R. Cole, K. P. N., and T. R. E. A Linear-Work Parallel Algorithm for Finding Minimum Spanning Trees. In Proc. SPAA: Annual ACM Symposium on Parallel Algorithms and Architectures, pages 11--15, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithm for this problem under both measures. Here is a brief summary of related results. Following the linear time sequential MSF algorithm of Karger, Klein, and Tarjan [KKT95] and building on it) came linear work parallel MSF algorithms for the concurrent read concurrent write (CRCW) PRAM [CKT94, CKT96] and the EREW PRAM [PR97] The best CRCW PRAM algorithm known to date [CKT96] runs in logarithmic time and linear work, but the time bound is not known to be optimal. The best EREW PRAM algorithm known prior to our work is the result of Poon and Ramachandran which runs in O(log n log log n2 ) ....

....CRCW PRAM algorithm of [CKT96] In phase 1, the algorithm reduces the number of vertices in the graph from n to n k 0 vertices, where n is the number of vertices in the input graph, and k 0 = log . To perform this reduction the algorithm uses the familiar recursion tree of depth log # n [CKT94, CKT96, PR97], which gives rise to O(2 ) recursive calls; however, the time needed per invocation in our algorithm is well below O(log n 2 ) Thus the total time for phase 1 is O(log n) We accomplish this by requiring phase 1 to find only a subset of the MSF. By contracting this subset of the MSF we ....

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R. Cole, P. N. Klein, and R. E. Tarjan, A linear-work parallel algorithm for finding minimum spanning trees, in Proceedings of the 6th Annual Symposium on Parallel Algorithms and Architectures (SPAA'94), Cape May, NJ, ACM, pp. 11--15.

....is the first provably optimal parallel algorithm for this problem under both measures. Here is a brief summary of related results. Following the linear time sequential MSF algorithm of Karger, Klein and Tarjan [KKT95] and building on it) came linear work parallel MST algorithms for the CRCW PRAM [CKT94, CKT96] and the EREW PRAM [PR97] The best CRCW PRAM algorithm known to date [CKT96] runs in logarithmic time and linear work, but the time bound is not known to be optimal. The best EREW PRAM algorithm known prior to our work is the result of Poon and Ramachandran which runs in O(log n log log n2 log ....

....PRAM algorithm of [CKT96] In Phase 1, the algorithm reduces the number of vertices in the graph from n to n=k vertices, where n is the number of vertices in the input graph, and k = log (2) n) 2 . y To perform this reduction the algorithm uses the familiar recursion tree of depth log n [CKT94, CKT96, PR97], which gives rise to O(2 log n ) recursive calls, but the time needed per invocation in our algorithm is well below O(log n=2 log n ) Thus the total time for Phase 1 is O(log n) We accomplish this by requiring Phase 1 to find only a subset of the MSF. By contracting this subset of the MSF ....

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R. Cole, P.N. Klein, and R.E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proc. SPAA, 1994, pp. 11--15.

.... Boruvka Algorithm has been combined with the linear verification algorithm to obtain the first linear randomized algorithm for MST problem, see Klein, Tarjan [29] and Karger, Klein, Trajan [27] Also an optional randomized parallel algorithm has been recently found by Cole, Klein and Tarjan [10]. In all these results the Boruvka Algorithm plays the key role. Indeed, in order to simplify their complicated parallel algorithm and its analysis Cole, Klein and Tarjan [10] call each iteration of Boruvka Algorithm (i.e. each iteration of edge selection and subsequent contraction) Boruvka Step. ....

....Klein, Trajan [27] Also an optional randomized parallel algorithm has been recently found by Cole, Klein and Tarjan [10] In all these results the Boruvka Algorithm plays the key role. Indeed, in order to simplify their complicated parallel algorithm and its analysis Cole, Klein and Tarjan [10] call each iteration of Boruvka Algorithm (i.e. each iteration of edge selection and subsequent contraction) Boruvka Step. This seems to be standard by now. The Combinatorial Optimization has gone a long way in its relatively short history. But it is a bit surprising how persistent are the ....

Cole, R., Klein, P. N., Tarjan, R. E.: A linear-work parallel algorithm for finding minimum spanning tress. Proc. of SPAA, 1994.

....forest is the problem of finding a minimum spanning forest. Karger [Kar94] obtained a randomised EREW PRAM algorithm that finds a minimum spanning forest of a graph with n vertices and m edges in O(log n) time using O(m= log n n 1 ffl ) processors, for any ffl 0. Cole, Klein and Tarjan [CKT94] CKT96] recently obtained a parallel version of the recently discovered randomised linear time algorithm for finding minimum spanning trees of Karger, Klein and Tarjan [KKT95] Their algorithm runs in O(log n) time in the CRCW PRAM model. Acknowlegement The authors would like to thank two ....

R. Cole, P.N. Klein, and R.E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proceedings of the 6th Annual ACM Symposium on Parallel algorithms and architectures, Cape May, New Jersey, pages 11--15, 1994.

