Orbital Systolic Algorithms and Array Processors for Solution of the Algebraic Path Problem
The algebraic path problem (APP) is a general framework which unifies several solution procedures for a number of well-known matrix and graph problems. In this paper, we present a new 3-dimensional (3-D) orbital algebraic path algorithm and corresponding 2-D toroidal array processors which solve the...
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| Published in | IEICE Transactions on Information and Systems Vol. E93.D; no. 3; pp. 534 - 541 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Oxford
The Institute of Electronics, Information and Communication Engineers
2010
Oxford University Press |
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| Online Access | Get full text |
| ISSN | 0916-8532 1745-1361 1745-1361 |
| DOI | 10.1587/transinf.E93.D.534 |
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| Abstract | The algebraic path problem (APP) is a general framework which unifies several solution procedures for a number of well-known matrix and graph problems. In this paper, we present a new 3-dimensional (3-D) orbital algebraic path algorithm and corresponding 2-D toroidal array processors which solve the n × n APP in the theoretically minimal number of 3n time-steps. The coordinated time-space scheduling of the computing and data movement in this 3-D algorithm is based on the modular function which preserves the main technological advantages of systolic processing: simplicity, regularity, locality of communications, pipelining, etc. Our design of the 2-D systolic array processors is based on a classical 3-D→2-D space transformation. We have also shown how a data manipulation (copying and alignment) can be effectively implemented in these array processors in a massively-parallel fashion by using a matrix-matrix multiply-add operation. |
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| AbstractList | The algebraic path problem (APP) is a general framework which unifies several solution procedures for a number of well-known matrix and graph problems. In this paper, we present a new 3-dimensional (3-D) orbital algebraic path algorithm and corresponding 2-D toroidal array processors which solve the n × n APP in the theoretically minimal number of 3n time-steps. The coordinated time-space scheduling of the computing and data movement in this 3-D algorithm is based on the modular function which preserves the main technological advantages of systolic processing: simplicity, regularity, locality of communications, pipelining, etc. Our design of the 2-D systolic array processors is based on a classical 3-D→2-D space transformation. We have also shown how a data manipulation (copying and alignment) can be effectively implemented in these array processors in a massively-parallel fashion by using a matrix-matrix multiply-add operation. |
| Author | MIYAZAKI, Toshiaki SEDUKHIN, Stanislav G. KURODA, Kenichi |
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| Keywords | orbital systolic algorithms Array processor data manipulation Step method Scheduling Systolic network Algorithm Implementation Pipeline processing Three dimensional model array processors Posterior probability algebraic path problem Integrated circuit Massive parallelism |
