Optimality of the Ω(n/k^2) lower bound for Chained Index

Determine whether the existing Ω(n/k^2) lower bound on the total communication for the Chained Index problem CI_{n,k} (k+1 players pass messages sequentially and the last must output the common bit shared across k correlated Index instances) is optimal as a function of n and k in the standard one-way blackboard model.

Background

Chained Index (CI_{n,k}) generalizes the classical two-party Index problem to k correlated instances arranged in a chain across k+1 players with one-way communication. Cormode et al. (ICALP 2019) established a lower bound of Ω(n/k^2) for CI_{n,k} and used it to derive streaming lower bounds, but did not resolve whether this dependence on k is optimal.

Clarifying the optimal communication complexity of CI_{n,k} directly impacts several streaming lower bounds obtained via reductions from CI_{n,k}, motivating the question of whether Ω(n/k^2) is tight or can be strengthened.