.... Boruvka Algorithm has been combined with the linear verification algorithm to obtain the first randomized linear algorithm for MST problem, see Klein, Tarjan [KT] and Karger, Klein and Tarjan [KKT] Also an optimal randomized parallel algorithm has been recently found by Cole, Klein and Tarjan [CKT]. In all these results the Boruvka Algorithm plays a key role. Indeed in order to simplify their complicated parallel algorithm and its analysis Cole, Klein and Tarjan [CKT] call each iteration of Boruvka Algorithm (i.e. each iteration of edge selection and subsequent contraction) Boruvka ....

....Klein and Tarjan [KKT] Also an optimal randomized parallel algorithm has been recently found by Cole, Klein and Tarjan [CKT] In all these results the Boruvka Algorithm plays a key role. Indeed in order to simplify their complicated parallel algorithm and its analysis Cole, Klein and Tarjan [CKT] call each iteration of Boruvka Algorithm (i.e. each iteration of edge selection and subsequent contraction) Boruvka Step . We need one more definition (related to the MST verification algorithm) Given a weighted graph G = V; E) and a spanning forest F (i.e. V; F ) contains no circuits but ....

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R. Cole, P. N. Klein, R. E. Tarjan, A linear-work parallel algorithm for finding minimum spanning trees, Proc. of SPAA, 1994.

....undirected graph is a fundamental problem that has received much attention. There have been many algorithms designed for the MSF problem that run in close to linear time (see, e.g. CLR91] Recently a randomized linear time algorithm for this problem was presented in [KKT95] Based on this work [CKT94] presented a randomized parallel algorithm on the CRCW PRAM which runs in O(2 log n log n) expected time while performing linear work. The expected time was later improved to logarithmic by [CKT96] In this paper we consider the design of a linear work parallel algorithm on a more restricted ....

....assume that edges in G have distinct weights, so the graph has a unique minimum spanning forest (MSF) We also assume that G does not contain isolated vertices, so we have m n Gamma 1. Our algorithm can be viewed as a parallelization of [KKT95] and has a recursive structure similar to that of [CKT94]. As in these two algorithms, our algorithm makes use of the following well known properties (see [Tar83] Cycle Property: For any cycle C in a graph, the heaviest edge in C does not appear in the MSF. Cut Property: For any proper nonempty subset X of the vertices, the lightest edge with exactly ....

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R. Cole, P.N. Klein, and R.E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proceedings of the 1994 ACM Symposium on Parallel Algorithms and Architectures, pages 11--15, 1994. 11

....is the first provably optimal parallel algorithm for this problem under both measures. Here is a brief summary of related results. Following the linear time sequential MSF algorithm of Karger, Klein and Tarjan [KKT95] and building on it) came linear work parallel MST algorithms for the CRCW PRAM [CKT94, CKT96] and the EREW PRAM [PR97] The best CRCW PRAM algorithm known to date [CKT96] runs in logarithmic time and linear work, but the time bound is not known to be optimal. The best EREW PRAM algorithm known prior to our work is the result of Poon and Ramachandran which runs in O(log n log log n2 log ....

....n is the number of vertices in the input graph, and k = log (2) n) 2 . y To perform this reduction y We use log (r) n to denote the log function iterated r times, and log n to denote the minimum r s.t. log (r) n 1. 1 the algorithm uses the familiar recursion tree of depth log n [CKT94, CKT96, PR97], which gives rise to O(2 log n ) recursive calls, but the time needed per invocation in our algorithm is well below O(log n=2 log n ) Thus the total time for Phase 1 is O(log n) We accomplish this by requiring Phase 1 to find only a subset of the MSF. By contracting this subset of the MSF ....

[Article contains additional citation context not shown here]

R. Cole, P.N. Klein, and R.E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proc. SPAA, 1994, pp. 11--15.

....to both time and work. Finding an MSF is an important problem and there has been considerable prior work on parallel algorithms for the MSF problem. Following the linear time sequential MSF algorithm of Karger, Klein and Tarjan [KKT95] came linear work parallel MSF algorithms for the CRCW PRAM [CKT94,CKT96] and the EREW PRAM [PR97] The best CRCW PRAM algorithm known to date [CKT96] runs in logarithmic time and linear work, but the time bound is not known to be optimal. The best EREW PRAM algorithm known prior to our work is the result of Poon and Ramachandran which runs in O(log n log log n 2 log ....

....PRAM algorithm of [CKT96] In Phase 1, the algorithm reduces the number of vertices in the graph from n to n=k vertices, where n is the number of vertices in the input graph, and k = log (2) n) 2 . y To perform this reduction the algorithm uses the familiar recursion tree of depth log n [CKT94,CKT96,PR97], which gives rise to O(2 log n ) recursive calls, but the time needed per invocation in our algorithm is well below O(log n=2 log n ) Thus the total time for Phase 1 is O(log n) We accomplish this by requiring Phase 1 to find only a subset of the MSF. By contracting this subset of the MSF ....

[Article contains additional citation context not shown here]

R. Cole, P.N. Klein, R.E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proc. SPAA, 1994, pp. 11--15.

....occur. Randomization can decrease the time complexity of sequential algorithms and the total work of parallel algorithms. Karger, Klein, and Tarjan [17] describe a randomized sequential algorithm that determines an MST in O(m) time with high probability. Similarly, Cole, Klein, and Tarjan [18] present a randomized parallel algorithm that finds an MST with O(m) expected work. 5 3 FNR 3.1 Overview of FNR FNR, our sequential algorithm for ANR, determines R(v) for all v in O(m log n) time. FNR takes only O(mff(m; n) time if the edges are presorted by weight. Recall that wt(e) denotes ....

R. Cole, P. N. Klein, and R. E. Tarjan. "A linear-work parallel algorithm for finding minimum spanning trees." In Proceedings of the 6th Annual ACM Symposium on Parallel Algorithms and Architectures (1994), 11--15.

.... 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 feature that they serialize into ....

R. Cole, P. N. Klein, and R. E. Tarjan, A linear-work parallel algorithm for finding minimum spanning trees, Proc. 6th ACM Symp. on Parallel Algorithms and Architectures, 1994, pp. 11--15.

....stronger CRCW PRAM. Another interesting open problem is whether there exists an O(logn) time, optimal speed up PRAM algorithm for finding a minimum spanning forest in a weighted graph. The fastest optimal seepdup CRCW PRAM algorithm for the problem to date, obtained by Cole, Klein and Tarjan [CKT94] runs in O(2 log n log n) time. An O(log n) time optimal speed up EREW PRAM algorithm for finding minimum spanning forests in dense graphs was obtained by Karger [Kar94] ....

R. Cole, P.N. Klein, and R.E. Tarjan. A linearwork parallel algorithm for finding minimum spanning trees. In Proceedings of the 6th Annual ACM Symposium on Parallel algorithms and architectures, Cape May, New Jersey, pages 11--15, 1994.

....m) by a Chernoff bound. Combining this with the previous high probability bound of O(m) on m 0 , we find that the total number of edges in the original problem and in all subproblems is O(m) with probability 1 Gamma e Gamma Omega Gamma m) 5 Remarks In work with Richard Cole [5], Klein and Tarjan have adapted the randomized algorithm to run in parallel. The parallel algorithm does linear expected work and runs in O(log n 2 log n ) expected time on a CRCW PRAM [16] This is the first parallel algorithm for minimum spanning trees that does linear work. In contrast, ....

R. Cole, P. N. Klein, and R. E. Tarjan, "A linear-work parallel algorithm for finding minimum spanning trees," to appear in Proc., 6th Symposium on Parallel Algorithms and Architectures, 1994.

....to the MSF. Discard these edges, and recursively calculate the MSF of the remaining graph. Identifying the edges to be discarded seems to require Theta(log n) time; thus the time required by an algorithm having the above form is O(log n) times the number of recursive invocations. Previously [5], by varying the sampling probability depending on the recursion depth, we were able to bound the number of invocations by O(2 log n ) but there seems no way to reduce it further to a constant, which is This research was supported in part by NSF grants CCR 8906949 and CCR 9202900. y This ....

....if T has an incident edge then the probability is at least 1 8 that T is one of the trees selected. Such a Boruvka step can be implemented to run in O(1) expected time using a number of processors equal to the number of active in edge trees plus the number of neutral edges incident to such trees [5]. 3 The Phase I Algorithm: Overview We now give the recursive procedure FindForest. It refers to m, which denotes the number of edges in the original input graph. It also refers to a global parameter k whose value is p log log m. We use log (i) m to denote the application to m of the i fold ....

R. Cole, P. N. Klein, and R. E. Tarjan, "A linear-work parallel algorithm for finding minimum spanning trees," 6th Annual ACM Symposium on Parallel Algorithms and Architectures (1994), pp. 11-15.

No context found.

R. Cole, K. P. N., and T. R. E. A Linear-Work Parallel Algorithm for Finding Minimum Spanning Trees. In Proc. SPAA: Annual ACM Symposium on Parallel Algorithms and Architectures, pages 11--15, 1994.

No context found.

R. Cole, K. P. N., and T. R. E., "A Linear-Work Parallel Algorithm for Finding Minimum Spanning Trees," in Proc. SPAA: Annual ACM Symp. on Parall. Alg. and Arch., pp. 11--15, 1994.

No context found.

R. Cole, K. P. N., and T. R. E. A Linear-Work Parallel Algorithm for Finding Minimum Spanning Trees. Anual SPAA, 11--15, 1994.

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R. Cole, P. N. Klein, and R. E. Tarjan. A linear-work parallel algorithm for finding minimum spanning trees. In Proceedings of the 6th Annual ACM Symposium on Parallel Algorithms and Architectures (1994), pp. 11--15.

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