Pointer-chasing is a central problem in two-party communication complexity: given input size n and a parameter k, the two players Alice and Bob are given functions $$N_A, N_B: [n] \rightarrow [n]$$ , respectively, and their goal is to compute the value of $$p_k$$ , where $$p_0 = 1$$ , $$p_1 = N_A(p_0)$$ , $$p_2 = N_B(p_1) = N_B(N_A(p_0))$$ , $$p_3 = N_A(p_2) = N_A(N_B(N_A(p_0)))$$ and so on, applying $$N_A$$ in even steps and $$N_B$$ in odd steps, for a total of k steps. In some versions of the problem, the final output is not $$p_k$$ itself, but rather some fixed function $$f(p_k)$$ of $$p_k$$ . It is trivial to solve the problem using k communication rounds, with Alice speaking first, by simply “chasing the function” for k steps. Many works have studied the communication complexity of pointer chasing, although the focus has always been on protocols with $$k-1$$ communication rounds, or with k rounds where Bob (the “wrong player”) speaks first. Many works have studied this setting giving sometimes tight or near-tight results. In this paper we study the communication complexity of the pointer chasing problem when the interaction between the two players is unlimited, i.e., without any restriction on the number of rounds. Perhaps surprisingly, this question was not studied before, to the best of our knowledge. Our main result is that the trivial k-round protocol is nearly tight (even) when the number of rounds is not restricted: we give a lower bound of $$\varOmega (k \log (n/k))$$ on the randomized communication complexity of the pointer chasing problem with unlimited interaction, and a somewhat stronger lower bound of $$\varOmega (k \log \log {k})$$ for protocols with zero error. When combined with prior work, our results also give a nearly-tight bound on the communication complexity of protocols using at most $$k-1$$ rounds, across all regimes of k; for $$k > \sqrt{n}$$ there was previously a significant gap between the upper and lower bound.

In the multi-party set disjointness problem, k players receive private inputs in the form of sets X1, ..., Xk ⊆ [n], and their goal is to check whether their sets intersect. The set intersection problem is similar, except that the players are required to output the full intersection of their sets rather than just checking whether it is empty. We study the communication complexity of these two problems in the shared-blackboard model of communication complexity, where players communicate with one another by broadcast.

In the multiparty number-in-hand set disjointness problem, we have k players, with private inputs X1,…,Xk ⊆ [n]. The players’ goal is to check whether ∩ℓ=1k Xℓ = ∅. It is known that in the shared blackboard model of communication, set disjointness requires Ω(n logk + k) bits of communication, and in the coordinator model, it requires Ω(kn) bits. However, these two lower bounds require that the players’ inputs can be highly correlated. We study the communication complexity of multiparty set disjointness under product distributions, and ask whether the problem becomes significantly easier, as it is known to become in the two-party case. Our main result is a nearly-tight bound of Θ̃(n1−1/k + k) for both the shared blackboard model and the coordinator model. This shows that in the shared blackboard model, as the number of players grows, having independent inputs helps less and less; but in the coordinator model, when k is very large, having independent inputs makes the problem much easier. Both our upper and our lower bounds use new ideas, as the original techniques developed for the two-party case do not scale to more than two players.