The streaming complexity of cycle counting, sorting by reversals, and other problems

In this paper we introduce a new technique for proving streaming lower bounds (and one-way communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a generalization of the well-known Boolean Hidden Matching problem, which was used by Gavinsky et al. to prove an exponential separation between quantum communication complexity and one-way randomized communication complexity. We are the first to introduce BHH, and to prove a lower bound for it.

The hardness of the BHH problem is inherently oneway: it is easy to solve BHH using logarithmic two-way communication, but it requires √n communication if Alice is only allowed to send messages to Bob, and not vice-versa. This one-wayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently one-way.

By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, of approximately counting the number of cycles in a 2-regular graph, and of other problems.

For example, here is one lower bound that we prove, for a cycle-counting problem: Alice gets a perfect matching E_A on a set of n nodes, and Bob gets a perfect matching E_B on the same set of nodes. The union E_A U E_B is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o(√n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is "natively" one-way: With 4 rounds of communication, it is easy to distinguish these two cases.