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Communicationoptimal parallel 2.5D matrix multiplication and LU factorization algorithms
"... One can use extra memory to parallelize matrix multiplication by storing p 1/3 redundant copies of the input matrices on p processors in order to do asymptotically less communication than Cannon’s algorithm [2], and be faster in practice [1]. We call this algorithm “3D ” because it arranges the p pr ..."
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Cited by 34 (16 self)
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One can use extra memory to parallelize matrix multiplication by storing p 1/3 redundant copies of the input matrices on p processors in order to do asymptotically less communication than Cannon’s algorithm [2], and be faster in practice [1]. We call this algorithm “3D ” because it arranges the p processors in a 3D array, and Cannon’s algorithm “2D ” because it stores a single copy of the matrices on a 2D array of processors. We generalize these 2D and 3D algorithms by introducing a new class of “2.5D algorithms”. For matrix multiplication, we can take advantage of any amount of extra memory to store c copies of the data, for any c ∈{1, 2,..., ⌊p 1/3 ⌋}, to reduce the bandwidth cost of Cannon’s algorithm by a factor of c 1/2 and the latency cost by a factor c 3/2. We also show that these costs reach the lower bounds [13, 3], modulo polylog(p) factors. We similarly generalize LU decomposition to 2.5D and 3D, including communicationavoiding pivoting, a stable alternative to partialpivoting [7]. We prove a novel lower bound on the latency cost of 2.5D and 3D LU factorization, showing that while c copies of the data can also reduce the bandwidth by a factor of c 1/2, the latency must increase by a factor of c 1/2, so that the 2D LU algorithm (c = 1) in fact minimizes latency. Preliminary results of 2.5D matrix multiplication on a Cray XT4 machine also demonstrate a performance gain of up to 3X with respect to Cannon’s algorithm. Careful choice of c also yields up to a 2.4X speedup over 3D matrix multiplication, due to a better balance between communication costs.
Communicationoptimal parallel algorithm for Strassen’s matrix multiplication
 In Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’12
, 2012
"... Parallel matrix multiplication is one of the most studied fundamental problems in distributed and high performance computing. We obtain a new parallel algorithm that is based on Strassen’s fast matrix multiplication and minimizes communication. The algorithm outperforms all known parallel matrix mul ..."
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Cited by 28 (17 self)
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Parallel matrix multiplication is one of the most studied fundamental problems in distributed and high performance computing. We obtain a new parallel algorithm that is based on Strassen’s fast matrix multiplication and minimizes communication. The algorithm outperforms all known parallel matrix multiplication algorithms, classical and Strassenbased, both asymptotically and in practice. A critical bottleneck in parallelizing Strassen’s algorithm is the communication between the processors. Ballard, Demmel, Holtz, and Schwartz (SPAA’11) prove lower bounds on these communication costs, using expansion properties of the underlying computation graph. Our algorithm matches these lower bounds, and so is communicationoptimal. It exhibits perfect strong scaling within the maximum possible range.
The Design and Analysis of BulkSynchronous Parallel Algorithms
, 1998
"... The model of bulksynchronous parallel (BSP) computation is an emerging paradigm of generalpurpose parallel computing. This thesis presents a systematic approach to the design and analysis of BSP algorithms. We introduce an extension of the BSP model, called BSPRAM, which reconciles sharedmemory s ..."
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Cited by 19 (2 self)
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The model of bulksynchronous parallel (BSP) computation is an emerging paradigm of generalpurpose parallel computing. This thesis presents a systematic approach to the design and analysis of BSP algorithms. We introduce an extension of the BSP model, called BSPRAM, which reconciles sharedmemory style programming with efficient exploitation of data locality. The BSPRAM model can be optimally simulated by a BSP computer for a broad range of algorithms possessing certain characteristic properties: obliviousness, slackness, granularity. We use BSPRAM to design BSP algorithms for problems from three large, partially overlapping domains: combinatorial computation, dense matrix computation, graph computation. Some of the presented algorithms are adapted from known BSP algorithms (butterfly dag computation, cube dag computation, matrix multiplication). Other algorithms are obtained by application of established nonBSP techniques (sorting, randomised list contraction, Gaussian elimination without pivoting and with column pivoting, algebraic path computation), or use original techniques specific to the BSP model (deterministic list contraction, Gaussian elimination with nested block pivoting, communicationefficient multiplication of Boolean matrices, synchronisationefficient shortest paths computation). The asymptotic BSP cost of each algorithm is established, along with its BSPRAM characteristics. We conclude by outlining some directions for future research.
Improving communication performance in dense linear algebra via topology aware collectives
, 2011
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A PolyAlgorithm for Parallel Dense Matrix Multiplication on TwoDimensional Process Grid Topologies
, 1995
"... In this paper, we present several new and generalized parallel dense matrix multiplication algorithms of the form C = αAB + βC on twodimensional process grid topologies. These algorithms can deal with rectangular matrices distributed on rectangular grids. We classify these algori ..."
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Cited by 14 (1 self)
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In this paper, we present several new and generalized parallel dense matrix multiplication algorithms of the form C = &alpha;AB + &beta;C on twodimensional process grid topologies. These algorithms can deal with rectangular matrices distributed on rectangular grids. We classify these algorithms coherently into three categories according to the communication primitives used and thus we offer a taxonomy for this family of related algorithms. All these algorithms are represented in the data distribution independent approach and thus do not require a specific data distribution for correctness. The algorithmic compatibility condition result shown here ensures the correctness of the matrix multiplication. We define and extend the data distribution functions and introduce permutation compatibility and algorithmic compatibility. We also discuss a permutation compatible data distribution (modified virtual 2D data distribution). We conclude that no single algorithm always achieves the best performance...
Trading Replication For Communication In Parallel DistributedMemory Dense Solvers
, 2002
"... We present new communicationefficient parallel dense linear solvers: a solver for triangular linear systems with multiple righthand sides and an LU factorization algorithm. These solvers are highly parallel and they perform a factor of 0.4P 1/6 less communication than existing algorithms, where ..."
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Cited by 13 (1 self)
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We present new communicationefficient parallel dense linear solvers: a solver for triangular linear systems with multiple righthand sides and an LU factorization algorithm. These solvers are highly parallel and they perform a factor of 0.4P 1/6 less communication than existing algorithms, where P is number of processors. The new solvers reduce communication at the expense of using more temporary storage. Previously, algorithms that reduce communication by using more memory were only known for matrix multiplication. Our algorithms are recursive, elegant, and relatively simple to implement. We have implemented them using MPI, a messagepassing libray, and tested them on a cluster of workstations. 1