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

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.

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-dev.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

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

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.

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-dev.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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