This Article 
 Bibliographic References 
 Add to: 
A Modular Systolic Linearization of the Warshall-Floyd Algorithm
May 1996 (vol. 7 no. 5)
pp. 449-455

Abstract—In this paper, we use a variant of the geometric method to derive efficient modular linear systolic algorithms for the transitive closure and shortest path problems. Furthermore, we show that partially-pipelined modular linear systolic algorithms with an output operation, for matrix multiplication, can be as fast as the fully-pipelined existing ones and, moreover, they need less cells.

[1] A. Benaini and M. Tchuente, "Matrix Product on Linear Systolic Arrays," Parallel and Distributed Algorithms, M. Cosnard, P. Quinton, M. Raynal and Y. Robert, eds., NorthHolland, 1989.
[2] A.L. Fischer and H.T. Kung, "Synchronizing Large VLSI Processor Arrays," Proc. Tenth Ann. IEEE/ACM Symp. Computer Architecture, pp. 54-58, June 1983.
[3] L.J. Guibas, H.T. Kung, and C.D. Thompson, "Direct VLSI Implementation of Combinatorial Algorithms, Proc. Conf. Very Large Scale Integration: Architecture, Design, Fabrication, pp. 509-525, California Institute of Technology Inst., Jan. 1979.
[4] H.T. Kung, "Why Systolic Architecture," IEEE Computer, vol. 15, no. 1, pp. 37-46, Jan. 1980.
[5] F.T. Leighton and C.E. Leiserson, Wafer-Scale Integration of Systolic Arrays," Proc. 23rd Symp. Foundations of Computer Science, pp. 297-311, Nov. 1982.
[6] S.Y. Kung,S.C. Lo,, and P.S. Lewis,“Optimal systolic design for the transitive closure and the shortest path problems,” IEEE Trans. Computers, vol. 36, pp. 603-614, May 1987.
[7] P.Z. Lee and Z.M. Kedem,“Synthesizing linear array algorithms from nested for loop algorithms,” IEEE Trans. Computers, vol. 37, pp. 1,578-1,598, Dec. 1988.
[8] P. Lee and Z. Kedem, "On High Speed Computing with a Programmable Linear Array Synthesizing Linear Array Algorithms From Nested For Loop Algorithms," J. Supercomputing. vol. 4, pp. 223-249, 1990.
[9] D.I. Moldovan and J.A.B. Fortes, “Partitioning and Mapping Algorithms into Fixed Size Systolic Arrays,” IEEE Trans. Computers, vol. 35, no. 1, pp.1-12, Jan. 1986.
[10] J.F. Myoupo, "A Linear Systolic Array for Transitive Closure Problems, Proc. Int'l Conf. Parallel Process. (ICPP), vol. 1, pp. 617-618, 1990.
[11] J. F. Myoupo and A.C. Fabret, "Designing Modular Linear Systolic Arrays Using Dependence Graph Regular Partitions, Rapport Interne, L.R.I. N°760, UniversitéParis XI, 1992.
[12] V.K. Prasanna Kumar and Y.C. Tsai, "Designing Linear Systolic Arrays, J. Parallel Distributed Computers, vol. 7, pp. 441-463, 1989.
[13] S.K. Prasanna Kumar and Y.C. Tsai, "On Mapping Algorithms to Linear and Fault-Tolerant Systolic Arrays," IEEE Trans. Computers, vol. 38, no. 3, pp. 470-478, 1989.
[14] I.V. Ramakrishnan, P.J. Varman, "Dynamic Programming and Transitive Closure on Linear Pipelines," Proc. Int'l. Conf. on Parallel Processing, (ICPP) 1984.
[15] I.V. Ramakrishnan, D.S. Fussel, and A. Silberschatz, "Mapping Homogenous Graphs on Linear Arrays," IEEE Trans. Computers, vol. 35, pp. 189-209, 1986.
[16] P.J. Varman and I.V. Ramakrishnan, "Synthesis of an Optimal Family of Matrix Multiplication Algorithms on Linear Arrays," IEEE Trans. Computers, vol. 35, no. 11, pp. 989-996, Nov. 1986.
[17] T. Risset, "Linear Systolic Arrays for Matrix Multiplication: Comparison of Existing Synthesis Methods and New Results," Algorithms and Parallel VLSI Architectures II, pp. 163-174, P. Quinton and Y. Robert eds. NorthHolland: Elsevier, 1991.
[18] G. Rote, "A Systolic Array Algorithm for the Algebraic Path Problem," Computing, vol. 34, pp. 192-219, 1985.
[19] W. Shang and J.A.B. Fortes, "On Time Mapping of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays," IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 5, pp. 350-363, May 1992.
[20] U. Schwiegelshohn and L. Thiele, "Linear Systolic Arrays for Matrix Computation," J. Parallel Distributed Computers, vol. 7, pp. 28-39, 1989.
[21] J. Teich and L. Thiele, "A Transformative Approach to the Partitioning of Processor Arrays," Proc. ASAP 1992, IEEE C. S. Press, pp. 4-20.
[22] J.D. Ullman, Computational aspects of VLSI, IEEE C. S. Press, 1984.
[23] J.L.A. Van de Snepscheut, "A Derivation of a Distributed Implementation of Warshall's Algorithm," Science of Computer Programming, vol. 7, pp. 55-60, 1986.
[24] S.W. Warman Jr., "A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations," CACM, vol. 18, no. 4, pp. 218-220, 1975.
[25] S. Warshall, "A Theorem on Boolean Matrices," J. ACM, vol. 9 no. 1, pp. 11-12, 1972.

Index Terms:
Modular linear systolic algorithms, transitive closure, shortest path, matrix multiplication.
Jean Frédéric Myoupo, Anne Cécile Fabret, "A Modular Systolic Linearization of the Warshall-Floyd Algorithm," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 5, pp. 449-455, May 1996, doi:10.1109/71.503769
Usage of this product signifies your acceptance of the Terms of Use.