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DOI: 10.4230/LIPIcs.ESA.2025.24

Fréchet Distance in Unweighted Planar Graphs

Authors: Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The Fréchet distance is a distance measure between trajectories in ℝ^d or walks in a graph G. Given constant-time shortest path queries, the Discrete Fréchet distance D_G(P, Q) between two walks P and Q can be computed in O(|P|⋅|Q|) time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a 1.01-approximation of D_G(P, Q) unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fréchet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist (1+ε)-approximations, both in weighted graphs and in ℝ^d, that take near-linear time for c-packed or κ-straight walks in the graph. In ℝ^d there also exists a near-linear time algorithm to compute the Fréchet distance whenever all input edges are long compared to the distance. We consider computing the Fréchet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic 1.25-approximations of the discrete Fréchet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an Õ((|P|+|Q|)^{1.5})-time algorithm to compute D_G(P, Q). Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate (1+ε)-approximations of their discrete Fréchet distance in subquadratic time.

Cite as

Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf. Fréchet Distance in Unweighted Planar Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2025.24,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Rotenberg, Eva and Wulf, Lasse},
  title =	{{Fr\'{e}chet Distance in Unweighted Planar Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.24},
  URN =		{urn:nbn:de:0030-drops-244924},
  doi =		{10.4230/LIPIcs.ESA.2025.24},
  annote =	{Keywords: Fr\'{e}chet distance, planar graphs, lower bounds, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.35

The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon

Authors: Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The Fréchet distance is a popular similarity measure that is well-understood for polygonal curves in ℝ^d: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat et al. (2002) were able to give a near-linear time 2-approximation algorithm. In this paper, we significantly improve upon their result: we present a (1+ε)-approximation algorithm, for any ε > 0, that runs in 𝒪(1/(ε) (n+m log n) log nm log 1/(ε)) time for a simple polygon bounded by two curves with n and m vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once, by interpreting the free space as one between separated one-dimensional curves. We solve this one-dimensional problem in near-linear time, generalizing a result by Bringmann and Künnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

Cite as

Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann. The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhorst_et_al:LIPIcs.ESA.2025.35,
  author =	{van der Horst, Thijs and van Kreveld, Marc and Ophelders, Tim and Speckmann, Bettina},
  title =	{{The Geodesic Fr\'{e}chet Distance Between Two Curves Bounding a Simple Polygon}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.35},
  URN =		{urn:nbn:de:0030-drops-245038},
  doi =		{10.4230/LIPIcs.ESA.2025.35},
  annote =	{Keywords: Fr\'{e}chet distance, approximation, geodesic, simple polygon}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.84

Property Testing of Curve Similarity

Authors: Peyman Afshani, Maike Buchin, Anne Driemel, Marena Richter, and Sampson Wong

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fréchet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query returns the set of vertices of the curve that lie within a radius δ of a specified vertex of the other curve. The goal is to use a small number of queries to determine with constant probability whether the two curves are similar (i.e., their discrete Fréchet distance is at most δ) or they are "ε-far" (for 0 < ε < 2) from being similar, i.e., more than an ε-fraction of the two curves must be ignored for them to become similar. We present two algorithms which are sublinear assuming that the curves are t-approximate shortest paths in the ambient metric space, for some t ≪ n. The first algorithm uses O(t/ε log t/ε) queries and is given the value of t in advance. The second algorithm does not have explicit knowledge of the value of t and therefore needs to gain implicit knowledge of the straightness of the input curves through its queries. We show that the discrete Fréchet distance can still be tested using roughly O({t³+t² log n}/ε) queries ignoring logarithmic factors in t. Our algorithms work in a matrix representation of the input and may be of independent interest to matrix testing. Our algorithms use a mild uniform sampling condition that constrains the edge lengths of the curves, similar to a polynomially bounded aspect ratio. Applied to testing the continuous Fréchet distance of t-straight curves, our algorithms can be used for (1+ε')-approximate testing using essentially the same bounds as stated above with an additional factor of poly(1/(ε')).

Cite as

Peyman Afshani, Maike Buchin, Anne Driemel, Marena Richter, and Sampson Wong. Property Testing of Curve Similarity. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2025.84,
  author =	{Afshani, Peyman and Buchin, Maike and Driemel, Anne and Richter, Marena and Wong, Sampson},
  title =	{{Property Testing of Curve Similarity}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{84:1--84:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.84},
  URN =		{urn:nbn:de:0030-drops-245522},
  doi =		{10.4230/LIPIcs.ESA.2025.84},
  annote =	{Keywords: Fr\'{e}chet distance, Trajectory Analysis, Curve Similarity, Property Testing, Monotonicity Testing}
}

Document

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.

