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DOI: 10.4230/LIPIcs.ISAAC.2016.30

Space-Efficient Plane-Sweep Algorithms

Authors: Amr Elmasry and Frank Kammer

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

Abstract

We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of n items and that the available workspace is Theta(s) bits, where lg n <= s <= n * lg n. Three techniques that can be used as general tools in different space-efficient algorithms are introduced and employed within our algorithms. In particular, we give an almost-optimal algorithm for finding the closest pair among a set of n points that runs in O(n^2 /s + n * lg s) time. We also give a simple algorithm to enumerate the intersections of n line segments that runs in O((n^2 /s^{2/3}) * lg s + k) time, where k is the number of intersections. The counting version can be solved in O((n^2/s^{2/3}) * lg s) time. When the segments are axis-parallel, we give an O((n^2/s) * lg^{4/3} s + n^{4/3} * lg^{1/3} n)-time algorithm that counts the intersections and an O((n^2/s) * lg s * lg lg s + n * lg s + k)-time algorithm that enumerates the intersections, where k is the number of intersections. We finally present an algorithm that runs in O((n^2 /s + n * lg s) * sqrt{(n/s) * lg n}) time to calculate Klee's measure of axis-parallel rectangles.

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Amr Elmasry and Frank Kammer. Space-Efficient Plane-Sweep Algorithms. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{elmasry_et_al:LIPIcs.ISAAC.2016.30,
  author =	{Elmasry, Amr and Kammer, Frank},
  title =	{{Space-Efficient Plane-Sweep Algorithms}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{30:1--30:13},
  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.2016.30},
  URN =		{urn:nbn:de:0030-drops-68009},
  doi =		{10.4230/LIPIcs.ISAAC.2016.30},
  annote =	{Keywords: closest pair, line-segments intersection, Klee's measure}
}

Document

DOI: 10.4230/LIPIcs.STACS.2015.288

Space-efficient Basic Graph Algorithms

Authors: Amr Elmasry, Torben Hagerup, and Frank Kammer

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract

We reconsider basic algorithmic graph problems in a setting where an n-vertex input graph is read-only and the computation must take place in a working memory of O(n) bits or little more than that. For computing connected components and performing breadth-first search, we match the running times of standard algorithms that have no memory restrictions, for depth-first search and related problems we come within a factor of \Theta(\log\log n), and for computing minimum spanning forests and single-source shortest-paths trees we come close for sparse input graphs.

Cite as

Amr Elmasry, Torben Hagerup, and Frank Kammer. Space-efficient Basic Graph Algorithms. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 288-301, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{elmasry_et_al:LIPIcs.STACS.2015.288,
  author =	{Elmasry, Amr and Hagerup, Torben and Kammer, Frank},
  title =	{{Space-efficient Basic Graph Algorithms}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{288--301},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.288},
  URN =		{urn:nbn:de:0030-drops-49217},
  doi =		{10.4230/LIPIcs.STACS.2015.288},
  annote =	{Keywords: graph algorithms, depth-first search, single-source shortest paths, register input model}
}

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Documents authored by Elmasry, Amr

Space-Efficient Plane-Sweep Algorithms

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Space-efficient Basic Graph Algorithms

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