An O(logn) parallel connectivity algorithm

Shiloach, Yossi; Vishkin, Uzi

doi:10.1016/0196-6774(82)90008-6

Cited by 521 publications

(252 citation statements)

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“…The spanning tree of Step 1 is found using a modified version of the hypercube connected components algorithm of [WOO88]. This performs better than the hypercube adaptation of the algorithm of [SHIL82]. While the preorder number and number of descendants can be found in O(log 2 n) time on an n node hypercube using the steps outlined in [GOPA85], we did not attempt to map this O(log 2 n) algorithm onto a p node hypercube.…”

Section: The Tarjan-vishkin Biconnected Components Algorithmmentioning

confidence: 99%

“…In the CRCW implementation of Figure 1 described in [TARJ85], the spanning tree T of Step 1 is found by using a modified version of Shiloach and Vishkin's connected components algorithm [SHIL82]. The preorder number and number of descendants is found using a doubling technique ([WYLL79], [NASS80]) and an Eulerian tour.…”

Section: The Tarjan-vishkin Biconnected Components Algorithmmentioning

confidence: 99%

“…The CRCW PRAM implementation of Tarjan and Vishkin runs in O(logn) time and uses O(n +m) processors. Here n and m are, respectively,connected components algorithm for the NCUBE hypercube developed in [WOO88] is far removed from the asymptotically fast algorithm of [SHIL82].…”

Section: Introductionmentioning

confidence: 99%

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Computing biconnected components on a hypercube

Woo

Sahni

1991

J Supercomput

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We describe two hypercube algorithms to find the biconnected components (i.e., blocks) of a connected undirected graph. One is a modified version of the Tarjan-Vishkin algorithm. The two hypercube algorithms were experimentally evaluated on an NCUBE/7 MIMD hypercube computer. The two algorithms have comparable performance and efficiencies as high as 0.7 were observed. Keywords and phrasesHypercube computing, MIMD computer, parallel programming, biconnected components __________________ * This research was supported in part by the National Science Foundation under grants DCR84-20935 and MIP 86-17374 1 2 INTRODUCTIONIn this paper we develop two biconnected component (i.e., block) algorithms suitable for medium grained MIMD hypercubes. The first algorithm is an adaptation of the algorithm of Tarjan and Vishkin [TARJ85]. Tarjan and Vishkin provide parallel CRCW and CREW PRAM implementations of their algorithm. The CRCW PRAM implementation of Tarjan and Vishkin runs in O(logn) time and uses O(n +m) processors. Here n and m are, respectively, the number of vertices and edges in the input connected graph. The CREW PRAM implementation runs in O(log 2 n) time using O(n 2 /log 2 n) processors.A PRAM algorithm that use p processors and t time can be simulated by a p processor hypercube in O(tlog 2 p) time using the random access read and write algorithms of Nassimi and Sahni [NASS81]. The CREW PRAM algorithm of [TARJ85] therefore results in an O(log 3 n) time O(n +m) processor hypercube algorithm. The CREW PRAM algorithm results in an O(log 4 n) time O(n 2 /log 2 n) processor hypercube algorithm. Using the results of Dekel, Nassimi, and Sahni [DEKE81] the biconnected components can be found in O(log 2 n) time using O(n 3 /logn) processors.The processor-time product of a parallel algorithm is a measure of the total amount of work done by the algorithm. For the three hypercube algorithms just mentioned, the processor-time product is, respectively, O(n 2 log 3 n) (assuming m ∼ ∼ O(n 2 )), O(n 2 log 2 n), and O(n 3 logn). In each case the processor-time product is larger than that for the single processor biconnected components algorithm (O(n 2 ) when m= O(n 2 )). As a result of this, we do not expect any of the above three hypercube algorithms to outperform the single processor algorithm unless the number of available processors, p, is sufficiently large. For example, if n = 1024, then the CRCW simulation on a hypercube does O(log 3 n) ∼ ∼1000 times more work than the uniprocessor algorithm. So we will need approximately 1000 processors just to break even.In fact, the processor-time product for many of the asymptotically fastest parallel hypercube algorithms exceeds that of the fastest uniprocessor algorithm by at least a multiplicative factor of log k n for some k, k≥1. As a result, the simulation of these algorithms on commercially available hypercubes with a limited number of processors does not yield good results. Consequently, there is often a wide disparity between the asymptotic algorithms developed for PRAM...

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Section: The Tarjan-vishkin Biconnected Components Algorithmmentioning

confidence: 99%

Section: The Tarjan-vishkin Biconnected Components Algorithmmentioning

confidence: 99%

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Computing biconnected components on a hypercube

Woo

Sahni

1991

J Supercomput

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“…In an earlier thesis Shiloach and Vishkin [80] have presented a deterministic algorithm for connectivity that has the same resource bounds. But, the random mating lemma has been used by Gazit [24] to obtain an optimal randomized algorithm for connectivity that runs in time O(log |V |) and uses (|V | + |E|)/ log(|V |) processors.…”

Section: The Random Mating Lemma and Optimal Connectivitymentioning

confidence: 99%

Randomized Parallel Computation

Rajasekaran

1988

Concurrent Computations

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Informally, a randomized algorithm (in the sense of [59] and [82]) is one which bases some of its decisions on the outcomes of coin flips. We can think of the algorithm with one possible sequence of outcomes for the coin flips to be different from the same algorithm with a different sequence of outcomes for the coin flips. Therefore, a randomized algorithm is really a family of algorithms.For a given input, some of the algorithms in this family might run for an indefinitely long time.

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“…Connectivity is a fundamental graph problem with a range of applications and can be building blocks for higher-level algorithms. The research community has produced a rich collection of theoretic deterministic [28,21,30,26,8,9,7,18,24,34,1,12,14] and randomized [17,29] parallel algorithms for connected components. Yet for implementations and experimental studies, although several fast PRAM algorithms exist, to our knowledge there is no parallel implementation of connected components (other than our own [4,6]) that achieves significant parallel speedup on sparse, irregular graphs when compared against the best sequential implementation.…”

Section: Connected Componentsmentioning

confidence: 99%