The Relationship between Inner Product and Counting Cycles

  • Xiaoming Sun
  • Chengu Wang
  • Wei Yu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

Cycle-Counting is the following communication complexity problem: Alice and Bob each holds a permutation of size n with the promise there will be either a cycles or b cycles in their product. They want to distinguish between these two cases by communicating a few bits. We show that the quantum/nondeterministic communication complexity is roughly \(\tilde \Omega((n-b)/(b-a))\) when \(a \equiv b \pmod 2\). It is proved by reduction from a variant of the inner product problem over ℤ m . It constructs a bridge for various problems, including In-Same-Cycle [10], One-Cycle [14], and Bipartiteness on constant degree graph [9]. We also give space lower bounds in the streaming model for the Connectivity, Bipartiteness and Girth problems [7]. The inner product variant we used has a quantum lower bound of Ω(nlogp(m)), where p(m) is the smallest prime factor of m. It implies that our lower bounds for Cycle-Counting and related problems still hold for quantum protocols, which was not known before this work.

Keywords

Hamiltonian Cycle Communication Complexity Black Vertex White Vertex Matroid Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xiaoming Sun
    • 1
  • Chengu Wang
    • 2
  • Wei Yu
    • 3
  1. 1.Institute of Computing TechnologyChinese Academy of SciencesChina
  2. 2.IIISTsinghua UniversityChina
  3. 3.Aarhus UniversityDenmark

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