The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The SAC-1 system: An introduction and survey
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Algorithms for polynomial factorization.
Algorithms for polynomial factorization.
Factoring multivariate polynomials over the integers
ACM SIGSAM Bulletin
Parallel evaluation of division-free arithmetic equations
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
A Survey of Parallel Machine Organization and Programming
ACM Computing Surveys (CSUR)
On Parsing and Compiling Arithmetic Expressions on Vector Computers
ACM Transactions on Programming Languages and Systems (TOPLAS)
Almost control-free (indeterministic) parallel computation on permit schemes
POPL '78 Proceedings of the 5th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
The delay of circuits whose inputs have specified arrival times
Discrete Applied Mathematics
Reduction of Depth of Boolean Networks with a Fan-In Constraint
IEEE Transactions on Computers
On unlimited parallelism of DSP arithmetic computations
ICASSP'93 Proceedings of the 1993 IEEE international conference on Acoustics, speech, and signal processing: plenary, special, audio, underwater acoustics, VLSI, neural networks - Volume I
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Let E be an arithmetic expression involving n variables, each of which appears just once, and the possible operations of addition, multiplication, and division. Although other cases are considered, when these three operations take unit time the restructuring algorithms presented in this paper yield evaluation times no greater than 2.88 log2n + 1 and 2.08 log2n for general expressions and division-free expressions, respectively. The coefficients are precisely given by 2/log2&agr; ≈ 2.88 and 1/log2&bgr; ≈ 2.08, where &agr; and &bgr; are the positive real roots of the equations z2 = z + 1 and z4 = 2z + 1, respectively. While these times were known to be of order log2n, the best previously known coefficients were 4 and 2.15 for the two cases.The authors conjecture that the present coefficients are the best possible, since they have exhibited expressions which seem to require these times within an additive constant.The paper also gives upper bounds to the restructuring time of a given expression E and to the number of processors required for its parallel evaluation. It is shown that at most O(n1.44) and O(n1.82) operations are needed for restructuring general expressions and division-free expression, respectively. It is pointed out that, since the order of the compiling time is greater than n log n, the numbers of required processors exhibit the same rate of growth in n as the corresponding compiling times.