Tight Dynamic Problem Lower Bounds from Generalized BMM and OMv
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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 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 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.
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