parallel prefix computation

(algorithm)

Definition: Calculate an associative function, f, on all prefixes of an n-element array, that is, s[0], f(s[0], s[1]), f(s[0], f(s[1], s[2])), ..., f(s[0], f(s[1], ... f(s[n-2], s[n-1])...)), using Θ(n) processors in Θ(log n) time. The algorithm is

 for j := 0 to lg(n)-1 do 
     for i := 2^j to n-1 parallel-do
         s[i] := f(s[i-2^j], s[i])

where lg is the logarithm base 2, and parallel-do does the innermost computations in parallel.

Note: In particular, this calculates any associative function, such as sum, maximum, or concatenate, over a list of values in logarithmic time. Note the write-after-read hazards in the parallel-do loop: old values of s[2^j] to s[n-1-2^j] must be read before being overwritten. Since this algorithm overwrites the initial values, the n processors can copy input values to a result array in parallel in one additional step.
From Yair Tuaff (r56409@email.sps.mot.com), 29 December 1999.

Author: PEB

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Entry modified 15 March 2007.
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Cite this as:
Paul E. Black, "parallel prefix computation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 15 March 2007. (accessed TODAY) Available from: http://www.nist.gov/dads/HTML/parallelPrefix.html