Randomized parallel selection

Abstract

We show that selection on an input of size N can be performed on a P-node hypercube (P = N/(log N)) in time O(n/P) with high probability, provided each node can process all the incident edges in one unit of time (this model is called the parallel model and has been assumed by previous researchers (e.g., [17])). This result is important in view of a lower bound of Plaxton that implies selection takes Ω((N/P)loglog P+log P) time on a P-node hypercube if each node can process only one edge at a time (this model is referred to as the sequential model). ABSTRACTWe show that selection on an input of size N can be performed on a P-node hypercube ( P = N/(log N)) in time O(N/P) with high probability, provided each node can process all the incident edges in one unit of time (this model is called the parallel model and has been assumed by previous researchers (e.g ., [17])). This result is important in view of a lower bound of Plaxton that implies selection takes Q((N/P) log log P+log P) time on a P-node hypercube if each node can process only one edge at a time (this model is referred to as the sequential model).

Unifying Themes for Selection on Any Network

“…But if a better sorting algorithm is discovered, the run time of our algorithm will improve somewhat, whereas [7] significant improvement over the deterministic algorithm. On the hypercube also, our algorithm has a better run time than that of [7], as has already been shown in [8].…”

“…Floyd and Rivest's [2] sequential algorithm takes n + min(i, n − i) + o(n) time. In [8], Rajasekaran has presented randomized algorithms for selection on the hypercube (on both the sequential and parallel versions). In [10], Rajasekaran also presents optimal or very nearly optimal randomized algorithms for selection on the mesh with fixed as well as reconfigurable buses.…”

“…The algorithm we present can be thought of as an extension of the algorithms given in [2,8]. Given is a set X of n keys and an integer i, 1 ≤ i ≤ n. Assume the keys are distinct (without loss of generality).…”

Randomized Priority Queues for Fast Parallel Access

“…For n = 256 and m = 2 24 the parallel priority queue needs an average of 3:73 ms for inserting plus deleting n elements. This is about 7:5 times faster than the centralized code with m = 2 21 which needs about 110 s for inserting plus deleting one element, i.e. 28.16 ms for 256 elements .…”

Designing efficient distributed algorithms using sampling techniques

“…This demonstrates that sampling techniques (see e.g [12][13]) are very useful tools for designing efficient algorithms for processing very large distributed files. We also use a randomized sampling technique which is a variant of [3,10] to design a much more efficient randomized selection algorithm. Given a file of size n and a p-processor de Bruijn network or hypercube, n is polynomial in p, our algorithm can perform a selection using only Op messages and delay O log p with high probability.…”