Integer multiplication in PARSAC-2 on stock microprocessors

Kuechlin, Wolfgang; Lutz, David; Nevin, Nicholas

doi:10.1007/3-540-54522-0_109

Cited by 7 publications

(7 citation statements)

References 3 publications

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“…Similarly, parallel versions of the Karatsuba's algorithm are mostly presented for the shared memory setting [25], or focus on the experimental analysis of specific implementations without formally analyzing the scalability and the communication cost of the proposed algorithms [11]. In [10], Cesari and Maeder introduce three parallel Karatusba-based algorithms for the distributed memory setting: The first two algorithms have time complexity O(n), where n denotes the number of digits of the input integers, when using n log 2 3 processors.…”

Section: Related Workmentioning

confidence: 99%

Communication-Optimal Parallel Standard and Karatsuba Integer Multiplication in the Distributed Memory Model

Stefani¹

2020

Preprint

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We present COPSIM a parallel implementation of standard integer multiplication for the distributed memory setting, and COPK a parallel implementation of Karatsuba's fast integer multiplication algorithm for a distributed memory setting. When using P processors, each equipped with a local memory, to compute the product of tho n-digits integer numbers, under mild conditions, our algorithms achieve optimal speedup of the computational time. That is, O n 2 /P for COPSIM, and O n log 2 3 /P for COPK. The total amount of memory required across the processors is O (n), that is, within a constant factor of the minimum space required to store the input values. We rigorously analyze the Input/Output (I/O) cost of the proposed algorithms. We show that their bandwidth cost (i.e., the number of memory words sent or received by at least one processors) matches asymptotically corresponding known I/O lower bounds, and their latency (i.e., the number of messages sent or received in the algorithm's critical execution path) is asymptotically within a multiplicative factor O log 2 2 P of the corresponding known I/O lower bounds. Hence, our algorithms are asymptotically optimal with respect to the bandwidth cost and almost asymptotically optimal with respect to the latency cost.

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Section: Related Workmentioning

confidence: 99%

Communication-Optimal Parallel Standard and Karatsuba Integer Multiplication in the Distributed Memory Model

Stefani¹

2020

Preprint

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“…CLN instead movides also a Newton method used for lame irmut We are not aware of any multiple precision package studied for distributed memory machines. Some preliminary results for specific algorithms are reported in [13,29,26]. For parallel shared memories machines few librariea exist, which parallelize high level algorithms and use sequential arith-metic [20].…”

Section: Basic Algorithmsmentioning

confidence: 99%

Calypso

Cesari

1997

Proceedings of the Second International Symposium on Parallel Symbolic Computation - PASCO '97

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We present a package of parallel algorithms for multiple precision arithmetic, implemented by using a mesaagepassing model of computation.These algorithms are organized in an object oriented library and perform parallel arithmetic in Z, Q, and ZP.The library hss a layered structure which provides ports bility and the possibility of extending the code e=ily. In the bottom layer we implement fundamental parallel algorithms which are machine dependent. Starting from integer mu]triplication, we developed different parallel Karatsub&type and FFT-b=ed schemes, including the integer 3-primes and floating-point FFT algorithms. These multiplication algorithms allowed us to design a pamllel Newton method for division. Parallel algorithms for modular arithmetic result by using both Montgomery's and classical arithmetic.We designed these algorithms under a messag~passing model of computation.These are implemented on different parallel platforms, targeting both distributed memory machhes, such as mwiively parallel processing systems and networks of workst ations, and shared memory architectures. We developed both architecture dependent and architecture independent algorithms by using standard interfaces such as the Message Passing Interface, MPI.On top of this layer we provide a convenient interface to the user which allows a sequential programming style, whale each algorithm is executed in parallel. At this level it is possible for the user to implement high level parallel algorithms, such m the GCD, which are machine independent.Finally we integrated our package in a computer algebra system, which uses a sequential frontend for interactive use, and a parallel backend for computation.We have called this package CALYPSO, a sort of acronym for computer Algebru Libmry for Pamllel Symbolic Computation.Permission to make digital/hard copy of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. PASCO'97, Wailea, Maui, Hawaii; @1997 ACM 0-89791-951-3/97/0007. . .US$3.50 ComputationThe motivation of our work, that is writing a package for parallel multiple precision arithmetic, originates from a more general instance d~igning and implementing a parallel computer algebra system to solve practical problems which are still intractable with the existing software. We are thinkingof problems such as integer and polynomial factorization, Grobner basis computation, symbolic solution of large systems of equations, solution of differential equations. These problems are definitely not solved either by general purpose systems such as MATHEMATICAL [43] or MAPLE [12], nor by dedicated systems. On the other hand, with the lat@ advances in parallel computer hardware, the large amo...

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“…Parallel implementations of Karatsuba's algorithm have been reported in Kuechlin et al (1991) on a shared memory architecture and in Char et al (1994) on a network of workstations. However, Kuechlin et al (1991) do not discuss the implementation and do not analyse the scalability of the algorithm.…”

Section: Introductionmentioning

confidence: 97%

“…However, Kuechlin et al (1991) do not discuss the implementation and do not analyse the scalability of the algorithm. Char et al (1994) give experimental data of two different implementations without providing satisfactory explanation of the results due to unpredictable behavior of the network.…”

Section: Introductionmentioning

confidence: 97%

Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures

Cesari

Maeder

1996

Journal of Symbolic Computation

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We present three parallel implementations of the Karatsuba algorithm for long integer multiplication on a distributed memory architecture and discuss the experimental results obtained on a Paragon computer. The first two implementations have both time complexity O(n) on n log 2 3 processors, but present different behavior for inputs of practical size. The third algorithm has complexity O(n log 2 n) on n processors, offering therefore better asymptotic efficiency. A refinement of the asymptotic analysis for the important case of a constant number of processors takes into account sequential parts of the algorithm and communications overhead. It is shown that the theoretically best speed-up and efficiency can be obtained with two of the algorithms for sufficient problem size. The experimental results confirm the analysis.

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Integer multiplication in PARSAC-2 on stock microprocessors

Cited by 7 publications

References 3 publications

Communication-Optimal Parallel Standard and Karatsuba Integer Multiplication in the Distributed Memory Model

Communication-Optimal Parallel Standard and Karatsuba Integer Multiplication in the Distributed Memory Model

Calypso

Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures

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