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 \emph{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.

Prove a new lower bound on dynamic connectivity

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 it presents Bob with the next piece. This model has a closer and more natural correspondence to dynamic data structures than the classic communication models do and hence presents a new perspective on data structures.

We first present a lower bound for the \emph{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. Then we apply the online communication model to data structure lower bounds by studying the Group Range Problem, a dynamic data structure problem. This problem admits an $O(\log n)$-time worst-case data structure. Using online communication complexity, we prove a tight cell- probe lower bound: 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.