Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC’15]. In recent work, Larsen and Williams [SODA’17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in Õ(n) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time Õ(n) storing just Õ(n) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n must have query time t satisfying tr = Ω̃(n). For r ≤ n, any data structure must have t = Ω̃(n). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over F2. Computer Science Institute of Charles University, Prague. [email protected]. Supported by the funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/20072013)/ERC Grant Agreement no. 616787. MADALGO. Aarhus University. [email protected]. Supported by a Villum Young Investigator Grant. MADALGO. Aarhus University. [email protected]. Supported by a Villum Young Investigator Grant and an AUFF Starting Grant.