S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. Journal of Computer and System Sciences, 53:395--416, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....MSF algorithm in [CKT96] and also is about twice as fast. 2. As modified for the CRCW PRAM, our algorithm is simpler than the linearwork logarithmic time CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ96, HZ01] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm also is simpler than the algorithms in [HZ96, HZ01] In the following we use the notation S T to denote the union of sets S and T , and we use S e to denote the set formed by adding the element e to ....

....CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ96, HZ01] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm also is simpler than the algorithms in [HZ96, HZ01]. In the following we use the notation S T to denote the union of sets S and T , and we use S e to denote the set formed by adding the element e to the set S. We say that a result holds with high probability (or w.h.p. in n if the probability that it fails to hold is less than 1 n c for ....

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S. Halperin and U. Zwick, An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM, J. Comput. System Sci., 53 (1996), pp. 395--416.

....is considerably simpler than the MSF algorithm in [CKT96] 2. As modified for the CRCW PRAM, our algorithm is simpler than the linear work logarithmictime CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94, HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is simpler than the algorithms in [HZ94, HZ96] In the following we use the notation S T to denote union of sets S and T , and we use S e to denote the set formed by adding the element e to the set S. ....

....logarithmictime CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94, HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is simpler than the algorithms in [HZ94, HZ96]. In the following we use the notation S T to denote union of sets S and T , and we use S e to denote the set formed by adding the element e to the set S. We say that a result holds with high probability (or w.h.p. in n if the probability that it fails to hold is less than 1=n c , for any ....

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S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proc. SPAA, 1994, pp. 1-10.

....on the EREW PRAM model. The algorithm improved upon some of the ideas presented in the Johnson and Metaxas CREW algorithm. This algorithm is the main subject of study in the project. An optimal O(lg n) time randomised algorithm for the EREW model has been presented in 1994 by Halperin and Zwick [13]. 2.2.4 SVM Algorithms Several CRCW PRAM connected components algorithms on the scan vector model have been designed for the SVM by Greiner [11] Blelloch has described a randomised algorithm for the SVM [2] These algorithms require O(lg n) steps and O(m lg n) work. To the best of our ....

....the sequence [ 20, 6, 10 ] Functions on sequences NESL has many powerful functions to operate on sequences such as scans, permute, pack (compact) distribute (broadcast) functions to work on sequences. For example vPrefixSums = plusscan( 10, 3, 5, 6 ] would assign vPrefixSums the sequence [0, 10, 13, 18 ]. Functions for nesting sequences NESL provides several functions to assist work with nested parallelism : partition( which converts a sequence to a sequence of sequences, flatten( which takes a nested sequence and reduces its level of nesting by 1, and other functions which take a ....

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. Proc. 6th ACM SPAA, p1--10 1994 47

....is considerably simpler than the MSF algorithm in [CKT96] 2. As modified for the CRCW PRAM, our algorithm is simpler than the linear work logarithmictime CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94, HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is arguably simpler than the algorithms in [HZ94, HZ96] The rest of this paper describes and analyzes our algorithm. In the following we use the notation S T to denote union of sets S and T , and we ....

....CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94, HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is arguably simpler than the algorithms in [HZ94, HZ96]. The rest of this paper describes and analyzes our algorithm. In the following we use the notation S T to denote union of sets S and T , and we use S e to denote the set formed by adding the element e to the set S. We say that a result holds with high probability (or w.h.p. in n if the ....

[Article contains additional citation context not shown here]

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proc. SPAA, 1994, pp. 1-10.

....is considerably simpler than the MSF algorithm in [CKT96] 2. As modified for the CRCW PRAM, our algorithm is simpler than the linearwork logarithmic time CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94,HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is arguably simpler than the algorithms in [HZ94,HZ96] In the following we say that a result holds with high probability (or w.h.p. in n if the probability that it fails to hold is less than 1=n c , ....

....CRCW algorithm for connected components given in [Gaz91] 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94,HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is arguably simpler than the algorithms in [HZ94,HZ96]. In the following we say that a result holds with high probability (or w.h.p. in n if the probability that it fails to hold is less than 1=n c , for any constant c 0. 2 The High Level Algorithm Our algorithm is divided into two phases along the lines of the CRCW PRAM algorithm of [CKT96] ....

[Article contains additional citation context not shown here]

S. Halperin, U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proc. SPAA, 1994, pp. 1-10.

....for sufficiently dense graphs, and is within a fractional polynomial factor for very sparse graphs. A related but simpler problem is that of finding connected components. Gazit [10] discovered a randomized logarithmic time, linear work CRCW PRAM connectedcomponents algorithm. Halperin and Zwick [12] discovered how to test connectivity on a CREW PRAM in the same bounds using Gazit s approach, and later refined their algorithm to actually find connected components [13] Gazit s approach was the inspiration for our two phase MSF algorithm: Gazit s algorithm first builds pieces of components and ....

S. Halperin, U. Zwick, "an optimal randomized logarithmic time connectivity algorithm for the EREW PRAM.", 6th Annual ACM Symposium on Parallel Algorithms and Architectures (1994) pp. 1-10.