Cite as

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}
}

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.2025.54

The Fréchet Distance Unleashed: Approximating a Dog with a Frog

Authors: Sariel Har-Peled, Benjamin Raichel, and Eliot W. Robson

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

Abstract

We show that a variant of the continuous Fréchet distance between polygonal curves can be computed using essentially the same algorithm used to solve the discrete version. The new variant is not necessarily monotone, but this shortcoming can be easily handled via refinement. Combined with a Dijkstra/Prim type algorithm, this leads to a realization of the Fréchet distance (i.e., a morphing) that is locally optimal (aka locally correct), that is both easy to compute, and in practice, takes near linear time on many inputs. The new morphing has the property that the leash is always as short as possible. These matchings/morphings are more natural, and are better than the ones computed by standard algorithms - in particular, they handle noise more graciously. This should make the Fréchet distance more useful for real world applications. We implemented the new algorithm, and various strategies to obtain fast practical performance. We performed extensive experiments with our new algorithm, and released publicly available (and easily installable and usable) Julia and Python packages. In particular, the Julia implementation, for computing the regular Fréchet distance, seems to be {significantly faster} than other currently available implementations. See Table 2.2. Our algorithms can be used to compute the almost-exact Fréchet distance between polygonal curves. Implementations and numerous examples are available here: https://frechet.xyz.

Cite as

Sariel Har-Peled, Benjamin Raichel, and Eliot W. Robson. The Fréchet Distance Unleashed: Approximating a Dog with a Frog. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2025.54,
  author =	{Har-Peled, Sariel and Raichel, Benjamin and Robson, Eliot W.},
  title =	{{The Fr\'{e}chet Distance Unleashed: Approximating a Dog with a Frog}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{54:1--54:13},
  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.2025.54},
  URN =		{urn:nbn:de:0030-drops-232066},
  doi =		{10.4230/LIPIcs.SoCG.2025.54},
  annote =	{Keywords: Curve similarity, Fr\'{e}chet distance}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2025.22

Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

Authors: Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter

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

Abstract

The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of 𝒪(n^{8/3} log³ n). To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of 𝒪(n^{8/3} log³ n) for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of 𝒪̃(n⁴). Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.

Cite as

Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter. Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blank_et_al:LIPIcs.SoCG.2025.22,
  author =	{Blank, Lotte and Conradi, Jacobus and Driemel, Anne and Kolbe, Benedikt and Nusser, Andr\'{e} and Richter, Marena},
  title =	{{Transforming Dogs on the Line: On the Fr\'{e}chet Distance Under Translation or Scaling in 1D}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{22:1--22:16},
  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.2025.22},
  URN =		{urn:nbn:de:0030-drops-231746},
  doi =		{10.4230/LIPIcs.SoCG.2025.22},
  annote =	{Keywords: Fr\'{e}chet distance under translation, Fr\'{e}chet distance under scaling, time series, shape matching}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2024.25

The Complexity of Geodesic Spanners Using Steiner Points

Authors: Sarita de Berg, Tim Ophelders, Irene Parada, Frank Staals, and Jules Wulms

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

A geometric t-spanner 𝒢 on a set S of n point sites in a metric space P is a subgraph of the complete graph on S such that for every pair of sites p,q the distance in 𝒢 is a most t times the distance d(p,q) in P. We call a connection between two sites a link. In some settings, such as when P is a simple polygon with m vertices and a link is a shortest path in P, links can consist of Θ (m) segments and thus have non-constant complexity. The spanner complexity is a measure of how compact a spanner is, which is equal to the sum of the complexities of all links in the spanner. In this paper, we study what happens if we are allowed to introduce k Steiner points to reduce the spanner complexity. We study such Steiner spanners in simple polygons, polygonal domains, and edge-weighted trees. Surprisingly, we show that Steiner points have only limited utility. For a spanner that uses k Steiner points, we provide an Ω(nm/k) lower bound on the worst-case complexity of any (3-ε)-spanner, and an Ω(mn^{1/(t+1)}/k^{1/(t+1)}) lower bound on the worst-case complexity of any (t-ε)-spanner, for any constant ε ∈ (0,1) and integer constant t ≥ 2. These lower bounds hold in all settings. Additionally, we show NP-hardness for the problem of deciding whether a set of sites in a polygonal domain admits a 3-spanner with a given maximum complexity using k Steiner points. On the positive side, for trees we show how to build a 2t-spanner that uses k Steiner points of complexity O(mn^{1/t}/k^{1/t} + n log (n/k)), for any integer t ≥ 1. We generalize this result to forests, and apply it to obtain a 2√2t-spanner in a simple polygon with total complexity O(mn^{1/t}(log k)^{1+1/t}/k^{1/t} + nlog² n). When a link in the spanner can be any path between two sites, we show how to improve the spanning ratio in a simple polygon to (2k+ε), for any constant ε ∈ (0,2k), and how to build a 6t-spanner in a polygonal domain with the same complexity.

