By Zadeh N.
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After log2 p steps, the root processor will have the computation result. All processors can then be notified of the result through a broadcasting operation from the root. Total time: 2 log 2 p steps. A TASTE OF PARALLEL ALGORITHMS 35 Parallel Prefix Computation. Again, this is quite simple and can be done optimally in 2 log 2 p steps (recall that the diameter of a binary tree is 2 log 2 p or 2 log 2 p – 1). The algorithm consists of an upward propagation phase followed by downward data movement.
Schematic representation of single-processor solution for the sieve of Eratosthenes. 4 shows a single-processor implementation of the algorithm. The variable “current prime” is initialized to 2 and, in later stages, holds the latest prime number found. For each prime found, “index” is initialized to the square of this prime and is then incremented by the current prime in order to mark all of its multiples. 5 shows our first parallel solution using p processors. The list of numbers and the current prime are stored in a shared memory that is accessible to all processors.
Computation graph for finding the sum of 16 numbers. Example. Finding the sum of 16 numbers can be represented by the binary-tree computation graph of Fig. 14 with T(1) = W(1) = 15. Assume unit-time additions and ignore all else. 76 Essentially, the 8 processors perform all of the additions at the same tree level in each time unit, beginning with the leaf nodes and ending at the root. The relatively low efficiency is the result of limited parallelism near the root of the tree. Now, assuming that addition operations that are vertically aligned in Fig.