Cell-Probe Lower Bounds from Online Communication Complexity

In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players, Bob, his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures. We first present a tight lower bound for the online set intersection problem in the online communication model, demonstrating a general approach for proving online communication lower bounds. The online communication model prevents a batching trick that classic communication complexity allows, and yields a stronger lower bound. We then apply the online communication model to prove data structure lower bounds for two dynamic data structure problems: the Group Range problem and the Dynamic Connectivity problem for forests. Both of the problems admit a worst case $O(\log n)$-time data structure. Using online communication complexity, we prove a tight cell-probe lower bound for each: spending $o(\log n)$ (even amortized) time per operation results in at best an $\exp(-\delta^2 n)$ probability of correctly answering a $(1/2+\delta)$-fraction of the $n$ queries.