Cite as

Sarita de Berg, Tim Ophelders, Irene Parada, Frank Staals, and Jules Wulms. The Complexity of Geodesic Spanners Using Steiner Points. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2024.25,
  author =	{de Berg, Sarita and Ophelders, Tim and Parada, Irene and Staals, Frank and Wulms, Jules},
  title =	{{The Complexity of Geodesic Spanners Using Steiner Points}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.25},
  URN =		{urn:nbn:de:0030-drops-221527},
  doi =		{10.4230/LIPIcs.ISAAC.2024.25},
  annote =	{Keywords: spanner, simple polygon, polygonal domain, geodesic distance, complexity}
}

Document

Graph Drawing Contest Report

DOI: 10.4230/LIPIcs.GD.2024.41

Graph Drawing Contest Report (Graph Drawing Contest Report)

Authors: Sara Di Bartolomeo, Fabian Klute, Debajyoti Mondal, and Jules Wulms

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract

This report describes the 31st Annual Graph Drawing Contest, held in conjunction with the 32nd International Symposium on Graph Drawing and Network Visualization (GD'24) at TU Wien, Vienna, Austria. The mission of the Graph Drawing Contest is to monitor and challenge the current state of the art in graph-drawing technology. This year’s edition featured two categories, a creative track in which participants visualized a dataset based on the Olympic medal track-record of countries and a live challenge held at the conference where participants had to draw a graph on a given point-set with as few crossings as possible.

Cite as

Sara Di Bartolomeo, Fabian Klute, Debajyoti Mondal, and Jules Wulms. Graph Drawing Contest Report (Graph Drawing Contest Report). In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dibartolomeo_et_al:LIPIcs.GD.2024.41,
  author =	{Di Bartolomeo, Sara and Klute, Fabian and Mondal, Debajyoti and Wulms, Jules},
  title =	{{Graph Drawing Contest Report}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{41:1--41:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.41},
  URN =		{urn:nbn:de:0030-drops-213256},
  doi =		{10.4230/LIPIcs.GD.2024.41},
  annote =	{Keywords: Information Visualization, Graph Drawing Contest}
}

Document

DOI: 10.4230/LIPIcs.COSIT.2024.8

Scalable Harmonious Simplification of Isolines

Authors: Steven van den Broek, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann

Published in: LIPIcs, Volume 315, 16th International Conference on Spatial Information Theory (COSIT 2024)

Abstract

Isolines visually characterize scalar fields by connecting all points of the same value by a closed curve at repeated intervals. They work only as a set which gives the viewer an indication of the shape of the underlying field. Hence, when simplifying isolines it is important that the correspondence - the harmony - between adjacent isolines is preserved whenever it is present. The majority of state-of-the-art simplification methods treat isolines independently; at best they avoid collisions between adjacent simplified isolines. A notable exception is the work by Van Goethem et al. (2021) who were the first to introduce the concept of harmony between adjacent isolines explicitly as an algorithmic design principle. They presented a proof-of-concept algorithm that harmoniously simplifies a sequence of polylines. However, the sets of isolines of scalar fields, most notably terrain, consist of closed curves which are nested in arbitrarily complex ways and not of an ordered sequence of polylines. In this paper we significantly extend the work by Van Goethem et al. (2021) to capture harmony in general sets of isolines. Our new simplification algorithm can handle sets of isolines describing arbitrary scalar fields and is more efficient, allowing us to harmoniously simplify terrain with hundreds of thousands of vertices. We experimentally compare our method to the results of Van Goethem et al. (2021) on bundles of isolines and to general simplification methods on isolines extracted from DEMs of Antartica. Our results indicate that our method efficiently preserves the harmony in the simplified maps, which are thereby less noisy, cartographically more meaningful, and easier to read.