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Tsay, “A family of efficient regular arrays for algebraic path problem,” IEEE Trans. Comput., vol.43, no.7, pp.769-777, July 1994. [5] G. Rote, “Path problems in graphs,” Computing Supplementum, vol.7, pp.155-189, 1990. [7] E. Fink, “A survey of sequential and systolic algorithms for the algebraic path problem,” Technical Report CS-92-37, Department of Computer Science, University of Waterloo, 1992. [12] S.Y. Kung, S.C. Lo, and P.S. Lewis, “Optimal systolic design for the transitive closure and the shortest path problems,” IEEE Trans. Comput., vol.36, no.5, pp.603-614, 1987. [20] G. Loh, “3D-stacked memory architectures for multi-core processors,” Computer Architecture, 2008. ISCA '08. 35th International Symposium, pp.453-464, June 2008. [9] C.T. Djamégni, P. Quinton, S. Rajopadhye, and T. Risset, “Derivation of systolic algorithms for the algebraic path problem by recurrence transformations,” Parallel Comput., vol.26, no.11, pp.1429-1445, 2000. [29] J.D. Ullmann, Computational aspects of VLSI, Computer Science Press, New York, 1984. [1] D.J. Lehmann, “Algebraic structures for transitive closure,” Theor. Comput. Sci., vol.4, no.1, pp.59-76, 1977. [14] T. Takaoka and K. Umehara, “An efficient vlsi algorithms for the all pairs shortest path problem,” J. Parallel Distrib. Comput., vol.16, no.3, pp.265-270, 1992. [8] D. Cachera, S. Rajopadhye, T. Risset, and C. Tadonki, “Parallelization of the algebraic path problem on linear SIMD/SPMD arrays,” Technical Report 1346, Irisa, 2001. [25] L.J. Guibas, H.T. Kung, and C.D. Thompson, “Direct VLSI implementation of combinatorial algorithms,” Proc. First Caltech Conference on VLSI, pp.509-525, Pasadena, CA, California Institute of Technology, Jan. 1979. [6] S.G. Sedukhin, “Design and analysis of systolic algorithms for the algebraic path problem,” Computers and Artificial Intelligence, vol.11, pp.269-292, 1992. [27] G.H. Golub and C.F.V. Loan, Matrix Computations, Third ed., The John Hopkins University Press, Baltimore, Maryland, 1996. [28] M. Penner, J.S. Park, and V.K. Prasanna, “Optimizing graph algorithms for improved cache performance,” IEEE Trans. Parallel Distrib. Syst., vol.15, no.9, pp.769-782, 2004. [24] S.M. Chai and D.S. Wills, “Systolic opportunities for multidimensional data streams,” IEEE Trans. Parallel Distrib. Syst., vol.13, no.4, pp.388-398, 2002. [15] A. Benaini and Y. Robert, “Space-time minimal systolic arrays for Gaussian elimination and the algebraic path problem,” Parallel Comput., vol.15, no.1-3, pp.211-225, 1990. [16] C. Scheiman and P. Cappello, “A processor-time-minimal systolic array for transitive closure,” IEEE Trans. Parallel Distrib. Syst., vol.3, no.3, pp.257-269, 1992. [21] S. Vangal, J. Howard, G. Ruhl, S. Dighe, H. Wilson, J. Tschanz, D. Finan, P. Iyer, A. Singh, T. Jacob, S. Jain, S. Venkataraman, Y. Hoskote, and N. Borkar, “An 80-tile 1.28TFLOPS network-on-chip in 65nm CMOS,” Solid-State Circuits Conference, 2007. ISSCC 2007. Digest of Technical Papers. pp.98-589, IEEE International, Feb. 2007. [11] S. Rajopadhye, “An improved systolic algorithm for the algebraic path problem,” Integr. VLSI J., vol.14, no.3, pp.279-296, 1993. [2] G. Rote, “A systolic array algorithm for the algebraic path problem,” Computing, vol.34, pp.191-219, 1985. [18] M. Koyanagi, T. Fukushima, and T. Tanaka, “Three-dimensional integration technology and integrated systems,” Design Automation Conference, 2009. ASP-DAC 2009. Asia and South Pacific, pp.409-415, Jan. 2009. [3] Y. Robert and D. Trystram, “Parallel implementation of the algebraic path problem,” Proc. Conference on Algorithms and Hardware for Parallel Processing on CONPAR 86, pp.149-156, New York, NY, USA, Springer-Verlag New York, 1986. [26] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms, Second ed., The MIT Press and McGraw-Hill Book Company, 2001. [31] L. Cannon, A cellular computer to implement the Kalman filter algorithm, Ph.D. Thesis, Montana State Univ., 1969. [17] D. Lavenier, P. Quinton, and S. Rajopadhye, “Advanced systolic design,” in Digital signal processing for multimedia systems, ed. K.K. Parhi and T. Nishitani, pp.657-692, CRC Press, 1999. 22 23 24 25 26 27 (6) 1992; 11 28 29 30 31 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 7 8 9 20 21 |
