广义BMM和OMv的紧动力问题下界
Tight Dynamic Problem Lower Bounds from Generalized BMM and OMv
摘要：
利用组合布尔矩阵乘法（BMM）假设和相关的在线矩阵向量乘法（OMv）假设的$k$维推广，证明了动力学问题的新的紧条件下界.组合$k$-团假设是文献中的标准假设，自然地推广了组合BMM假设。本文在组合$k$-Clique假设下证明了几个动力学问题的紧下界。例如，我们表明：* 动态范围模式问题没有组合算法与$\mathrm{poly}(n)$预处理时间，$O(n^{2/3-ε})$更新时间和$O(n^{2/3-ε})$查询时间的任何$ε> 0$，匹配已知的上限为这个问题。以前的下界只排除了在OMv假设下具有$O(n^{1/2-ε})$更新和查询时间的算法。其他示例包括用于动态子图连接性、动态2D正交范围颜色计数、动态2模式文档检索和更高维度中的动态范围模式的紧密组合下限。此外，我们提出OuMv $_k$假设作为OMv假设的自然推广。在这个假设下，我们证明了各种动态问题的紧下界。例如，我们表明：* 动态天际线点计数问题在$(2k-1)$维空间没有算法与$\mathrm{poly}(n)$预处理时间和$O(n^{1-1/k-ε})$更新和查询时间为$ε> 0$，即使更新是半在线的。其他例子包括单位立方体的（半在线）动态克利测度的紧条件下界，以及埃里克森问题和朗格曼问题的高维推广。
英文摘要：
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, $O(n^{2/3-ε})$ update time and $O(n^{2/3-ε})$ query time for any $ε> 0$, matching the known upper bounds for this problem. Previous lower bounds only ruled out algorithms with $O(n^{1/2-ε})$ 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 $O(n^{1-1/k-ε})$ update and query time for $ε> 0$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
