Gazit,H., `An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph,' Proc. IEEE Symposium on Foundations Of Computer Science, 1986, pp.492-501.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....It is therefore critical for the domain scientists to have a discrimination function that determines which points will be needed during the post processing phase. 18 C.1. Clustering Clustering also called connected component labeling is a common technique in the fields of computer science [15], electrical engineering (VLSI layout) 4] and physics (percolation clustering) 13] It takes as input an array of points, and returns a label for each point, dividing the points into sets. Each set has the following property: For every member P in a set S, P corresponds to a unique point in the ....

H. Gazit, "An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph," 27 Symposium on the Foundations of Computer Science, pp. 492-501 (1986).

....algorithm runs in logarithmic time and linear work but is considerably simpler than the 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 ....

H. Gazit, An optimal randomized parallel algorithm for finding connected components in a graph, SIAM J. Comput., 20 (1991), pp. 1046--1067.

....best bound known for testing graph connectivity if the input graph is specified as a list of edges that are not organized as adjacency lists for vertices. 50 The connected components of a graph can be obtained optimally in logarithmic time on a CRCW PRAM using the randomized algorithm of Gazit [Ga86]. The required adjacency lists can also be obtained optimally in logarithmic time using a randomized algorithm for bucket sort in the range [1: n] given in [RR89b] 2) Sorting Our algorithms use sorting in several places. All of the sorting that is needed can be performed using an algorithm for ....

H. Gazit, "An optimal randomized parallel algorithm for finding connected components in a graph," Proc. 27th Ann. IEEE Symp. on Foundations of Comp. Sci., 1986, pp. 492-501.

.... [45] uses less operations than the deterministic algorithm of the same paper, and runs more quickly than the newer deterministic algorithm of Goldberg and Spencer [27] Other problems with randomized algorithms more e#cient than their deterministic counterparts include finding connected components [24], and sorting integers [59] Finally, a randomized algorithm may have the same bounds as a deterministic algorithm, but may be much simpler. Also, randomized algorithms may be discovered before any equally e#cient deterministic solution. Examples include algorithms for list ranking and for tree ....

H. Gazit, An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph. 27th Symp. Found. Comput. Sci., 1986, 492--501.

....Vishkin [8] this algorithm is considered complex. Recently, simpler, near optimal, O(lg n) time deterministic algorithms using (n m)ff(n; m) lg n processor were presented by Hagerup [12] and by Iwama and Kambayashi [16] An optimal O(lg n) time randomised algorithm has been presented by Gazit [10]. 2.2.2 CREW PRAM Algorithms The first algorithm for this model to achieve O(lg 2 n) running time was introduced in 1979 by Hirschberg, Chandra and Sarwate [14] this algorithm used n 2 = lg n processors. In 1982 Chin, Lam and Chen [5] published an O(lg 2 n) time CREW algorithm using only ....

....are logical functions, comparison functions, predicates, arithmetic functions, functions to convert data types, and a random number generator function. An apply to each construct can be used execute these scalar functions in parallel on sequences of data. For example, 22 vDoubled= s 2 : s in [ 10, 3, 5 ] ; would assign vDoubled 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 ....

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. Proc. 27th IEEE FOCS, p492--501, 1986.

.... O( m n)ff(m; n) log n) processors, where ff(m; n) is a functional inverse of the Ackermann function, were obtained by Cole and Vishkin [CV91] and Iwama and Kambayashi [IK94] The number of processors can be reduced to the optimal number of O( m n) log n) if randomisation is allowed (Gazit [Gaz91] Until not long ago, the best CREW PRAM connectivity algorithm used O(log 2 n) time (Hirschberg, Chandra and Sarwate [HCS79] Chin, Lam and Chen [CLC82] The problem of designing an o(log 2 n) time CREW PRAM algorithm was open for over a decade (cf. KR90] until a deterministic O(log ....

