D.R. Karger, N. Nisan, and M. Parnas, Fast Connected Components Algorithms for the EREW PRAM, SPAA'92, pp. 373-381.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....or fat tree, the algorithm completes in time O(log N ) etc. In specifying and analyzing this GC algorithm, I build on the simplest possible approach to parallel connected components. There is a lot of work on improving the exponent for various types of parallel architecture (see, e.g. [28]) It is quite possible that one or more of these superior algorithms could be adapted to suit this GC algorithm. However, as will be discussed in the next section, this GC algorithm is presently conservative to the point of unusability; additional thought spent on this algorithm should be ....

David R. Karger, Noam Nisan, and Michal Parnas. Fast connected components algorithms for the EREW PRAM. SIAM Journal on Computing, 28(3):1021--1034, 1999. 121

....or fat tree, the algorithm completes in time O(log N3) etc. In specifying and analyzing this GC algorithm, I build on the simplest possible approach to parallel connected components. There is a lot of work on improving the exponent for various types of parallel architecture (see, e.g. [28]) It is quite possible that one or more of these superior algorithms could be adapted to suit this GC algorithm. However, as will be discussed in the next section, this GC algorithm is presently conservative to the point of unusability; additional thought spent on this algorithm should be ....

David R. Karger, Noam Nisan, and Michal Pamas. Fast connected components algorithms for the EREW PRAM. SIAM Journal on Computing, 28(3): 1021-1034, 1999.

....[27] The breakthrough in this field was due to Johnson and Metaxas [20] who gave a CREW PRAM algorithm for computing the connected components in O(log 3=2 n) time using O(n m) processors. Later, other results have been developed in this direction, attaining the same time and processors bounds [26, 25, 31]. Recently, Johnson and Metaxas [23] provided the first o(log 2 n) time algorithm, running on the weakest EREW PRAM model, for computing the MST of an undirected graph. The algorithm uses the growth control scheduling of the connectivity algorithm in [20] and it also makes use of an ....

D. R. Karger, N. Nissan, and M. Parnas. Fast connected component algorithm for the EREW PRAM. In ACM Symposium on Parallel Algorithms and Architectures, pages 373--381, 1992.

....until a deterministic O(log 1:5 n) time CREW PRAM algorithm that uses m n processors was obtained by Johnson and Metaxas [JM91] Johnson and Metaxas have later shown [JM92] that their algorithm can also be implemented in the EREW PRAM model. At about the same time, Karger, Nisan and Parnas [KNP92] used the interesting technique of short random walks on graphs, developed initially by Aleliunas, Karp, Lipton, Lovasz and Rackoff [AKL 79] to develop a randomised EREW PRAM algorithm that runs in either O(log n) time using O( n 1 ffl m) log n) processors, for any ffl 0, or in O(log n ....

....n) Our result is therefore optimal. Sparse graphs usually pose the greatest difficulty to algorithms for finding connected components. Note, for example, that the algorithm of Cole and Vishkin [CV91] is optimal if m = Omega Gamma n log n) and that the algorithm of Karger, Nisan and Parnas [KNP92] is optimal if m = Omega Gamma n 1 ffl ) for some ffl 0. Our algorithm is unusual in the sense that it reduces the problem of finding the connected components of a graph G = V; E) to the problem of finding the connected components of a sparse graph G 0 = V 0 ; E 0 ) with O(m n) ....

[Article contains additional citation context not shown here]

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proceedings of the 4th Annual ACM Symposium on Parallel algorithms and architectures, San Diego, California, pages 373--381, 1992.

....to the recursive algorithm of Section 2.3 the graph is repeatedly contracted by associating a set of vertices with one vertex. The set of vertices is found using the original pseudorandom generator of Nisan [Nis90] 4.2. 1 A Variation of the O(log 1:5 n) Space Algorithm Karger et al. KNP92] and Sinha and Tompa [ST] show that the resemblance between Nisan et al. s O(log 1:5 n) space algorithm and the recursive algorithm of Section 2.3 is more than superficial. They adapt the recursive algorithm s scheme of landmarks and neighborhoods to devise an algorithm with the same time and ....

