Optimal Parallel Approximation Algorithms for Prefix Sums and Integer Sorting (Extended Abstract)

Ragde. Triply-logarithmic upper and lower bounds for minimum, range minima, and related problems with integer inputs. Optimal doubly logarithmic parallel algorithms based on nding all nearest smaller values. Optimal sub-logarithmic time integer sorting on a CRCW PRAM (note). Submitted for publication, 1991. 27] Sandeep Sen. Finding an approximate median with high probability in constant parallel time. 9 Let j be the set of elements of value j, j = 1; : : :; n. The integer-chain sorting algorithm in 11, Sec. 9] consists of t = O(lg n) iterations. At iteration i, an array A i of size cn=2 i is used. For each set j an interval D i j of size d i j = O(i j) may be allocated in A i , where i j is an estimate for j j j computed in this step, and a subset i j j is injectively mapped into D i j. At the end of the t = O(lg n) iterations, each element in j is mapped into a private cell in one of the intervals D i j. Let d i j = P i k=1 d i j , and d j = d t i. The preex sums d i j i , j = 1; : : :; n, can be obtained by straightforward modiications of the integer chain sorting algorithm. After the execution of this algorithm, we can obtain the padded integer sorting sequence in an array B of size 2cn, as follows: Step 1. Allocate from B to each set j a private interval B j of size d j , so that the allocation is ordered according to j. Step 2. For each j, allocate from B j to each subset i j a private sub-interval of size d i j , so that the allocation is ordered according to i. B i j Step 3. Copy the contents of each interval D i j into interval B i j. It is easy to verify that the input elements are indeed sorted in the array B. Step 1 is implemented by the ordered allocation algorithm of Theorem 5.1. Step 2 is implemented using the preex sums sequences d i j i , j = 1; : : :; n. Step 3 is trivial. We have Theorem 5.2. (optimal padded integer sorting) The padded integer sorting problem can be solved in time …