...., we mean that the probability that it does not occur is O(n Gammac ) for some fixed constant c 0. By appropriately setting some relevant parameters, we can usually make the constant c arbitrarily large. The first stage (the size reduction) uses an adaptation of a method developed by Gazit [Gaz91] to reduce the number of non isolated vertices, followed by an application of a random sampling method Karger [Kar93] used also by Karger et al. KNP92] to reduce the number of edges. The method of Gazit was developed for the CRCW PRAM model and adapting it to run in the EREW PRAM model ....

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, 1991.

....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 ....

Gazit, H., An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph, SIAM Journal of Computing, 20(6), 1991, 1046-1067.

....and m= log n n 1 ffl processors for any constant ffl 0. Thus his algorithm is within a constant factor of optimal 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] ....

H. Gazit, "An optimal randomized parallel algorithm for finding connected components in a graph," SIAM Journal on Computing, 6(1991), 1046--1067.

....Lemma 2.1 ( 11] Given a string of length n, its suffix tree can be constructed in O(log n) expected time and O(n) expected work if the string has constant sized alphabet, and in O(logn) expected time and O(n log log n) expected work if it has a polynomial sized alphabet. Lemma 2. 2 ( [14]) Given a graph on n vertices and m edges, its connected components can be determined in O(log n) time and O(m) work. Lemma 2.3 ( 7] Given an array A of n numbers, it can be pre processed in O(log n) time and O(n) work so any range maxima query (that is, given [i; j] return the maximum value ....

H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. In Proc. of the 27th IEEE Annual Symp. on Foundation of Computer Science, pages 492--501, 1986.

....that T is connected and has n Gamma 1 edges. Then, the checker confirms that all non tree edges (v; u) are backedges; i.e. that either v is an ancestor of u or u is an ancestor of v. Connectivity can be determined in O(log n) expected time and O( m n log n ) processors using the techniques of [13]. Finding lowest common ancestors for all edges can be done in O(log n) time using O(n= log n) processors [26] In the problem of constructing a breadth first search tree, the input is an undirected graph G and some node r in G. The output is a rooted (directed) tree T which can be obtained by ....

Gazit, H., "An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph", in Proceedings Foundations of Computer Science, pp. 492-501, 1986.

....as a linear work parallel algorithm for computing connected components. Indeed, in this case the verification step becomes trivial, as does the edge selection method in Contract. The resulting randomized connected components algorithm, while not as fast as the O(log n) time algorithm of Gazit [9], is conceptually quite simple. ....

H. Gazit, "An optimal randomized parallel algorithm for finding connected components in a graph," Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1986, pp. 492-501.

....problem therefore does not work. Instead each vertex checks if it belongs to any tree after hooking. If it does not then it can hook to any neighbor, even if it has a larger index. This is called an unconditional hook. The randomized algorithm can be improved to run in optimal work O(n m) [29]. The basic idea is to not use all the edges for hooking on each step, and instead use a sample of the 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 ....

Hillel Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--67, December 1991.

.... 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 ....

H. Gazit, An optimal randomized parallel algorithm for finding connected components in a graph, SIAM J. Comput. 20 (1991), no. 6, 1046--1067.

....to read and write to a block of shared memory. Many algorithms also assume the CRCW PRAM model, in which any number of processors can read from or write to a single location of the shared memory at the same time. These include the optimal parallel graph connectivity algorithm by Hillel Gazit [3], which achieves an asymptotic running time of log(N ) using O( N M) log(N) processors on a CRCW PRAM; N is the number of vertices in the graph, and M is the number of edges. Shiloach and Vishkin [4] conjecture that this limit can not be improved by any polynomial number of processors. A ....

....of the graph. Processor 0 keeps a list of processor id numbers. At each step, this list is scrambled and broadcast to the other processors. This list determines which processors communicate with each other on each step: p list[0] communicates with p list[1] p list[2] communicates with p list[3] and so on. Suppose that p i is communicating with p j . p i and p j each select 4 vertices that have not yet chosen three neighbors. A single vertex will not be selected more than once in this operation. Then p i and p j select 4 vertices at random from the vertices that have fewer edges than ....

