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DOI: 10.4230/LIPIcs.SoCG.2018.70

Fractal Dimension and Lower Bounds for Geometric Problems

Authors: Anastasios Sidiropoulos, Kritika Singhal, and Vijay Sridhar

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

Abstract

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension. More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type: - For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}. - For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014].

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Anastasios Sidiropoulos, Kritika Singhal, and Vijay Sridhar. Fractal Dimension and Lower Bounds for Geometric Problems. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{sidiropoulos_et_al:LIPIcs.SoCG.2018.70,
  author =	{Sidiropoulos, Anastasios and Singhal, Kritika and Sridhar, Vijay},
  title =	{{Fractal Dimension and Lower Bounds for Geometric Problems}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{70:1--70:14},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.70},
  URN =		{urn:nbn:de:0030-drops-87831},
  doi =		{10.4230/LIPIcs.SoCG.2018.70},
  annote =	{Keywords: fractal dimension, treewidth, spanners, lower bounds, exponential time hypothesis}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2017.58

Algorithmic Interpretations of Fractal Dimension

Authors: Anastasios Sidiropoulos and Vijay Sridhar

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

Abstract

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta). Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings. We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension. However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space.

Cite as

Anastasios Sidiropoulos and Vijay Sridhar. Algorithmic Interpretations of Fractal Dimension. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{sidiropoulos_et_al:LIPIcs.SoCG.2017.58,
  author =	{Sidiropoulos, Anastasios and Sridhar, Vijay},
  title =	{{Algorithmic Interpretations of Fractal Dimension}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{58:1--58:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.58},
  URN =		{urn:nbn:de:0030-drops-72126},
  doi =		{10.4230/LIPIcs.SoCG.2017.58},
  annote =	{Keywords: fractal dimension, exact algorithms, fixed parameter tractability, approximation schemes, spanners}
}

Document

DOI: 10.4230/LIPIcs.ICALP.2016.85

Quasimetric Embeddings and Their Applications

Authors: Facundo Mémoli, Anastasios Sidiropoulos, and Vijay Sridhar

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

Cite as

Facundo Mémoli, Anastasios Sidiropoulos, and Vijay Sridhar. Quasimetric Embeddings and Their Applications. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{memoli_et_al:LIPIcs.ICALP.2016.85,
  author =	{M\'{e}moli, Facundo and Sidiropoulos, Anastasios and Sridhar, Vijay},
  title =	{{Quasimetric Embeddings and Their Applications}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{85:1--85:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.85},
  URN =		{urn:nbn:de:0030-drops-62007},
  doi =		{10.4230/LIPIcs.ICALP.2016.85},
  annote =	{Keywords: metric embeddings, quasimetrics, outliers, random embeddings, treewidth, Directed Sparsest-Cut, Directed Multicut}
}

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Fractal Dimension and Lower Bounds for Geometric Problems

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Algorithmic Interpretations of Fractal Dimension

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Quasimetric Embeddings and Their Applications

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