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proceedings of the 1992 ACM Symposium on Parallel Algorithms and Architectures, pages 373--382, San Diego, CA, June 1992.

....as the algorithm presented in Section 3 but it is defined for the EREW PRAM model. This fact has several significant aspects which are discussed in Section 9 where a comparative study of the cost of our algorithm is given together with an overview of the algorithms for PRAM currently proposed by [14, 26, 4, 24, 27, 22, 1, 12, 7, 17, 20, 25, 5]. 2 Preliminaries A graph is a pair (V; E) where V is the set of vertices and E V Theta V is the set of edges. Let v = jVj, i.e. cardinality of V, and e = jE j. Then the size of a graph is e v. Given an edge a = x; z) vertex x is the source and vertex z is the target of a 2 . A ....

....for graphs which are not sparse) However, this algorithm too is for the CRCW model. Using a schedule technique to delay hookings, an algorithm for CCug on the EREW model was presented in [17] The algorithm has depth O(log 1:5 (v) with 2e v processors. The same bound is also obtained by [20, 25]. More recently, using a more sophisticated scheduling scheme, this bound has been significantly reduced to O(log(v)log(log(v) with 2e v processors in [5] This algorithm has the best known speedup and also the best TS for dense graphs. However, for sparse graphs the algorithm in [12] could ....

Karger D. R., N. Nisan and Parnas, M. Fast Connected Components Algorithms for the EREW PRAM. 4th Annual ACM Symp. on Parallel Architectures and Algorithms, (1992), 373-381.

.... algorithms for undirected connectivity [4, 8] derandomization [1] recycling of random bits [10, 15] approximation algorithms [6, 12, 17] efficient constructions in cryptography [14] and self stabilizing distributed computing [11, 16] Frequently (see, for example, Karger et al. [19] and Nisan et al. 20] we are interested in E[T (N ) the expected time before a simple random walk on an undirected connected graph, G, visits its N th distinct vertex, N n. The corresponding question for edges is also interesting, and arises in the work of Broder et al. 8] how large is ....

.... example, much stronger results are already known about the properties of short random walks on the special class of graphs known as expanders (see, for example, Ajtai et al. 1] and Jerrum and Sinclair [17] One might hope our results would dramatically improve the algorithms of Karger et al. [19] and Nisan et al. 20] for undirected connectivity. As mentioned above, both require an estimate of E[T (N ) and both used the estimate E[T (N ) O(N 4 ) Unfortunately, substituting our bound only improves the constants for the algorithms, since the running times of both depend on the ....

D. R. Karger, N. Nisan, and M. Parnas, Fast connected components algorithms for the EREW PRAM, in Proceedings of the 1992 ACM Symposium on Parallel Algorithms and Architectures, San Diego, CA, June--July 1992, pp. 373--381.

....2D40 and 3D20 have a large number of components, graphs of type 2D60, 3D40 and AD3 are highly connected, and most graphs of type AD3E consist of one component only. 3. 1 Parallel algorithm There has been a lot of publications on parallel distributed algorithms for the connected components problem [16, 17, 18, 7, 2]. Most of these algorithms were designed for PRAM. Algorithm. We chose the algorithm by Krishnamurthy e.a. 7] because it seemed easy to implement and practical results for comparison were provided. It is a refinement of the algorithm by Shiloach and Vishkin [2] In the following we only give an ....

