The Relationship between Inner Product and Counting Cycles

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.