On the "log rank"-conjecture in communication complexity

Ran Raz, Boris Spieker

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between two n-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove: 1) The rank of the input matrix over the reals for this problem is 2O(n). 2) The non-deterministic communication complexity of the problem is Ω(nloglog n). Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglog n) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.

Original languageEnglish (US)
Pages (from-to)567-588
Number of pages22
Issue number4
StatePublished - Dec 1995
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


  • Mathematics Subject Classification (1991): 68Q15, 05C50, 68R10, 05C15


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