Research output: Contribution to book/anthology/report/proceeding › Article in proceedings › Research › peer-review

**Tight cell probe bounds for succinct boolean matrix-Vector multiplication.** / Chakraborty, Diptarka; Kamma, Lior; Larsen, Kasper Green.

Research output: Contribution to book/anthology/report/proceeding › Article in proceedings › Research › peer-review

Chakraborty, D, Kamma, L & Larsen, KG 2018, Tight cell probe bounds for succinct boolean matrix-Vector multiplication. in *STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing.* Association for Computing Machinery, A C M Symposium on the Theory of Computing. Annual Proceedings, no. 2018, pp. 1297-1306, 50th Annual ACM Symposium on Theory of Computing, STOC 2018, Los Angeles, United States, 25/06/2018. https://doi.org/10.1145/3188745.3188830

Chakraborty, D., Kamma, L., & Larsen, K. G. (2018). Tight cell probe bounds for succinct boolean matrix-Vector multiplication. In *STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing *(pp. 1297-1306). Association for Computing Machinery. A C M Symposium on the Theory of Computing. Annual Proceedings, No. 2018 https://doi.org/10.1145/3188745.3188830

Chakraborty D, Kamma L, Larsen KG. 2018. Tight cell probe bounds for succinct boolean matrix-Vector multiplication. In STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. pp. 1297-1306. (A C M Symposium on the Theory of Computing. Annual Proceedings; No. 2018). https://doi.org/10.1145/3188745.3188830

Chakraborty, Diptarka, Lior Kamma, and Kasper Green Larsen "Tight cell probe bounds for succinct boolean matrix-Vector multiplication". *STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing.* Association for Computing Machinery. (A C M Symposium on the Theory of Computing. Annual Proceedings; Journal number 2018). 2018, 1297-1306. https://doi.org/10.1145/3188745.3188830

Chakraborty D, Kamma L, Larsen KG. Tight cell probe bounds for succinct boolean matrix-Vector multiplication. In STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. 2018. p. 1297-1306. (A C M Symposium on the Theory of Computing. Annual Proceedings; No. 2018). https://doi.org/10.1145/3188745.3188830

Chakraborty, Diptarka ; Kamma, Lior ; Larsen, Kasper Green. / **Tight cell probe bounds for succinct boolean matrix-Vector multiplication**. STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery, 2018. pp. 1297-1306 (A C M Symposium on the Theory of Computing. Annual Proceedings; No. 2018).

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title = "Tight cell probe bounds for succinct boolean matrix-Vector multiplication",

abstract = "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{\textquoteright}15]. In recent work, Larsen and Williams [SODA{\textquoteright}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 {\~O}(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional {\~O}(n7/4) 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 {\~O}(n3/2) storing just {\~O}(n3/2) 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 < n2 must have query time t satisfying tr = Ω (n3). For r ≤ n, any data structure must have t = Ω (n2). 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.",

keywords = "Boolean matrix-vector multiplication, Cell probe model, Lower bound, Succinct data structure",

author = "Diptarka Chakraborty and Lior Kamma and Larsen, {Kasper Green}",

year = "2018",

month = jun,

day = "20",

doi = "10.1145/3188745.3188830",

language = "English",

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number = "2018",

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N2 - 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(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n7/4) 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(n3/2) storing just Õ(n3/2) 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 < n2 must have query time t satisfying tr = Ω (n3). For r ≤ n, any data structure must have t = Ω (n2). 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.

AB - 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(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n7/4) 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(n3/2) storing just Õ(n3/2) 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 < n2 must have query time t satisfying tr = Ω (n3). For r ≤ n, any data structure must have t = Ω (n2). 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.

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KW - Succinct data structure

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