D. D. Gajski. An algorithm for solving linear recurrence systems on parallel and pipeline machines. IEEE Transactions on Computers, C-3(3):190--206, Mar. 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... their hardware overhead, but also introduces a number of new application domains [Par89] page 8 of 38 Since the early seventies the treatment of linear recurrences has been widely studied in the compiler community, mainly when compilation to vector and highly pipelined computers are targeted [Che75, Gaj81]. 3.1 What is new It is instructive and important to compare the new technique with the related research. The maximally fast implementation of linear computation (Section 5) is a widely applicable topic which has not been studied previously. The proposed arbitrarily fast solution maintains the ....

D.D. Gajski: "An Algorithm for Solving Linear Recurrences Systems on Parallel and Pipelined Machines", Vol. 30, No. 3, pp. 190-206, 1981.

.... which is regularly used in almost all optimizing compilers and many high level synthesis systems [Rab91] Since the early seventies the treatment of linear recurrences has been widely studied in the compiler community, mainly when compilation to vector and highly pipelined computers are targeted [Pot92, Che75, Gaj81]. Another popular approach, particularly in logic synthesis [Bra84] is static ordering where the order of the transformations is given a priori, most often in the form of a script. Hennessy has categorized and presented models to describe the effects of local and global optimizations on symbolic ....

D.D. Gajski. An Algorithm for Solving Linear Recurrences Systems on Parallel and Pipelined Machines. Vol. 30, (no.3), pp.190-206, 1981.

....that differ in several attributes (e.g. functionality, cost, speed) We briefly review some results from parallel computing. Chen and Kuck developed an algorithm [1] for computing N m th order band linear recurrence with a fixed number of p processors, independent of the problem size N . Gajski[6] lowered the time bound of Chen and Kuck s for computing the band linear recurrences with p processors. Wang and Nicolau[27] proposed the Harmonic Scheduling technique for deriving optimal schedules for band linear recurrences with p processors and their schedules improved upon Gajski s in ....

.... for RhN;mi) number of periods)T p (period of size p 2 ) N p 2 (m(2m 1) p 0 (m 1) 2) 2mp) N p ( 2m 2 3m) 0 m 1 2p (2m 2 m) 2 The regular schedule, based on matrix chain multiplication, was derived using the Harmonic Scheduling technique[27] Gajski s algorithm[6] achieves the same execution time as the Regular Schedule, under concurrent read and exclusive write(CREW) parallel random access machine(PRAM) model, but uses a different formulation and organizes the computation differently. When mapped onto real machines, the Regular Schedule will perform ....

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....code. where Phi and Omega are binary associative operators, and Omega distributes over Phi. Such linear recurrences 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, ....

.... 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 ....

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Daniel D. Gajski. An algorithm for solving linear recurrence systems on parallel and pipeline machines. IEEE Transactions on Computers, C-3(3):190--206, March 1981.

.... 1=2) and 3N= p 1=2) O(logp) time steps respectively with p processors for parallel evaluation of the Horner expression, which is equivalent to evaluating the last equation in a first order band linear recurrence, i.e. evaluating xN only without having to compute x 1 ; xN01 . Gajski [4] lowered the time bound of Chen further to (2m 2 3m)N= p m 1=2) for p m 1 and N p 2 for computing the band linear recurrences with p MIMD processors. Recently, a novel approach, Harmonic Scheduling [24] has been proposed for generating scalable parallel schedules for band linear ....

....= c[ k 3] 3 a[ k 4] k 3] t[6] c[ k 5] 3 a[ k 6] k 5] t[13] c[ k 8] 3 a[ k 9] k 8] t[23] c[ k 3] 3 a[ k 4] k 3] 2. x[ k 2] x[ k 2] c[ k 2] t[3] t[2] c[ k 4] t[7] t[6] c[ k 6] t[14] t[13] c[ k 9] t[24] t[23] c[ k 13] t[4] = a[ k 5] k 4] a[ k 6] k 5] 3. x[ k 4] t[1] 3 x[ k 2] t[8] t[7] 3 a[ k 7] k 6] t[15] t[14] 3 a[ k 10] k 9] t[25] t[24] 3 a[ k 14] k 13] t[5] t[4] 3 a[ k 7] k 6] t[10] a[ k 8] k 7] 3 a[ k 9] k 8] 4. x[ k 4] x[ k 4] t[3] ....

[Article contains additional citation context not shown here]

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....to evaluating the last equation in a first order LR, i.e. evaluating xN only without having to compute x 1 ; xN01 , Note that the Horner expression only requires the final value of the last equation, while the BLR evaluation computes the solutions of all equations in the recurrences. Gajski [15] improved the time bound of [6] for computing BLR with resource constraints. Recently, Wang and Nicolau [33] further improved on Gajski s results for general BLR s (i.e. mth order LR s) with resource constraints and in particular found the strict time optimal schedules for (i=1;i =N;i ) for ....

....one position right with second to last element x k0m shifted out. The matrix chain multiplication in Definition 3. 2 allows us to treat the parallel computation of RhN;mi in similar fashion to parallel prefix computation [23] Different forms of matrix chain multiplication for BLR were used in [6, 17, 15, 33]. However, our Regular Schedule is new and different in its regular and periodic organization of the computation that uses the matrix chain multiplication. Algorithm 3.1 describes the generic Regular Schedule in terms of matrix chain multiplication for parallel computing RhN;mi. It is also called ....

