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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.435
URN: urn:nbn:de:0030-drops-53160
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5316/
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Sun, Xiaoming ; Woodruff, David P.

Tight Bounds for Graph Problems in Insertion Streams

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Abstract

Despite the large amount of work on solving graph problems in the data stream model, there do not exist tight space bounds for almost any of them, even in a stream with only edge insertions. For example, for testing connectivity, the upper bound is O(n * log(n)) bits, while the lower bound is only Omega(n) bits. We remedy this situation by providing the first tight Omega(n * log(n)) space lower bounds for randomized algorithms which succeed with constant probability in a stream of edge insertions for a number of graph problems. Our lower bounds apply to testing bipartiteness, connectivity, cycle-freeness, whether a graph is Eulerian, planarity, H-minor freeness, finding a minimum spanning tree of a connected graph, and testing if the diameter of a sparse graph is constant. We also give the first Omega(n * k * log(n)) space lower bounds for deterministic algorithms for k-edge connectivity and k-vertex connectivity; these are optimal in light of known deterministic upper bounds (for k-vertex connectivity we also need to allow edge duplications, which known upper bounds allow). Finally, we give an Omega(n * log^2(n)) lower bound for randomized algorithms approximating the minimum cut up to a constant factor with constant probability in a graph with integer weights between 1 and n, presented as a stream of insertions and deletions to its edges. This lower bound also holds for cut sparsifiers, and gives the first separation of maintaining a sparsifier in the data stream model versus the offline model.

BibTeX - Entry

@InProceedings{sun_et_al:LIPIcs:2015:5316,
  author =	{Xiaoming Sun and David P. Woodruff},
  title =	{{Tight Bounds for Graph Problems in Insertion Streams}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{435--448},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5316},
  URN =		{urn:nbn:de:0030-drops-53160},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.435},
  annote =	{Keywords: communication complexity, data streams, graphs, space complexity}
}

Keywords: communication complexity, data streams, graphs, space complexity
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 28.07.2015


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