov Wiratma, Maarten Löffler, and Frank Staals

Published in: LIPIcs, Volume 114, 10th International Conference on Geographic Information Science (GIScience 2018)

Abstract
Two of the grouping definitions for trajectories that have been developed in recent years allow a continuous motion model and allow varying shape groups. One of these definitions was suggested as a refinement of the other. In this paper we perform an experimental comparison to highlight the differences in these two definitions on various data sets.
Cite as

Lionov Wiratma, Maarten Löffler, and Frank Staals. An Experimental Comparison of Two Definitions for Groups of Moving Entities (Short Paper). In 10th International Conference on Geographic Information Science (GIScience 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 114, pp. 64:1-64:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{wiratma_et_al:LIPIcs.GISCIENCE.2018.64,
  author =	{Wiratma, Lionov and L\"{o}ffler, Maarten and Staals, Frank},
  title =	{{An Experimental Comparison of Two Definitions for Groups of Moving Entities}},
  booktitle =	{10th International Conference on Geographic Information Science (GIScience 2018)},
  pages =	{64:1--64:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-083-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{114},
  editor =	{Winter, Stephan and Griffin, Amy and Sester, Monika},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GISCIENCE.2018.64},
  URN =		{urn:nbn:de:0030-drops-93928},
  doi =		{10.4230/LIPIcs.GISCIENCE.2018.64},
  annote =	{Keywords: Trajectories, grouping algorithms, experimental comparison}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.56
On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance

Authors: Marc van Kreveld, Maarten Löffler, and Lionov Wiratma

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification.
Cite as

Marc van Kreveld, Maarten Löffler, and Lionov Wiratma. On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{vankreveld_et_al:LIPIcs.SoCG.2018.56,
  author =	{van Kreveld, Marc and L\"{o}ffler, Maarten and Wiratma, Lionov},
  title =	{{On Optimal Polyline Simplification Using the Hausdorff and Fr\'{e}chet Distance}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.56},
  URN =		{urn:nbn:de:0030-drops-87694},
  doi =		{10.4230/LIPIcs.SoCG.2018.56},
  annote =	{Keywords: polygonal line simplification, Hausdorff distance, Fr\'{e}chet distance, Imai-Iri, Douglas-Peucker}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.48
A Refined Definition for Groups of Moving Entities and its Computation

Authors: Marc van Kreveld, Maarten Löffler, Frank Staals, and Lionov Wiratma

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
One of the important tasks in the analysis of spatio-temporal data collected from moving entities is to find a group: a set of entities that travel together for a sufficiently long period of time. Buchin et al. [JoCG, 2015] introduce a formal definition of groups, analyze its mathematical structure, and present efficient algorithms for computing all maximal groups in a given set of trajectories. In this paper, we refine their definition and argue that our proposed definition corresponds better to human intuition in certain cases, particularly in dense environments. We present algorithms to compute all maximal groups from a set of moving entities according to the new definition. For a set of n moving entities in R^1, specified by linear interpolation in a sequence of tau time stamps, we show that all maximal groups can be computed in O(tau^2 n^4) time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of alpha(n) in the running time. In higher dimensions, we can compute all maximal groups in O(tau^2 n^5 log n) time (for any constant number of dimensions). We also show that one tau factor can be traded for a much higher dependence on n by giving a O(tau n^4 2^n) algorithm for the same problem. Consequently, we give a linear-time algorithm when the number of entities is constant and the input size relates to the number of time stamps of each entity. Finally, we provide a construction to show that it might be difficult to develop an algorithm with polynomial dependence on n and linear dependence on tau.
Cite as

Marc van Kreveld, Maarten Löffler, Frank Staals, and Lionov Wiratma. A Refined Definition for Groups of Moving Entities and its Computation. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 48:1-48:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{vankreveld_et_al:LIPIcs.ISAAC.2016.48,
  author =	{van Kreveld, Marc and L\"{o}ffler, Maarten and Staals, Frank and Wiratma, Lionov},
  title =	{{A Refined Definition for Groups of Moving Entities and its Computation}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{48:1--48:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.48},
  URN =		{urn:nbn:de:0030-drops-68188},
  doi =		{10.4230/LIPIcs.ISAAC.2016.48},
  annote =	{Keywords: moving entities, trajectories, grouping, computational geometry}
}
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