| References_xml | – reference: [12] S.Y. Kung, S.C. Lo, and P.S. Lewis, “Optimal systolic design for the transitive closure and the shortest path problems,” IEEE Trans. Comput., vol.36, no.5, pp.603-614, 1987. – reference: [17] D. Lavenier, P. Quinton, and S. Rajopadhye, “Advanced systolic design,” in Digital signal processing for multimedia systems, ed. K.K. Parhi and T. Nishitani, pp.657-692, CRC Press, 1999. – reference: [10] T. Risset and Y. Robert, “Synthesis of processor arrays for the algebraic path problem: Unified old results and deriving new architectures,” Parallel Processing Letters, no.11, pp.19-28, 1991. – reference: [3] Y. Robert and D. Trystram, “Parallel implementation of the algebraic path problem,” Proc. Conference on Algorithms and Hardware for Parallel Processing on CONPAR 86, pp.149-156, New York, NY, USA, Springer-Verlag New York, 1986. – reference: [26] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms, Second ed., The MIT Press and McGraw-Hill Book Company, 2001. – reference: [8] D. Cachera, S. Rajopadhye, T. Risset, and C. Tadonki, “Parallelization of the algebraic path problem on linear SIMD/SPMD arrays,” Technical Report 1346, Irisa, 2001. – reference: [29] J.D. Ullmann, Computational aspects of VLSI, Computer Science Press, New York, 1984. – reference: [22] E.H. Heijne, “Gigasensors for an attoscope: Catching quanta in CMOS,” IEEE Solid-State Circuits Newsletter, vol.13, no.4, pp.28-34, Oct. 2008. – reference: [27] G.H. Golub and C.F.V. Loan, Matrix Computations, Third ed., The John Hopkins University Press, Baltimore, Maryland, 1996. – reference: [31] L. Cannon, A cellular computer to implement the Kalman filter algorithm, Ph.D. Thesis, Montana State Univ., 1969. – reference: [5] G. Rote, “Path problems in graphs,” Computing Supplementum, vol.7, pp.155-189, 1990. – reference: [24] S.M. Chai and D.S. Wills, “Systolic opportunities for multidimensional data streams,” IEEE Trans. Parallel Distrib. Syst., vol.13, no.4, pp.388-398, 2002. – reference: [4] B.M. Maggs and S.A. Poltkin, “Minimum-cost spanning tree as a path-finding problem,” Inf. Process. Lett., vol.26, no.6, pp.291-293, 1988. – reference: [13] P.Y. Chang and J.C. Tsay, “A family of efficient regular arrays for algebraic path problem,” IEEE Trans. Comput., vol.43, no.7, pp.769-777, July 1994. – reference: [2] G. Rote, “A systolic array algorithm for the algebraic path problem,” Computing, vol.34, pp.191-219, 1985. – reference: [16] C. Scheiman and P. Cappello, “A processor-time-minimal systolic array for transitive closure,” IEEE Trans. Parallel Distrib. Syst., vol.3, no.3, pp.257-269, 1992. – reference: [20] G. Loh, “3D-stacked memory architectures for multi-core processors,” Computer Architecture, 2008. ISCA '08. 35th International Symposium, pp.453-464, June 2008. – reference: [15] A. Benaini and Y. Robert, “Space-time minimal systolic arrays for Gaussian elimination and the algebraic path problem,” Parallel Comput., vol.15, no.1-3, pp.211-225, 1990. – reference: [7] E. Fink, “A survey of sequential and systolic algorithms for the algebraic path problem,” Technical Report CS-92-37, Department of Computer Science, University of Waterloo, 1992. – reference: [6] S.G. Sedukhin, “Design and analysis of systolic algorithms for the algebraic path problem,” Computers and Artificial Intelligence, vol.11, pp.269-292, 1992. – reference: [28] M. Penner, J.S. Park, and V.K. Prasanna, “Optimizing graph algorithms for improved cache performance,” IEEE Trans. Parallel Distrib. Syst., vol.15, no.9, pp.769-782, 2004. – reference: [11] S. Rajopadhye, “An improved systolic algorithm for the algebraic path problem,” Integr. VLSI J., vol.14, no.3, pp.279-296, 1993. – reference: [18] M. Koyanagi, T. Fukushima, and T. Tanaka, “Three-dimensional integration technology and integrated systems,” Design Automation Conference, 2009. ASP-DAC 2009. Asia and South Pacific, pp.409-415, Jan. 2009. – reference: [21] S. Vangal, J. Howard, G. Ruhl, S. Dighe, H. Wilson, J. Tschanz, D. Finan, P. Iyer, A. Singh, T. Jacob, S. Jain, S. Venkataraman, Y. Hoskote, and N. Borkar, “An 80-tile 1.28TFLOPS network-on-chip in 65nm CMOS,” Solid-State Circuits Conference, 2007. ISSCC 2007. Digest of Technical Papers. pp.98-589, IEEE International, Feb. 2007. – reference: [30] A.S. Zekri and S.G. Sedukhin, “Computationally efficient parallel matrix-matrix multiplication on the torus,” ISHPC-6th Int. Symp. on High Performance Computing, pp.219-226, Springer-Verlag, Sept. 2005. – reference: [23] Wikipedia, “FLOPS: Cost of computing,” Website, http://en.wikipedia.org/wiki/GFlop, 2009. – reference: [1] D.J. Lehmann, “Algebraic structures for transitive closure,” Theor. Comput. Sci., vol.4, no.1, pp.59-76, 1977. – reference: [19] Intel, “Tera-scale Computing Research Program,” Website, http://techresearch.intel.com/articles/Tera-Scale/1421.htm, 2009. – reference: [14] T. Takaoka and K. Umehara, “An efficient vlsi algorithms for the all pairs shortest path problem,” J. Parallel Distrib. Comput., vol.16, no.3, pp.265-270, 1992. – reference: [9] C.T. Djamégni, P. Quinton, S. Rajopadhye, and T. Risset, “Derivation of systolic algorithms for the algebraic path problem by recurrence transformations,” Parallel Comput., vol.26, no.11, pp.1429-1445, 2000. – reference: [25] L.J. Guibas, H.T. Kung, and C.D. Thompson, “Direct VLSI implementation of combinatorial algorithms,” Proc. First Caltech Conference on VLSI, pp.509-525, Pasadena, CA, California Institute of Technology, Jan. 1979. – ident: 15 doi: 10.1016/0167-8191(90)90044-A – ident: 17 doi: 10.1201/9781482276046-23 – ident: 21 doi: 10.1109/ISSCC.2007.373606 – ident: 2 doi: 10.1007/BF02253318 – ident: 31 – ident: 1 doi: 10.1016/0304-3975(77)90056-1 – ident: 14 doi: 10.1016/0743-7315(92)90038-O – ident: 9 – ident: 7 – ident: 26 – ident: 3 doi: 10.1007/3-540-16811-7_165 – ident: 24 doi: 10.1109/71.995819 – ident: 10 doi: 10.1142/S0129626491000173 – ident: 22 doi: 10.1109/N-SSC.2008.4785820 – ident: 4 doi: 10.1016/0020-0190(88)90185-8 – ident: 30 doi: 10.1007/978-3-540-77704-5_19 – ident: 20 doi: 10.1109/ISCA.2008.15 – ident: 13 doi: 10.1109/12.293256 – ident: 19 – ident: 5 doi: 10.1007/978-3-7091-9076-0_9 – ident: 28 doi: 10.1109/TPDS.2004.44 – ident: 29 – ident: 11 doi: 10.1016/0167-9260(93)90012-2 – ident: 16 doi: 10.1109/71.139200 – volume: 11 start-page: 269 issn: 0232-0274 issue: 3 year: 1992 ident: 6 – ident: 12 doi: 10.1109/TC.1987.1676945 – ident: 8 – ident: 27 – ident: 18 doi: 10.1109/ASPDAC.2009.4796515 – ident: 25 – ident: 23 |
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| SubjectTerms | algebraic path problem Applied sciences array processors data manipulation Design. Technologies. Operation analysis. Testing Electronics Exact sciences and technology Integrated circuits Integrated circuits by function (including memories and processors) orbital systolic algorithms Semiconductor electronics. Microelectronics. Optoelectronics. Solid state devices |
| Title | Orbital Systolic Algorithms and Array Processors for Solution of the Algebraic Path Problem |
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