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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

While the standard unweighted Voronoi diagram in the plane has linear worst-case complexity, many of its natural generalizations do not. This paper considers two such previously studied generalizations, namely multiplicative and semi Voronoi diagrams. These diagrams both have quadratic worst-case complexity, though here we show that their expected complexity is linear for certain natural randomized inputs. Specifically, we argue that the expected complexity is linear for: (1) semi Voronoi diagrams when the visible direction is randomly sampled, and (2) for multiplicative diagrams when either weights are sampled from a constant-sized set, or the more challenging case when weights are arbitrary but locations are sampled from a square.

Chenglin Fan and Benjamin Raichel. Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.ESA.2020.45, author = {Fan, Chenglin and Raichel, Benjamin}, title = {{Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {45:1--45:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.45}, URN = {urn:nbn:de:0030-drops-129111}, doi = {10.4230/LIPIcs.ESA.2020.45}, annote = {Keywords: Voronoi Diagrams, Expected Complexity, Computational Geometry} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In this paper we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. We define an uncertain curve as a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves.
We prove that both problems are NP-hard for the continuous Fréchet distance, and the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound discrete Fréchet distance can be computed in polynomial time using dynamic programming. Furthermore, we show that computing the expected discrete or continuous Fréchet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments.
On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly δ-separated. Finally, we study the setting with Sakoe - Chiba bands, restricting the alignment of the two curves, and give polynomial-time algorithms for upper bound and expected (discrete) Fréchet distance for point-set-modelled uncertainty regions.

Kevin Buchin, Chenglin Fan, Maarten Löffler, Aleksandr Popov, Benjamin Raichel, and Marcel Roeloffzen. Fréchet Distance for Uncertain Curves. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2020.20, author = {Buchin, Kevin and Fan, Chenglin and L\"{o}ffler, Maarten and Popov, Aleksandr and Raichel, Benjamin and Roeloffzen, Marcel}, title = {{Fr\'{e}chet Distance for Uncertain Curves}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.20}, URN = {urn:nbn:de:0030-drops-124276}, doi = {10.4230/LIPIcs.ICALP.2020.20}, annote = {Keywords: Curves, Uncertainty, Fr\'{e}chet Distance, Hardness} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Many modern data analysis algorithms either assume or are considerably more efficient if the distances between the data points satisfy a metric. However, as real data sets are noisy, they often do not possess this fundamental property. For this reason, Gilbert and Jain [A. Gilbert and L. Jain, 2017] and Fan et al. [C. Fan et al., 2018] introduced the closely related sparse metric repair and metric violation distance problems. Given a matrix, representing all distances, the goal is to repair as few entries as possible to ensure they satisfy a metric. This problem was shown to be APX-hard, and an O(OPT^{1/3})-approximation was given, where OPT is the optimal solution size.
In this paper, we generalize the problem, by describing distances by a possibly incomplete positively weighted graph, where again our goal is to find the smallest number of weight modifications so that they satisfy a metric. This natural generalization is more flexible as it takes into account different relationships among the data points. We demonstrate the inherent combinatorial structure of the problem, and give an approximation-preserving reduction from MULTICUT, which is hard to approximate within any constant factor assuming UGC. Conversely, we show that for any fixed constant ς, for the large class of ς-chordal graphs, the problem is fixed parameter tractable, answering an open question from previous work. Call a cycle broken if it contains an edge whose weight is larger than the sum of all its other edges, and call the amount of this difference its deficit. We present approximation algorithms, one depending on the maximum number of edges in a broken cycle, and one depending on the number of distinct deficit values, both quantities which may naturally be small. Finally, we give improved analysis of previous algorithms for complete graphs.