Karger, D.R., N. Nissan, M. Parnas, Fast Connected Components Algorithms for the EREW PRAM, Proc. 4th Symposium on Parallel Algorithms and Architectures, ACM-SIAM, 1992, 562-572. 302-153-7-

....The first breakthrough is due to Johnson and Metaxas; they devised O(log 1:5 n) time algorithms for the connected component problem [10] and the MST problem [11] Their results were later improved by Chong and Lam to O(log n loglog n) time [4, 3] Randomization can help again. Karger et al. [14, 12] devised a randomized algorithm running in O(log n) expected time; more recently, Poon and Ramachandran [16] gave the first randomized algorithm using linear expected work and polylog expected time (precisely, O(log n Delta log log n Delta 2 log n ) Until now, it has been open whether ....

D.R. Karger, N. Nisan, and M. Parnas, Fast Connected Components Algorithms for the EREW PRAM, SPAA'92, pp. 373-381.

....of the array N . For any starting vertex i visited at time t = 0, the array N stores a random walk such that if the vertex visited at time t, 0 t k is v it (note: v i0 = i) then the vertex visited at time t 1 is given by N v i;t ;t 1 . This immediate and natural algorithm was employed in [17]. 3.2.2 Uniform Generation of Spanning Trees. Given a graph G = V; E) and a random walk starting at s 2 V which covers the graph, a spanning tree T of G can be obtained by selecting for each v 2 V Gamma fsg the first entry edge for v in the walk. If p is an array storing successive vertices ....

D.R. Karger, N. Nisan, and M. Parnas. Fast Connected Components Algorithms for the EREW PRAM. 4th Symp. on Parallel Algorithms and Architectures, 1992.

.... theory of efficient, highly parallel graph algorithm design [25, 27, 31, 46] Parallel algorithms that run in polylog time with linear or sub linear number of processors have been developed for several fundamental problems on undirected graphs including connected components and spanning forest x [2, 5, 7, 13, 16, 17, 24, 26, 42], minimum spanning forest (MSF) 2, 5, 6] ear decomposition and 2 edge connectivity [32, 37, 43] open ear decomposition and biconnectivity [32, 37, 43, 52] triconnectivity [12, 36] and planarity [44] All of these algorithms (with the exception of some algorithms for MSF) have the additional ....

D. R. Karger, N. Nisan, and M. Parnas, Fast connected components algorithms for the EREW PRAM, Proc. 4th ACM Symp. on Parallel Algorithms and Architectures, 1992, pp. 373--381.

....and #UCB ERL 92 172. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of either organization. The inherent contention in the algorithm has made even EREW solutions much more challenging [5, 14, 15, 17]. Practical application of the theoretical work to parallel machines has been largely restricted to small shared memory machines and SIMD machines with very slow processors [11] Many practical solutions have been developed for modern MIMD massively parallel platforms, or MPP s [6, 9, 13, 22, 18] ....

D. R. Karger, N. Nisan, M. Parnas, "Fast Connected Components Algorithm for the EREW PRAM," 4th ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 373-381.

....findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of any organization. arbitrary bandwidth to any memory location, but the inherent contention in the algorithm makes even EREW solutions much more challenging [6, 15, 18, 20]. Implementation of the theoretical work has been restricted to shared memory machines [12] and SIMD machines with very slow processors [12, 17, 23] Many practical solutions have been developed independently of theoretical work for modern MIMD massively parallel platforms (MPP s) 7, 10, 13, 14, ....

D. R. Karger, N. Nisan, M. Parnas, "Fast Connected Components Algorithm for the EREW PRAM," 4th ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 373-381.

....until a deterministic O(log 1:5 n) time CREW PRAM algorithm that uses m n processors was obtained by Johnson and Metaxas [JM91] Johnson and Metaxas have later shown [JM92] that their algorithm can also be implemented in the EREW PRAM model. At about the same time, Karger, Nisan and Parnas [KNP92] used the interesting technique of short random walks on graphs, developed initially by Aleliunas, Karp, Lipton, Lovasz and Rackoff [AKL 79] to develop a randomized EREW PRAM algorithm that runs in either O(log n) time using O( n 1 ffl m) log n) processors, for any ffl 0, or in O(log ....