Hillel Gazit, An Optimal Randomized Parallel Algorithm for Finding Connected Components. SIAM Comput., 20(6)

....2 n) time computations. Accelerating centroid decomposition was the motivation for the tree contraction version of [CV88] Two logarithmic time connectivity algorithms were given: 1) a deterministic one which is optimal on all except very sparse graphs [CV86a] 2) a randomized optimal one [Gaz86] For Figure 1, the deterministic algorithms builds on a restricted union find problem, a scheduling problem, dubbed duration unknown task scheduling, and the Euler tour technique, as well as ideas from two previous connectivity algorithms [HCS79] and [SV82a] It should be pointed out that the ....

.... Randomization has shown to be very useful for both the simulation of PRAM like shared memory models of parallel computation by other models of parallel machines (e.g. in [KU86] KRS88] MV84] Ran87] KPS90] and [MSP90] and for the design of parallel algorithms (e.g. in [ABI86] AM90] Gaz86] GM91] KR87] Lub86] MR85] MV90] MV91] RR89a] RS89] Rei81] Sch80a] Sen89] and [Vis84b] All target al..gorithms in this section are randomized, and their running time is at the doubly logarithmic level, or faster. By the doubly logarithmic level, we mean O(f(n) log log n) ....

H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. In Proc. of the 27th IEEE Annual Symp. on Foundation of Computer Science, pages 492--501, 1986.

....the mesh. Thus the available connected component algorithms for rmesh are not suitable to our goal. Our approach is to take a pram algorithm for computing connected components, and simulate it on the p rmesh using the pram simulation algorithm from Section 3. We use a crcw pram algorithm by Gazit [18] for computing the connected components of a given graph G on v nodes and m edges on n = v m) lg v processors, in T = O(lg v) time w.h.p. In our case, v = O( N=p) 2 ) and m = O(N 2 =p) Using Theorem 3.8, Gazit s algorithm can be simulated on the p rmesh in time O(T (n=p lg lg p) which ....

H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, December 1991.

....computer science theory courses as an application of depth first and breadth first search. Parallel solutions have received a great deal of attention from both theorists and practical computer scientists, and have proven difficult. Theoretical work shows good results on the CRCW PRAM model[3, 10, 11, 24], which assumes uniform memory access time and arbitrary bandwidth to any memory location. This material is based upon work supported under a National Science Foundation Presidential Faculty Fellowship Award, a Graduate Research Fellowship, and Infrastructure Grant number CDA 8722788, as well ....

H. Gazit, "An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph," SIAM Journal of Computing 20(6), December 1991.

....solutions are well understood and commonly used as an application of depth first and breadth first search. Parallel solutions have received a great deal of attention from theorists, and have proven difficult. Algorithms such as Shiloach Vishkin obtain good results with the CRCW PRAM model [3, 11, 12, 28], which assumes uniform memory access time and Copyright 1995 by the Association for Computing Machinery, Inc. ACM) Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for ....

H. Gazit, "An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph," SIAM Journal of Computing 20(6), December 1991.

....[CV91] and Iwama and Kambayashi [IK94] The number of processors can be reduced This work forms a part of an M.Sc. thesis written by the first author. E mail addresses of authors: fhshay,zwickg math.tau.ac.il. to the optimal number of O( m n) log n) if randomization is allowed (Gazit [Gaz91] Until not long ago, the best CREW PRAM connectivity algorithm used O(log 2 n) time (Hirschberg, Chandra and Sarwate [HCD79] Chin, Lam and Chen [CLC82] The problem of designing an o(log 2 n) time CREW PRAM algorithm was open for over a decade (cf. KR90] until a deterministic O(log ....

....finds its connected components. Both stages take O(log n) time using O( m n) log n) processors. The failure probability of both stages is polynomially small in n (i.e. n Gammac for any desired c 0) This first stage (the size reduction) uses an adaptation of a method developed by Gazit [Gaz91] to reduce the number of non isolated vertices, followed by an application of a method of Karger, Nisan and Parnas [KNP92] to reduce the number of edges. The method of Gazit was developed for the CRCW model and adapting it to run in the EREW model requires subtle changes. The second stage ....

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, 1991.