Chenglin Fan, Anna C. Gilbert, Benjamin Raichel, Rishi Sonthalia, and Gregory Van Buskirk. Generalized Metric Repair on Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.SWAT.2020.25, author = {Fan, Chenglin and Gilbert, Anna C. and Raichel, Benjamin and Sonthalia, Rishi and Van Buskirk, Gregory}, title = {{Generalized Metric Repair on Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {25:1--25:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.25}, URN = {urn:nbn:de:0030-drops-122727}, doi = {10.4230/LIPIcs.SWAT.2020.25}, annote = {Keywords: Approximation, FPT, Hardness, Metric Spaces} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Fréchet gap distance. In the man and dog analogy, the Fréchet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve.
The Fréchet gap distance was originally introduced by Filtser and Katz (2015) in the context of the discrete Fréchet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task.
For this problem we give an O(n^5 log(n)) time exact algorithm and a more efficient O(n^2 log(n) + (n^2/epsilon) log(1/epsilon)) time (1+epsilon)-approximation algorithm, where n is the total number of vertices of the input curves. Note that for (small enough) constant epsilon and ignoring logarithmic factors, our approximation has quadratic running time, matching the lower bound, assuming SETH (Bringmann 2014), for approximating the standard Fréchet distance for general curves.

Chenglin Fan and Benjamin Raichel. Computing the Fréchet Gap Distance. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fan_et_al:LIPIcs.SoCG.2017.42, author = {Fan, Chenglin and Raichel, Benjamin}, title = {{Computing the Fr\'{e}chet Gap Distance}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {42:1--42:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.42}, URN = {urn:nbn:de:0030-drops-71849}, doi = {10.4230/LIPIcs.SoCG.2017.42}, annote = {Keywords: Frechet Distance, Approximation, Polygonal Curves} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS).
In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr'echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.

Chenglin Fan, Omrit Filtser, Matthew J. Katz, and Binhai Zhu. On the General Chain Pair Simplification Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fan_et_al:LIPIcs.MFCS.2016.37, author = {Fan, Chenglin and Filtser, Omrit and Katz, Matthew J. and Zhu, Binhai}, title = {{On the General Chain Pair Simplification Problem}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {37:1--37:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.37}, URN = {urn:nbn:de:0030-drops-64510}, doi = {10.4230/LIPIcs.MFCS.2016.37}, annote = {Keywords: chain simplification, discrete Frechet distance, dynamic programming, geometric arrangements, protein structural resemblance} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

The genomic scaffold filling problem has attracted a lot of attention recently. The problem is on filling an incomplete sequence (scaffold) I into I', with respect to a complete reference genome G, such that the number of adjacencies between G and I' is maximized. The problem is NP-complete and APX-hard, and admits a 1.2-approximation. However, the sequence input I is not quite practical and does not fit most of the real datasets (where a scaffold is more often given as a list of contigs). In this paper, we revisit the genomic scaffold filling problem by considering this important case when, (1) a scaffold S is given, the missing genes X = c(G) - c(S) can only be inserted in between the contigs, and the objective is to maximize the number of adjacencies between G and the filled S' and (2) a scaffold S is given, a subset of the missing genes X' subset X = c(G) - c(S) can only be inserted in between the contigs, and the objective is still to maximize the number of adjacencies between G and the filled S''. For problem (1), we present a simple NP-completeness proof, we then present a factor-2 greedy approximation algorithm, and finally we show that the problem is FPT when each gene appears at most d times in G. For problem (2), we prove that the problem is W[1]-hard and then we present a factor-2 FPT-approximation for the case when each gene appears at most d times in G.

Haitao Jiang, Chenglin Fan, Boting Yang, Farong Zhong, Daming Zhu, and Binhai Zhu. Genomic Scaffold Filling Revisited. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jiang_et_al:LIPIcs.CPM.2016.15, author = {Jiang, Haitao and Fan, Chenglin and Yang, Boting and Zhong, Farong and Zhu, Daming and Zhu, Binhai}, title = {{Genomic Scaffold Filling Revisited}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.15}, URN = {urn:nbn:de:0030-drops-60791}, doi = {10.4230/LIPIcs.CPM.2016.15}, annote = {Keywords: Computational biology, Approximation algorithms, FPT algorithms, NP- completeness} }

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