....our work, Radzik [Rad94] had recently obtained a randomized EREW PRAM connectivity algorithm that runs in O(log n) time using m n processors. In this work we combine methods from many of the previous works, including in particular the method of short random walks used by Karger, Nisan and Parnas [KNP92] to obtain a randomized EREW PRAM algorithm that runs in O(log n) time using O( m n) log n) processors. A running time of O(log n) is best possible in the EREW PRAM model (Cook, Dwork and Reischuk [CDR86] and Omega Gammad m n) log n) processors are clearly necessary to obtain a running ....

[Article contains additional citation context not shown here]

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proceedings of the 4th Annual ACM Symposium on Parallel algorithms and architectures, San Diego, California, pages 373--381, 1992.

....setting, by Aleliunas, Karp, Lipton, Lovasz and Rackoff [AKL 79] to obtain a randomized LOGSPACE connectivity algorithm. It was first used, in the parallel setting by Nisan, Szemer edi and Wigderson [NSW92] A much more efficient implementation was then presented by Karger, Nisan and Parnas [KNP92] Further improvements were then obtained, independently, by Radzik [Rad94] and Halperin and Zwick [HZ94] In this work we present the first optimal speedup parallel algorithm that uses the random walks method that does produce a spanning forest of the original graph. We do so while keeping the ....

....[CL95] O(logn log log n) m n deterministic Halperin, Zwick O(log n) O( m n) log n) randomized Table 1: PRAM algorithm for finding spanning forests. EREW PRAM authors ref. time processors det. ran. Nisan, Szemer edi, Wigderson [NSW92] O(log n) n O(1) randomized Karger, Nisan, Parnas [KNP92] O(log n) O( m n 1 ffl ) log n) randomized Radzik [Rad94] O(log n) m n randomized Halperin, Zwick [HZ94] O(log n) O( m n) log n) randomized Table 2: PRAM algorithm for finding connected components. space. The fact that we can now find a spanning forest, and not just the connected ....

[Article contains additional citation context not shown here]

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proceedings of the 4th Annual ACM Symposium on Parallel algorithms and architectures, San Diego, California, pages 373--381, 1992.

....n) time algorithm by Gazit [9] but it is quite complicated, and for the problem sizes tested here, it is probably not practical. All of these algorithms are concurrent read concurrent write (CRCW) While not investigated here, there has also been numerous exclusivewrite (EW) algorithms, e.g. [6, 13, 14, 15]. Two measures are used for making comparisons. Execution times on a Connection Machine 2 and a Cray YMP C90 are given for the algorithms, using various sizes and classes of graphs. For the algorithms which contract the graph, the number of edges remaining after each iteration of the algorithms ....

D. R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithm for the EREW PRAM. In 4th Symposium on Parallel Algorithms and Architectures, pages 373--381, 1992.

....degree first. Also, since the results of this paper were reported, Nisan, Szemer edi, and Wigderson [24] have described an O(log n 3=2 ) space algorithm for the single connectivity problem. This result subsumes our time bound, but not our processor bound. A paper by Karger, Nisan, and Parnas [18] which relates to this latter result has bounds equal to ours. Despite these several results, however, a conjecture posed by Wyllie [33] and Shiloach and Vishkin [26] remains open. The conjecture states that no O(log n) time algorithm exists for the exclusive write PRAM model. The techniques ....

D. R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proc. of 4th Symposium on Parallel Algorithms and Architectures, pages 373--381, June 1992.

No context found.

D.R. Karger, N. Nisan, and M. Parnas, Fast Connected Components Algorithms for the EREW PRAM, SPAA'92, pp. 373-381.

No context found.

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. In Proceedings of the 4th Annual ACM Symposium on Parallel algorithms and architectures, San Diego, California, pages 373-381, 1992.

No context found.

D.R. Karger, N. Nisan, and M. Parnas. Fast connected components algorithms for the EREW PRAM. SIAM Journal on Computing, 28:1021--1034, 1999.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC