Abstract
This paper presents two new time and/or processor bounds for solving certain classes of linear recurrences. The first result provides a parallel algorithm for solving Xi=aiXi−1+di for 1≤i≤n in 210gn units of time using only 3/4n processors. The second results relate to solving Xi=Xi−1+Xi−2+di for 1≤i≤n. It is shown that Xi's can be computed in at most 310gn units of time using 5/4n processors. In the special case when di=0 for all i, it is shown that Xi's (the first n-Fibonacci numbers) can be computed in parallel in 210gn-1 units of time using only n/2 processors. These time and processor bounds compare very favourably with the previously known results for these problems.
Research reported in this paper was supported in part by the grant from the Energy Resource Institute, University of Oklahoma
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References
R.W. Hockney and C.R. Jesshope, Parallel Computers, Adam and Hilger Ltd., Bristol 1981, Chapter 5, Section 5.2.
D.J. Kuck, The Structure of Computers and Computations, Vol. 1, Addison-Wesley, 1980, Chapter 2, Section 2.3.
L. Hyafil and H.T. Kung, "The Complexity of Parallel Evaluation of Linear Recurrences," Journal of ACM, Vol. 24, 1977, pp. 513–521.
D. Heller, "A Survey of Parallel Algorithms in Numerical Algebra," SIAM Review, Vol. 20, 1978, pp. 740–777.
J.T. Schwartz, "Ultracomputers," ACM Transactions on Programming Languages and Systems, Vol. 2, 1980, pp. 484–521.
J.L. Lambiotte, Jr. and R.G. Voigt, "The Solution of Tridiagonal Linear Systems on CDC STAR 100 Computer," ACM Transactions on Mathematica Software, Vol. 1, 1975, pp. 308–329.
H.S. Stone, "An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations," Journal of ACM, Vol. 20, 1973, pp. 27–38.
H.S. Stone, "Parallel Tridiagonal Equation Sovers," ACM Transactions on Mathematical Software, Vol. 1, 1975, pp. 289–307.
A.H. Sameh and R.P. Brent, "Solving Triangular Systems on a Parallel Computer," SIAM Journal of Numerical Analysis, Vol. 14, 1977, pp. 1101–1113.
M. Snir, "Depth-Size Trade-Offs for Parallel Prefix Computation," Technical Report, Hebrew University of Jerusalem, 1983.
F.E. Fich, "New Bounds for Parallel Prefix Circuits," Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 100–109.
S.C. Chen and D.J. Kuck, "Time and Parallel Processor Bounds for Linear Recurrence Systems," IEEE Transactions on Computers, Vol. 24, 1975, pp. 701–707.
S.C. Chen, "Speedup of Iterative Programs in Multiprocessing Systems," Ph.D. Dissertation, Computer Science, University of Illinois, Urbana, 1975.
R.W. Hockney, "A Fast Direct Solution of Poisson's Equation Using Fourrier Analysis," Journal of ACM, Vol. 12, 1965, pp. 95–113.
S. Fortune and J. Wylie, "Parallelism in Random Access Machines," Proceedings of the 10th Annual ACM Symposium on Theory of Computing, 1978, pp. 114–118.
B.L. Buzbee, G.H. Golub and C.W. Nelson, "On Direct Methods for Solving Poisson's Equations," SIAM Journal on Numerical Analysis, Vol. 7, 1970, pp. 627–656.
H.F. Jordan, "Experience with Pipelined Multiple Instruction Streams," IEEE Proceedings, 1984, pp. 113–123.
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Lakshmivarahan, S., Dhall, S.K. (1985). Parallel algorithms for solving certain classes of linear recurrences. In: Maheshwari, S.N. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1985. Lecture Notes in Computer Science, vol 206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16042-6_26
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DOI: https://doi.org/10.1007/3-540-16042-6_26
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