Tight Dynamic Problem Lower Bounds from Generalized BMM and OMv

TL;DR

This paper establishes tight conditional lower bounds for various dynamic problems using generalized BMM and OMv hypotheses, advancing understanding of their computational complexity.

Contribution

It introduces generalized hypotheses and proves tight lower bounds for multiple dynamic problems, extending prior results and providing a unified framework.

Findings

Tight lower bounds for Dynamic Range Mode match upper bounds.

New bounds for Dynamic Subgraph Connectivity and Range Color Counting.

Introduction of the OuMv_k hypothesis for high-dimensional problems.

Abstract

The main theme of this paper is using $k$-dimensional generalizations of the combinatorial Boolean Matrix Multiplication (BMM) hypothesis and the closely-related Online Matrix Vector Multiplication (OMv) hypothesis to prove new tight conditional lower bounds for dynamic problems. The combinatorial $k$-Clique hypothesis, which is a standard hypothesis in the literature, naturally generalizes the combinatorial BMM hypothesis. In this paper, we prove tight lower bounds for several dynamic problems under the combinatorial $k$-Clique hypothesis. For instance, we show that: * The Dynamic Range Mode problem has no combinatorial algorithms with $\mathrm{poly}(n)$ pre-processing time, $O(n^{2/3-\epsilon})$ update time and $O(n^{2/3-\epsilon})$ query time for any $\epsilon > 0$, matching the known upper bounds for this problem. Previous lower bounds only ruled out algorithms with $O(n^{1/2-\epsilon})$ update and query time under the OMv hypothesis. Other examples include tight combinatorial lower bounds for Dynamic Subgraph Connectivity, Dynamic 2D Orthogonal Range Color Counting, Dynamic 2-Pattern Document Retrieval, and Dynamic Range Mode in higher dimensions. Furthermore, we propose the OuMv$_k$ hypothesis as a natural generalization of the OMv hypothesis. Under this hypothesis, we prove tight lower bounds for various dynamic problems. For instance, we show that: * The Dynamic Skyline Points Counting problem in $(2k-1)$-dimensional space has no algorithm with $\mathrm{poly}(n)$ pre-processing time and $O(n^{1-1/k-\epsilon})$ update and query time for $\epsilon > 0$, even if the updates are semi-online. Other examples include tight conditional lower bounds for (semi-online) Dynamic Klee's measure for unit cubes, and high-dimensional generalizations of Erickson's problem and Langerman's problem.