Cite as

Steven van den Broek, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann. Scalable Harmonious Simplification of Isolines. In 16th International Conference on Spatial Information Theory (COSIT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 315, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vandenbroek_et_al:LIPIcs.COSIT.2024.8,
  author =	{van den Broek, Steven and Meulemans, Wouter and Reimer, Andreas and Speckmann, Bettina},
  title =	{{Scalable Harmonious Simplification of Isolines}},
  booktitle =	{16th International Conference on Spatial Information Theory (COSIT 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-330-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{315},
  editor =	{Adams, Benjamin and Griffin, Amy L. and Scheider, Simon and McKenzie, Grant},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.COSIT.2024.8},
  URN =		{urn:nbn:de:0030-drops-208230},
  doi =		{10.4230/LIPIcs.COSIT.2024.8},
  annote =	{Keywords: Simplification, isolines, harmony}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2024.24

Optimizing Symbol Visibility Through Displacement

Authors: Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy, Wouter Meulemans, Bettina Speckmann, and Miloš Stojaković

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider unit square symbols that need to be placed at specified y-coordinates. We optimize the drawing order of the symbols as well as their x-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most 2, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in O(nlog n) time. If the width is at most 2, there is a vertical line that stabs all squares, and in this case, we give a 2-approximation algorithm (assuming fixed container height) that runs in O(nlog n) time. As a minimum visible perimeter of 2 is always trivially achievable, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

Cite as

Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy, Wouter Meulemans, Bettina Speckmann, and Miloš Stojaković. Optimizing Symbol Visibility Through Displacement. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gartner_et_al:LIPIcs.SWAT.2024.24,
  author =	{G\"{a}rtner, Bernd and Kalani, Vishwas and M. Reddy, Meghana and Meulemans, Wouter and Speckmann, Bettina and Stojakovi\'{c}, Milo\v{s}},
  title =	{{Optimizing Symbol Visibility Through Displacement}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.24},
  URN =		{urn:nbn:de:0030-drops-200643},
  doi =		{10.4230/LIPIcs.SWAT.2024.24},
  annote =	{Keywords: symbol placement, visibility, jittering, stacking order}
}

Document

DOI: 10.4230/LIPIcs.GIScience.2023.10

Data-Spatial Layouts for Grid Maps

Authors: Nathan van Beusekom, Wouter Meulemans, Bettina Speckmann, and Jo Wood

Published in: LIPIcs, Volume 277, 12th International Conference on Geographic Information Science (GIScience 2023)

Abstract

Grid maps are a well-known technique to visualize data associated with spatial regions. A grid map assigns each region to a tile in a grid (often orthogonal or hexagonal) and then represents the associated data values within this tile. Good grid maps represent the underlying geographic space well: regions that are geographically close are close in the grid map and vice versa. Though Tobler’s law suggests that spatial proximity relates to data similarity, local variations may obscure clusters and patterns in the data. For example, there are often clear differences between urban centers and adjacent rural areas with respect to socio-economic indicators. To get a better view of the data distribution, we propose grid-map layouts that take data values into account and place regions with similar data into close proximity. In the limit, such a data layout is essentially a chart and loses all spatial meaning. We present an algorithm to create hybrid layouts, allowing for trade-offs between data values and geographic space when assigning regions to tiles. Our algorithm also handles hierarchical grid maps and allows us to focus either on data or on geographic space on different levels of the hierarchy. Leveraging our algorithm we explore the design space of (hierarchical) grid maps with a hybrid layout and their semantics.

Cite as

Nathan van Beusekom, Wouter Meulemans, Bettina Speckmann, and Jo Wood. Data-Spatial Layouts for Grid Maps. In 12th International Conference on Geographic Information Science (GIScience 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 277, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{vanbeusekom_et_al:LIPIcs.GIScience.2023.10,
  author =	{van Beusekom, Nathan and Meulemans, Wouter and Speckmann, Bettina and Wood, Jo},
  title =	{{Data-Spatial Layouts for Grid Maps}},
  booktitle =	{12th International Conference on Geographic Information Science (GIScience 2023)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-288-4},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{277},
  editor =	{Beecham, Roger and Long, Jed A. and Smith, Dianna and Zhao, Qunshan and Wise, Sarah},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2023.10},
  URN =		{urn:nbn:de:0030-drops-189052},
  doi =		{10.4230/LIPIcs.GIScience.2023.10},
  annote =	{Keywords: Grid map, algorithms, trade-offs}
}

