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DOI: 10.4230/LIPIcs.STACS.2017.20
URN: urn:nbn:de:0030-drops-70132
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7013/
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Carette, Titouan ; Laurière, Mathieu ; Magniez, Frédéric

Extended Learning Graphs for Triangle Finding

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LIPIcs-STACS-2017-20.pdf (0.5 MB)


Abstract

We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].

BibTeX - Entry

@InProceedings{carette_et_al:LIPIcs:2017:7013,
  author =	{Titouan Carette and Mathieu Lauri{\`e}re and Fr{\'e}d{\'e}ric Magniez},
  title =	{{Extended Learning Graphs for Triangle Finding}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Heribert Vollmer and Brigitte Vallée},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7013},
  URN =		{urn:nbn:de:0030-drops-70132},
  doi =		{10.4230/LIPIcs.STACS.2017.20},
  annote =	{Keywords: Quantum query complexity, learning graphs, triangle finding}
}

Keywords: Quantum query complexity, learning graphs, triangle finding
Seminar: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
Issue Date: 2017
Date of publication: 24.02.2017


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