....(using a di#erent method) by [JM92] After publication of the preliminary version of this paper [KNP92] several improvements were given. Chong and Lam [CL95] gave an O(log n log log n) time deterministic algorithm that uses m n processors. Halperin and Zwick improved our methods to yield first [HZ94] an optimal randomized algorithm for connected components that runs in O(log n) time with a linear number of processors, and subsequently [HZ6] an optimal randomized algorithm for finding a spanning forest of the graph (note that our algorithm does not find spanning forests) In the following ....

....duplicates from the set of m log n edges that we sample for the first application of the random walk algorithm, but the m log n processors that we have are su#cient to do this by sorting. 8. Conclusion. Since the publication of the preliminary version of this paper [KNP92] Halperin and Zwick [HZ94] have used it to derive a work and processoroptimal randomized EREW connected components algorithm. The obvious remaining open problem is to find a deterministic O(log n) time EREW algorithm for connected components. One way to work toward this goal is to improve on the bounded space universal ....

<F3.742e+05> S. Halperin and U.<F3.822e+05> Zwick,<F3.856e+05> An optimal randomized logarithmic time connectivity algorithm for the EREW<F3.822e+05> PRAM, in Proc. 6th Annual ACM-SIAM Symposium on Parallel Algorithms and Architectures, ACM Press, New York, 1994, pp. 1--10.

....edges. This basic technique developed for parallel algorithms has since been used to improve some sequential algorithms, such as deriving the first linear work algorithm for minimum spanning trees [43] Another improvement is to use the EREW model instead of requiring concurrent reads and writes [35]. However this comes at the cost of greatly complicating the algorithm. The basic idea is to keep circular linked lists of the neighbors of each vertex, and then to splice these lists when merging vertices. 4.3.4 Extentions to Spanning Trees and Minimum Spanning Trees The connected component ....

Shay Halperin and Uri Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proc. ACM Symposium on Parallel Algorithms and Architectures, pages 1--10, June 1994.

.... 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 ....

S. Halperin and U. Zwick, An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM, Proc. 6th ACM Symp. on Parallel Algorithms and Architectures, 1994, pp. 1--10.

....in Theorem 4.1, can be improved to O(lg N lg lg lg p lg p (N=p) 2 lg p) steps on the collision rmesh. This can be obtained by using the algorithm presented in Section 4, with the exception of replacing Gazit s crcw pram connectivity algorithm by the erew pram algorithm of Halperin and Zwick [22]. Very recently, Czumaj et al. 14] improved the results of Section 3 for the arbitrary rmesh, showing that an n processor crcw pram can be simulated on an arbitrary n rmesh in constant time w.h.p. This, however, does not imply improvement in simulating a crcw pram on the collision rmesh. As ....

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In 6th ACM Symp. on Parallel Algorithms and Architectures, pages 1--10, July 1994.

....given in Theorem 4.1, can be improved to O(lg N lg lg lg p (N=p) 2 lg p) steps on the collision rmesh. This can be obtained by using the algorithm presented in Section 4, with the exception of replacing Gazit s crcw pram connectivity algorithm by the erew pram algorithm of Halperin and Zwick [23]. Very recently, Czumaj et al. 14] improved the results of Section 3 for the arbitrary rmesh, showing that an n processor crcw pram can be simulated on an arbitrary n rmesh in constant time w.h.p. This, however, does not imply improvement in simulating a crcw pram on the collision rmesh. As a ....

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the erew pram. In 6th ACM Symp. on Parallel Algorithms and Architectures, pages 1--10, July 1994.

....structure in O(log 2 n) time in the EREWPRAM model. The construction uses O(m k log n) work. The time is O(log 2 n) because we construct planar line segment arrangements (Theorem 3. 2) The construction also uses a parallel algorithm for finding the connected components of a graph (see, e.g. [14, 33]) details will be provided in the full paper) Triangulating non intersecting (d Gamma 1) simplices in IR d . If the simplices in S are interior disjoint, then Pellegrini [49] notes that a slight modification of the method for building the incidence query data structure can be used to ....

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the erew pram. In Proc. ACM Symp. Parallel Algorithms and Architectures, pages 1--10, 1994.

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S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. Journal of Computer and System Sciences, 53:395--416, 1996.

....is described in Part I of the paper comprising Sections 5 to 9. The second stage of our algorithm is described in Part II of the paper comprising Sections 10 to 16. We then end in Section 17 with some concluding remarks and open problems. A preliminary version of this paper had appeared in [HZ94] 2 2 An overview of the algorithm The input to the connectivity algorithm is a graph G = V; E) with n vertices and m edges. The graph is specified using its adjacency lists. Our algorithm is composed of two main stages. The first stage takes the input graph G = V; E) and produces an image ....

S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proceedings of the 6th Annual ACM Symposium on Parallel algorithms and architectures, Cape May, New Jersey, pages 1--10, 1994.

....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 same resource bounds as in [HZ94] i.e. O(log n) time and O(m n) work. In addition, the algorithm ....

....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 same resource bounds as in [HZ94] i.e. O(log n) time and O(m n) work. In addition, the algorithm presented here uses linear space. The algorithm of [HZ94] needed more 1 Although Radzik presents his algorithm as an algorithm for finding connected components, his algorithm may also be used to find a spanning forest of the ....

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S. Halperin and U. Zwick. An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. In Proceedings of the 6th Annual ACM Symposium on Parallel algorithms and architectures, Cape May, New Jersey, pages 1--10, 1994. Journal version submitted for publication.

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