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 Ω̃((n - b)/(b - a)) when a ≡ b (mod 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(n log p(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. © 2012 Springer-Verlag Berlin Heidelberg.