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.
翻译:本文的主旨是使用 $k$- complia complication (BMM) 组合式的 美元字型 Boolean 矩阵乘数(BMM) 假设和密切相关的 在线矩阵矢量乘数(OMv) 假设来证明动态问题的新的严格条件下限。 组合式 $k$- Clique 假设是文献中的标准假设, 自然将组合式 BMM 假设简单化 组合式 BMM 假设。 在本文中, 我们证明, 在组合式的 美元基数( 美元- 基数- 逻辑乘数 乘数( BMM) 假设中, 动态模式没有以 $\ mathrem@ (n) 预处理时间、 $(n\\\\ 2/3\\\ \ \ \ \ liclicrial) 更新时间和 $(nexcialal- complia) 时间里, 内, 内有Oal- oria- oria- lademodeal 时间。