....the mesh. Thus the available connected component algorithms for rmesh are not suitable to our goal. Our approach is to take a pram algorithm for computing connected components, and simulate it on the p rmesh using the pram simulation algorithm from Section 3. We use a crcw pram algorithm by Gazit [18] for computing the connected components of a given graph G on n nodes and m edges in O(n m) work and O(lg n) time, with high probability. Using p 0 processors, the self simulation property of the crcw pram implies that Gazit s algorithm runs in O(lg n (n m) p 0 ) time w.h.p. In our case, ....

H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, December 1991.

....PRAM authors ref. time processors det. ran. Shiloach, Vishkin [SV82] O(logn) m n deterministic Awerbuch, Shiloach [AS87] O(logn) m n deterministic Cole, Vishkin [CV91] O(logn) O( m n)ff(m; n) log n) deterministic Iwama, Kambayashi [IK94] O(logn) O( m n)ff(m; n) log n) deterministic Gazit [Gaz91] O(logn) O( m n) log n) randomized CREW PRAM authors ref. time processors det. ran. Hirschberg, Chandra, Sarwate [HCS79] O(log 2 n) O(n 2 = log n) deterministic Chin, Lam, Chen [CLC82] O(log 2 n) O(n 2 = log 2 n) deterministic Han, Wagner [HW90] O(log 2 n) O( m n log n) log 2 ....

....previous connectivity and spanning forest algorithms. In the first stage, for example, we use the maximum hooking method of Chong and Lam [CL95] the growth control mechanisms and the idea of edge list plugging used by Johnson and Metaxas [JM91] JM92] and finally a sampling technique of Gazit [Gaz91] In the second phase, we use a sampling technique of Karger, Klein and Tarjan [KKT95] In the third stage we use short random walks, as was done by Karger, Nisan and Parnas [KNP92] A full description of the algorithms described here is quite lengthy. They involve many intricate details and ....

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, 1991.

....although for some graphs, including planar graphs, it only requires O(m) work. If scans are assumed to require only constant time, a reasonable assumption for the hardware used [3] only O(log n) time is needed. There is an optimal, randomized O(n) work, O(log 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 ....

....edges as nodes during the early iterations. But this is only a small fraction of the number of edges, which is initially proportional to the square of the initial number of nodes. Thus during contraction, the graph becomes increasingly dense. Note that a similar mating algorithm is used by Gazit [9] to transform sparse graphs into dense graphs for this reason. Once the graph is almost fully connected, edge contraction becomes very quick as the number of edges is bounded by the square of the current number of supernodes. For tertiary graphs, a similar phenomenon is seen, except that since the ....

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM J. Comput., 20(6), Dec. 1991.

....A path joining v 1 and v k in G is a sequence of vertices v 1 , v 2 , v k such that (v i , v i 1 ) 2 E for 1 i k: A graph G = V; E) is connected if there is a path from every vertex in V to every other vertex in V . We will use known parallel algorithms for the connected components [CV 91, G 91] which require O(log(n) time and n m log(n) processors for randomized algorithms and m n log(n) Delta ff(m; n) for the deterministic algorithm, where ff(m; n) is the inverse Ackerman function. By using a randomized connectivity algorithm for planar graphs, the processor bounds can be ....

.... be found in Hopcroft and Tarjan s papers [HT 73a, HT 73b] We can run this algorithm in parallel because we can find the 2 connected components [TV 85] and the 3 connected components [FRT 89] in O(log(n) time using the same processor complexity as the parallel con4 nectivity algorithms [CV 91, G 91] which use a sublinear number of processors. Summarizing these known results, we have: Lemma 2.1 Given a graph of n vertices and m edges, there is an algorithm to compute the 3connected components tree in O(log(n) time and n m log(n) processors by a randomized algorithm and m n log(n) ....

H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6), pages 1046-1067, December 1991. 21

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Gazit,H., `An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph,' Proc. IEEE Symposium on Foundations Of Computer Science, 1986, pp.492-501.

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H. Gazit. An optimal randomized parallel algorithm for finding connected components in a graph. SIAM Journal on Computing, 20(6):1046--1067, 1991.

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