A. C. Greenberg, R. E. Ladner, M. S. Paterson, and Z. Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, Aug. 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is O(log n) O(1) nc p = 1 o(1) nc p ; giving a total cost of the entire schedule of L(2 log L=g p 1) 1 o(1) nc p : 20) This cost is 1 optimal if n AE 2pL=c. To obtain a BSP schedule for the evaluation of a k level linear recurrence, we will use the strategy proposed in [31]. Thus, if f(a[i Gamma1] a[i Gamma2] a[i Gammak] P k j=1 x[k Gammaj ]a[i Gammaj] we build the k Theta k (recurrence) matrix M = 0 B B B B 0 : 0 x[0] x[1] I . x[k Gamma 1] 1 C C C C A (21) with I the (k Gamma 1) Theta (k Gamma 1) identity matrix, and compute ....

A. C. Greenberg et al. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, August 1982.

....an FNC 1 Boolean circuit family can compute the value of an O(log) depth algebraic formula (over N or Z) and therefore the number of accepting trees in an NC 1 Boolean circuit family. Thus at least in the P uniform case, #NC 1 = GapNC 1 = FNC 1 , and PNC 1 = C = NC 1 = NC 1 . [GLPG82] studies the problem of computing the first N terms of an M order linear recurrence, and gives O( log M) log N) depth arithmetic circuits to do the job. Since their technique involves reducing the linear recurrence problem to various products of large numbers of M Theta M matrices, and since ....

....) log N) The infinite sum in section 2. 3 increases linearly with c, which is O(log log(N 2 ) for a total depth of O( log T ) log log T ) log N) perhaps O( log T ) log N) could be achieved with more tweaking and tighter analysis, thus (among other things) improving the arithmetic results of [GLPG82] to work on Boolean circuits. Another intriguing open question is extending the result to a broader class of rings or semirings, e.g. rationals (with n bit numerators and denominators) polynomials (with n bit integer coefficients) or families of finite (semi)rings with at most 2 n elements. ....

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Albert C. Greenberg, Richard E. Ladner, Michael S. Paterson, and Zvi Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1), August 1982.

.... frequently appear in scientific applications [20] are very useful in the design of parallel algorithms [19, 1] and can be used to solve a much broader class of recurrences [18, 11] Researchers have been studying parallel and vector al gorithms to solve linear recurrences since the 1960 s [17, 3, 18, 31, 6, 16, 29, 22, 11, 35, 12, 23], and considerable effort has gone into producing fast implementations of these algorithms on parallel and vector machines [24, 21, 33, 32, 13, 26, 30] Some supercomputer manufacturers have considered the solution of linear recurrences important enough to warrant the addition of special hardware ....

....of Phi and Omega . With a variation of cyclic reduction, Chen and Kuck showed how to solve an m th order linear recurrence in O(lgn lg m) time [6] This algorithm is impractical, however, since it requires nm 2 processors. It has been shown that both with a variation of cyclic reduction [12] and with the partition method [5] the total number of operations can be reduced to nm(m 2) total calls to Omega and nm(m 1) total calls to Phi. When the number of processors p is significantly less than n, or the vector half performance length n 1=2 on a vector computer is significantly ....

Albert C. Greenberg, Richard E. Ladner, Machael S. Paterson, and Zvi Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, August 1982.

....1 m Gamma1 then we see immediately that fm can be calculated by an arithmetic circuit of size and depth O(log 2 m) if we compute the matrix power efficiently by repeated squaring. The above result for Fibonacci numbers is a special case of the following more general result by Greenberg et al. [108] on the parallel evaluation of k th order linear recurrences. If we have F = f 0 f 1 : f k Gamma1 ) and fm = P k j=1 a k Gammaj fm Gammaj for m k, then (f m Gammak 1 : fm ) F M m Gammak 1 where M is the k Theta k matrix 0 B B B B B 0 Delta Delta Delta 0 a 0 a 1 I . ....

A C Greenberg, R E Ladner, M S Paterson, and Z Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, August 1982.

.... frequently appear in scientific applications [19] are very useful in the design of parallel algorithms [18, 1] and can be used to solve a much broader class of recurrences [17, 10] Researchers have been studying parallel and vector algorithms to solve linear recurrences since the 1960 s [16, 3, 17, 29, 5, 15, 27, 21, 10, 33, 11, 22], and considerable effort has gone into producing fast implementations of these algorithms on parallel and vector machines [23, 20, 31, 30, 12, 24, 28, 26] Some supercomputer manufacturers have considered the solution of linear recurrences important enough to warrant the addition of special ....

....take advantage of the associativity of Phi and Omega . Chen and Kuck [5] showed how to solve an m th order linear recurrence in O(lg n lg m) time. This algorithm is impractical, however, since it requires nm 2 processors. It has been shown that both with a variation of cyclic reduction [11] and with the partition method [6] the total number of operations can be reduced to nm(m 2) total calls to Omega and nm(m 1) total calls to Phi. When the number of processors p is significantly less than n, or the vector half performance length n 1=2 on a vector computer is significantly ....

A. C. Greenberg, R. E. Ladner, M. S. Paterson, and Z. Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, Aug. 1982.

....= log n CREW processors, which was later improved to n 6 = log 5 n by Viswanathan et al. 18] Huang et al. 13] also gave an O( p n log n) time algorithm with n 3:5 = log n CREW processors. To the best of our knowledge, no work has explicitly dealt with Problems 1, 2, 4. Greenberg et al. [10] solved a linear recurrence in O(log 2 n) time with n 3 = log 2 n CREW processors, which solves the 1D problem as a special case in the same complexity. In this paper we present a unifying framework for the parallel computation of dynamic programming. We use two well know methods, the ....

....for each matrix product at level k is O(n2 2k ) Thus the number of operations at level k is O(n 2 2 k ) Since the number of operations decreases by a constant factor as level k goes down, computing B requires O(n 3 ) operations. Thus we get O(log 2 n) time using O(n 3 ) operations [10]. We can also improve the total number of operations. Since a r;i = 0 for i Gamma r 0 in the 1D problem, the leftmost matrix at each level has only one row of nontrivial values. Compute B from the bottom until the level k such that 2 k = Then there remain n= matrices. The number of ....

Greenberg, A. C., Ladner, R. E., Paterson, M. S., Galil, Z. Efficient parallel algorithms for linear recurrence computation. Inform. Process. Lett. 15 (1982), 31--35.

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A. C. Greenberg, R. E. Ladner, M. S. Paterson, and Z. Galil. Efficient parallel algorithms for linear recurrence computation. Information Processing Letters, 15(1):31--35, Aug. 1982.

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