Vector Multiplication

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Kasper Green Larsen - One of the best experts on this subject based on the ideXlab platform.

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    Symposium on the Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • STOC - Tight cell probe bounds for succinct Boolean matrix-Vector Multiplication
    Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    arXiv: Data Structures and Algorithms, 2017
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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'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 $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ 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 $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ 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 $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. 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 $\mathbb{F}_2$.

  • SODA - Faster online matrix-Vector Multiplication
    2017
    Co-Authors: Kasper Green Larsen, Ryan Williams
    Abstract:

    We consider the Online Boolean Matrix-Vector Multiplication (OMV) problem studied by Henzinger et al. [STOC'15]: given an n × n Boolean matrix M, we receive n Boolean Vectors v1,...,vn one at a time, and are required to output Mvi (over the Boolean semiring) before seeing the Vector vi+1, for all i. Previous known algorithms for this problem are combinatorial, running in O(n3 /log2 n) time. Henzinger et al. conjecture there is no O(n3−e) time algorithm for OMV, for all e > 0; their OMV conjecture is shown to imply strong hardness results for many basic dynamic problems. We give a substantially faster method for computing OMV, running in [EQUATION] randomized time. In fact, after seeing [EQUATION] Vectors, we already achieve [EQUATION] amortized time for matrix-Vector Multiplication. Our approach gives a way to reduce matrix-Vector Multiplication to solving a version of the Orthogonal Vectors problem, which in turn reduces to "small" algebraic matrix-matrix Multiplication. Applications include faster independent set detection, partial match retrieval, and 2-CNF evaluation. We also show how a modification of our method gives a cell probe data structure for OMV with worst case [EQUATION] time per query Vector, where w is the word size. This result rules out an unconditional proof of the OMV conjecture using purely information-theoretic arguments.

  • Faster Online Matrix-Vector Multiplication
    arXiv: Data Structures and Algorithms, 2016
    Co-Authors: Kasper Green Larsen, Ryan Williams
    Abstract:

    We consider the Online Boolean Matrix-Vector Multiplication (OMV) problem studied by Henzinger et al. [STOC'15]: given an $n \times n$ Boolean matrix $M$, we receive $n$ Boolean Vectors $v_1,\ldots,v_n$ one at a time, and are required to output $M v_i$ (over the Boolean semiring) before seeing the Vector $v_{i+1}$, for all $i$. Previous known algorithms for this problem are combinatorial, running in $O(n^3/\log^2 n)$ time. Henzinger et al. conjecture there is no $O(n^{3-\varepsilon})$ time algorithm for OMV, for all $\varepsilon > 0$; their OMV conjecture is shown to imply strong hardness results for many basic dynamic problems. We give a substantially faster method for computing OMV, running in $n^3/2^{\Omega(\sqrt{\log n})}$ randomized time. In fact, after seeing $2^{\omega(\sqrt{\log n})}$ Vectors, we already achieve $n^2/2^{\Omega(\sqrt{\log n})}$ amortized time for matrix-Vector Multiplication. Our approach gives a way to reduce matrix-Vector Multiplication to solving a version of the Orthogonal Vectors problem, which in turn reduces to "small" algebraic matrix-matrix Multiplication. Applications include faster independent set detection, partial match retrieval, and 2-CNF evaluation. We also show how a modification of our method gives a cell probe data structure for OMV with worst case $O(n^{7/4}/\sqrt{w})$ time per query Vector, where $w$ is the word size. This result rules out an unconditional proof of the OMV conjecture using purely information-theoretic arguments.

Rob H Bisseling - One of the best experts on this subject based on the ideXlab platform.

  • Cache-Oblivious Sparse Matrix-Vector Multiplication by Using Sparse Matrix Partitioning Methods
    SIAM Journal on Scientific Computing, 2009
    Co-Authors: A. N. Yzelman, Rob H Bisseling
    Abstract:

    In this article, we introduce a cache-oblivious method for sparse matrix-Vector Multiplication. Our method attempts to permute the rows and columns of the input matrix using a recursive hypergraph-based sparse matrix partitioning scheme so that the resulting matrix induces cache-friendly behavior during sparse matrix-Vector Multiplication. Matrices are assumed to be stored in row-major format, by means of the compressed row storage (CRS) or its variants incremental CRS and zig-zag CRS. The zig-zag CRS data structure is shown to fit well with the hypergraph metric used in partitioning sparse matrices for the purpose of parallel computation. The separated block-diagonal (SBD) form is shown to be the appropriate matrix structure for cache enhancement. We have implemented a run-time cache simulation library enabling us to analyze cache behavior for arbitrary matrices and arbitrary cache properties during matrix-Vector Multiplication within a $k$-way set-associative idealized cache model. The results of these simulations are then verified by actual experiments run on various cache architectures. In all these experiments, we use the Mondriaan sparse matrix partitioner in one-dimensional mode. The savings in computation time achieved by our matrix reorderings reach up to 50 percent, in the case of a large link matrix.

  • a two dimensional data distribution method for parallel sparse matrix Vector Multiplication
    Siam Review, 2005
    Co-Authors: Brendan Vastenhouw, Rob H Bisseling
    Abstract:

    A new method is presented for distributing data in sparse matrix-Vector Multiplication. The method is two-dimensional, tries to minimize the true communication volume, and also tries to spread the computation and communication work evenly over the processors. The method starts with a recursive bipartitioning of the sparse matrix, each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros. The communication volume caused by the split is minimized. After the matrix partitioning, the input and output Vectors are partitioned with the objective of minimizing the maximum communication volume per processor. Experimental results of our implementation, Mondriaan, for a set of sparse test matrices show a reduction in communication volume compared to one-dimensional methods, and in general a good balance in the communication work. Experimental timings of an actual parallel sparse matrix-Vector Multiplication on an SGI Origin 3800 computer show that a sufficiently large reduction in communication volume leads to savings in execution time.

Diptarka Chakraborty - One of the best experts on this subject based on the ideXlab platform.

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    Symposium on the Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • STOC - Tight cell probe bounds for succinct Boolean matrix-Vector Multiplication
    Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    arXiv: Data Structures and Algorithms, 2017
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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'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 $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ 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 $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ 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 $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. 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 $\mathbb{F}_2$.

Olaf O Storaasli - One of the best experts on this subject based on the ideXlab platform.

  • sparse matrix Vector Multiplication design on fpgas
    Field-Programmable Custom Computing Machines, 2007
    Co-Authors: Junqing Sun, Gregory D Peterson, Olaf O Storaasli
    Abstract:

    Creating a high throughput sparse matrix Vector Multiplication (SpMxV) implementation depends on a balanced system design. In this paper, we introduce the innovative SpMxV solver designed for FPGAs (SSF). Besides high computational throughput, system performance is optimized by reducing initialization time and overheads, minimizing and overlapping I/O operations, and increasing scalability. SSF accepts any matrix size and can be easily adapted to different data formats. SSF minimizes the control logic by taking advantage of the data flow via an innovative accumulation circuit which uses pipelined floating point adders. Compared to optimized software codes on a Pentium 4 microprocessor, our design achieves up to 20x speedup.

  • FCCM - Sparse Matrix-Vector Multiplication Design on FPGAs
    15th Annual IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM 2007), 2007
    Co-Authors: Junqing Sun, Gregory D Peterson, Olaf O Storaasli
    Abstract:

    Creating a high throughput sparse matrix Vector Multiplication (SpMxV) implementation depends on a balanced system design. In this paper, we introduce the innovative SpMxV solver designed for FPGAs (SSF). Besides high computational throughput, system performance is optimized by reducing initialization time and overheads, minimizing and overlapping I/O operations, and increasing scalability. SSF accepts any matrix size and can be easily adapted to different data formats. SSF minimizes the control logic by taking advantage of the data flow via an innovative accumulation circuit which uses pipelined floating point adders. Compared to optimized software codes on a Pentium 4 microprocessor, our design achieves up to 20x speedup.

Lior Kamma - One of the best experts on this subject based on the ideXlab platform.

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    Symposium on the Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • STOC - Tight cell probe bounds for succinct Boolean matrix-Vector Multiplication
    Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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’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 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

  • tight cell probe bounds for succinct boolean matrix Vector Multiplication
    arXiv: Data Structures and Algorithms, 2017
    Co-Authors: Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
    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'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 $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ 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 $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ 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 $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. 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 $\mathbb{F}_2$.