[Article contains additional citation context not shown here]

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....to evaluating the last equation in a first order LR, i.e. evaluating xN only without having to compute x 1 ; xN01 , Note that the Horner expression only requires the final value of the last equation, while the BLR evaluation computes the solutions of all equations in the recurrences. Gajski [7] improved the time bound of [4] for computing BLR with resource constraints. Recently, Wang and Nicolau [16] further improved on Gajski s results for general BLR s (i.e. mth order LR s) with resource constraints and in particular found the strict time optimal schedules for computing 1st and ....

....3.1) computes RhN;mi on p processors, where p m is a constant independent of N , in time Tp = N p ( 2m 2 3m) 0 m 1 2p (2m 2 m) 2) By Theorem 3.1, Algorithm 3.1 is scalable in terms of number of processors p. Different forms of matrix chain multiplication for BLR were used in [4, 8, 7, 16]. However, our Regular Schedule is new and different in its regular and periodic organization of the computationbased on matrix chain multiplication. The regular schedule can be derived using similar techniques for deriving the Harmonic Schedule presented in [16] Gajski s algorithm [7] achieves ....

[Article contains additional citation context not shown here]

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....; m i n where Phi and Omega are binary associative operators, and Omega distributes over Phi. Such linear recurrences 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, ....

.... 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 ....

[Article contains additional citation context not shown here]

D. D. Gajski. An algorithm for solving linear recurrence systems on parallel and pipeline machines. IEEE Transactions on Computers, C-3(3):190--206, Mar. 1981.

.... for RhN;mi) number of periods)T p (period of size p 2 ) N p 2 (m(2m 1) p 0 (m 1) 2) 2mp) N p ( 2m 2 3m) 0 m 1 2p (2m 2 m) 2 The regular schedule, based on matrix chain multiplication, was derived using the Harmonic Scheduling technique[27] Gajski s algorithm[15] achieves the same execution time as the Regular Schedule, under concurrent read and exclusive write(CREW) parallel random access machine(PRAM) model, but uses a different formulation and organizes the computation differently. When mapped onto real machines, the Regular Schedule will perform ....

....the last equation in a first order LR, i.e. evaluating xN only without having to compute x 1 ; xN01 , Note that the Horner expression does not require the values of all the equations other than the last, while BLR evaluation requires the values of all the equations in the recurrences. Gajski[15] lowered the time bound of [6] further for computing BLR with resource constraints. Recently, Wang and Nicolau[27] improved on Gajski s results for general BLR s(i.e. mth order LR s) with resource constraints and in particular found the strict time optimal schedules for computing 1st and ....

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....and with different types of functional units having different time constraints, has not been studied. The regular schedules presented in this paper aim to solve this problem. In the field of parallel computing, parallel evaluationof linear recurrences has been studied for quite some time [1, 5, 17], for which the review is given in [18] However, the underlyingmodel differs from high level synthesis, since parallel computing typically deals with a number of identical, multi function processors, while high level synthesis deals with a variety of functional units that differ in several ....

....that differ in several attributes (e.g. functionality, cost, speed) We briefly review some results from parallel computing. Chen and Kuck developed an algorithm [1] for computing N m th order band linear recurrence with a fixed number of p processors, independent of the problem size N . Gajski[5] lowered the time bound of Chen and Kuck s for computing the band linear recurrences with p processors. Wang and Nicolau[17] proposed the Harmonic Scheduling technique for deriving optimal schedules for band linear recurrences with p processors and their schedules improved upon Gajski s in ....

D. Gajski, "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

....last equation in a first order band linear recurrence, i.e. evaluating xN only without having to compute x 1 ; xN01 . Hyafil and Kung thus does not require the values of all the equations other than the last, while we require the values of all the equations in the recurrence system. Gajski[9] lowered the time bound of [5] further to (2m 2 3m)N= p m 1=2) for p m 1 and N p 2 for computing the band linear recurrence with resource constraints. For the analysis of our algorithms and to facilitate a comparison with previous techniques to be discussed shortly we use the ....

....= a[ k 5] k 4] a[ k 6] k 5] 3. x[ k 4] t[1] 3 x[ k 2] t[8] t[7] 3 a[ k 7] k 6] t[15] t[14] 3 a[ k 10] k 9] t[25] t[24] 3 a[ k 14] k 13] t[5] t[4] 3 a[ k 7] k 6] t[10] a[ k 8] k 7] 3 a[ k 9] k 8] 4. x[ k 4] x[ k 4] t[3] t[9] = t[8] c[ k 7] t[16] t[15] c[ k 10] t[26] t[25] c[ k 14] t[11] t[10] 3 a[ k 10] k 9] t[19] a[ k 12] k 11] 3 a[ k 13] k 12] 5. x[ k 6] t[4] 3 x[ k 4] x[ k 7] t[5] 3 x[ k 4] t[17] t[16] 3 a[ k 11] k 10] t[27] t[26] 3 a[ k 15] k ....

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Gajski, D., "An Algorithm for Solving Linear Recurrence Systems on Parallel and Pipelined Machines", IEEE Transactions on Computers, Vol. c-30, No.3, March 1981.

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D. D. Gajski. An algorithm for solving linear recurrence systems on parallel and pipeline machines. IEEE Transactions on Computers, C-3(3):190--206, Mar. 1981.

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