Document

DOI: 10.4230/LIPIcs.GIScience.2021.II.7

Coordinated Schematization for Visualizing Mobility Patterns on Networks

Authors: Bram Custers, Wouter Meulemans, Bettina Speckmann, and Kevin Verbeek

Published in: LIPIcs, Volume 208, 11th International Conference on Geographic Information Science (GIScience 2021) - Part II

Abstract

GPS trajectories of vehicles moving on a road network are a valuable source of traffic information. However, the sheer volume of available data makes it challenging to identify and visualize salient patterns. Meaningful visual summaries of trajectory collections require that both the trajectories and the underlying network are aggregated and simplified in a coherent manner. In this paper we propose a coordinated fully-automated pipeline for computing a schematic overview of mobility patterns from a collection of trajectories on a street network. Our pipeline utilizes well-known building blocks from GIS, automated cartography, and trajectory analysis: map matching, road selection, schematization, movement patterns, and metro-map style rendering. We showcase the results of our pipeline on two real-world trajectory collections around The Hague and Beijing.

Cite as

Bram Custers, Wouter Meulemans, Bettina Speckmann, and Kevin Verbeek. Coordinated Schematization for Visualizing Mobility Patterns on Networks. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part II. Leibniz International Proceedings in Informatics (LIPIcs), Volume 208, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{custers_et_al:LIPIcs.GIScience.2021.II.7,
  author =	{Custers, Bram and Meulemans, Wouter and Speckmann, Bettina and Verbeek, Kevin},
  title =	{{Coordinated Schematization for Visualizing Mobility Patterns on Networks}},
  booktitle =	{11th International Conference on Geographic Information Science (GIScience 2021) - Part II},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-208-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{208},
  editor =	{Janowicz, Krzysztof and Verstegen, Judith A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2021.II.7},
  URN =		{urn:nbn:de:0030-drops-147665},
  doi =		{10.4230/LIPIcs.GIScience.2021.II.7},
  annote =	{Keywords: Trajectories, Visualization, Schematization}
}

Document

DOI: 10.4230/LIPIcs.GIScience.2021.II.8

Harmonious Simplification of Isolines

Authors: Arthur van Goethem, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann

Published in: LIPIcs, Volume 208, 11th International Conference on Geographic Information Science (GIScience 2021) - Part II

Abstract

Current techniques for simplification focus on reducing complexity while maintaining the geometric similarity to the input. For isolines that jointly describe a scalar field, however, we postulate that geometric similarity of each isoline separately is not sufficient. Rather, we need to maintain the harmony between these isolines to make them visually relate and describe the structures of the underlying terrain. Based on principles of manual cartography, we propose an algorithm for simplifying isolines while considering harmony explicitly. Our preliminary visual and quantitative results suggest that our algorithm is effective.

Cite as

Arthur van Goethem, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann. Harmonious Simplification of Isolines. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part II. Leibniz International Proceedings in Informatics (LIPIcs), Volume 208, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vangoethem_et_al:LIPIcs.GIScience.2021.II.8,
  author =	{van Goethem, Arthur and Meulemans, Wouter and Reimer, Andreas and Speckmann, Bettina},
  title =	{{Harmonious Simplification of Isolines}},
  booktitle =	{11th International Conference on Geographic Information Science (GIScience 2021) - Part II},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-208-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{208},
  editor =	{Janowicz, Krzysztof and Verstegen, Judith A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2021.II.8},
  URN =		{urn:nbn:de:0030-drops-147675},
  doi =		{10.4230/LIPIcs.GIScience.2021.II.8},
  annote =	{Keywords: Simplification, isolines, harmony}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2021.56

Obstructing Classification via Projection

Authors: Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin Verbeek

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space ℝ^d and each point is labeled with k binary-valued properties. A priori we assume that it is "easy" to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space ℝ^m (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger’s Theorem to show that, under certain conditions, a simple projection to ℝ^{d-1} suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear "inseparability" of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities.

Cite as

Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin Verbeek. Obstructing Classification via Projection. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 56:1-56:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haghighatkhah_et_al:LIPIcs.MFCS.2021.56,
  author =	{Haghighatkhah, Pantea and Meulemans, Wouter and Speckmann, Bettina and Urhausen, J\'{e}r\^{o}me and Verbeek, Kevin},
  title =	{{Obstructing Classification via Projection}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{56:1--56:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.56},
  URN =		{urn:nbn:de:0030-drops-144965},
  doi =		{10.4230/LIPIcs.MFCS.2021.56},
  annote =	{Keywords: Projection, classification, models of learning}
}

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