Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, Michael Wagner
2020 Innovations in Theoretical Computer Science  
Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an NP oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the n-dimensional inner product problem with m queries, we prove that lower bounds on the query time imply
more » ... gidity lower bounds for the query set itself. In particular, an explicit lower bound of ω n r log m for r redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time Ω(n 3/2 /r) for redundancy r ≥ √ n in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Koucký, Loff, and Mukhopadhyay. ACM Subject Classification Theory of computation → Cell probe models and lower bounds; Theory of computation → Circuit complexity
doi:10.4230/lipics.itcs.2020.35 dblp:conf/innovations/RamamoorthyR20 fatcat:wfes7y25c5acdmvjcske5vnaoy