# Cell-Probe Lower Bounds from Online Communication Complexity

## TL;DR

This paper introduces an online communication complexity model that closely aligns with dynamic data structures, providing new lower bounds for problems like set intersection, Group Range, and Dynamic Connectivity, demonstrating fundamental limits on their efficiency.

## Contribution

The paper develops a novel online communication complexity model and applies it to establish tight cell-probe lower bounds for dynamic data structure problems, advancing understanding of their inherent computational limits.

## Key findings

- Established a tight lower bound for online set intersection.
- Proved lower bounds for Group Range and Dynamic Connectivity problems.
- Demonstrated that sub-logarithmic time per operation leads to exponentially low success probability.

## Abstract

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.