2 Search Results for "Rosenbaum, David J."

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.37

Fast Approximate Counting of Cycles

Authors: Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Cite as

Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Fast Approximate Counting of Cycles. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2024.37,
  author =	{Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia},
  title =	{{Fast Approximate Counting of Cycles}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.37},
  URN =		{urn:nbn:de:0030-drops-201809},
  doi =		{10.4230/LIPIcs.ICALP.2024.37},
  annote =	{Keywords: Approximate triangle counting, Approximate cycle counting Fast matrix multiplication, Fast rectangular matrix multiplication}
}

Document

DOI: 10.4230/LIPIcs.TQC.2013.294

Optimal Quantum Circuits for Nearest-Neighbor Architectures

Authors: David J. Rosenbaum

Published in: LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

Abstract

We show that the depth of quantum circuits in the realistic architecture where a classical controller determines which local interactions to apply on the kD grid Z^k where k >= 2 is the same (up to a constant factor) as in the standard model where arbitrary interactions are allowed. This allows minimum-depth circuits (up to a constant factor) for the nearest-neighbor architecture to be obtained from minimum-depth circuits in the standard abstract model. Our work therefore justifies the standard assumption that interactions can be performed between arbitrary pairs of qubits. In particular, our results imply that Shor's algorithm, controlled operations and fanouts can be implemented in constant depth, polynomial size and polynomial width in this architecture. We also present optimal non-adaptive quantum circuits for controlled operations and fanouts on a kD grid. These circuits have depth Theta(sqrt[k](n)), size Theta(n) and width Theta(n). Our lower bound also applies to a more general class of operations.

Cite as

David J. Rosenbaum. Optimal Quantum Circuits for Nearest-Neighbor Architectures. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 294-307, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{rosenbaum:LIPIcs.TQC.2013.294,
  author =	{Rosenbaum, David J.},
  title =	{{Optimal Quantum Circuits for Nearest-Neighbor Architectures}},
  booktitle =	{8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)},
  pages =	{294--307},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-55-2},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{22},
  editor =	{Severini, Simone and Brandao, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.294},
  URN =		{urn:nbn:de:0030-drops-43290},
  doi =		{10.4230/LIPIcs.TQC.2013.294},
  annote =	{Keywords: 2D, Nearest Neighbor, Quantum Architecture, Quantum Complexity, Quantum Computation}
}

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1 Censor-Hillel, Keren
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1 Mathematics of computing → Approximation algorithms
1 Mathematics of computing → Graph algorithms

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1 2D
1 Approximate cycle counting Fast matrix multiplication
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