Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model

TL;DR

This paper introduces a new communication model to establish lower bounds on the time complexity of dynamic data structures, particularly for the interval union problem, revealing fundamental computational limits.

Contribution

It presents a novel communication model and leverages it to prove tight lower bounds for dynamic interval union and related graph problems, advancing understanding of data structure complexity.

Findings

Any data structure for dynamic interval union with constant error probability requires (n  .01) time in expectation.

The new communication model enables proving lower bounds via reductions from nondeterministic communication games.

Lower bounds are extended to several dynamic graph problems through problem reductions.

Abstract

In this paper, we develop a new communication model to prove a data structure lower bound for the dynamic interval union problem. The problem is to maintain a multiset of intervals $\mathcal{I}$ over $[0, n]$ with integer coordinates, supporting the following operations: - insert(a, b): add an interval $[a, b]$ to $\mathcal{I}$, provided that $a$ and $b$ are integers in $[0, n]$; - delete(a, b): delete a (previously inserted) interval $[a, b]$ from $\mathcal{I}$; - query(): return the total length of the union of all intervals in $\mathcal{I}$. It is related to the two-dimensional case of Klee's measure problem. We prove that there is a distribution over sequences of operations with $O(n)$ insertions and deletions, and $O(n^{0.01})$ queries, for which any data structure with any constant error probability requires $\Omega(n\log n)$ time in expectation. Interestingly, we use the sparse set disjointness protocol of H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of nondeterministic communication games, for which we prove lower bounds. For applications, we prove lower bounds for several dynamic graph problems by reducing them from dynamic interval union.