Y. Shiloch and U. Vishkin, An O(log n) Parallel Connectivity Algorithm, J. Algorithms, 3(1) (1983), 57--67. F. Dehne, A. Ferreira, E. Caceres, S. W. Song, and A. Roncato  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the label of the component as its leader. There are many parallel algorithms for determining connected components. The algorithm by Chin, Lam and Chen [2] determines connected components in O(log n) time with O(n ) work on a common CRCW PRAM model. The algorithm by Shiloach and Vishkin [8] determines connected component in O(log n) time with O( m n) Delta log n) on an arbitrary CRCW model where m is the number of edges in the graph. Both of these algorithms assume that in one time step every edge can be examined (in parallel) While this is true in PRAM models, it does not apply ....

Y. Shiloach and U. Vishkin. An o(n log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57--67, 1982.

....graph G = V; E) jV j = n, and jEj = m. The problem of finding the connected components (CC) of G can be solved sequentially in O(n m) time by the depth first search. There exist several PRAM CC algorithms. The simplest ones require O(log n) parallel steps using n m CRCW PRAM processors [7]. Although a simulation of a CRCW PRAM algorithm on an EREW PRAM slows down the time by the factor of log n, there are some o(log n) time algorithms running on a linear number of EREW processors. In particular, an algorithm running in O(log n log log n) time on n m EREW PRAM processors [2] and ....

....in [1] is designed for graphs corresponding to 2 dimensional images. In the global phase, processors merge the vertical and horizontal borders of their local tiles in log p iterations, where p is the number of processors. The algorithm [6] simulates the Shiloach and Vishkin PRAM algorithm [7] on graphs consisting of remote edges and of roots of local components produced by local DFS s. It was implemented in Split C on CM 5. Our algorithm, denoted by MPI CC in the further text, is an extension and optimalization of the algorithm from [6] It was implemented in C and MPI, and ....

Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. Journal

.... rst search (BFS) or depth rst search (DFS) algorithms [8, 29] BFS has been used in discovering the Strongly Connected Components (SCC, directions of hyperlinks are taken into account) and Weakly Connected Components [4] A large number of graph algorithms for nding connected components exist [28, 1, 14, 15]. In this paper, we propose a new, algebraic method for nding disconnected and nearly disconnected components, such as those illustrated in Figure 1. The method follows the spectral graph partitioning framework, rst developed by Donath and Ho man [10] and Fiedler [11, 12] and recently populated ....

Y. Shiloach and U. Vishkin. An o(log n) parallel connectivity algorithm. J. Algorithm, pages 57-67, 1982.

....incident edges, and contract them. We use randomization in both identifying the lowest weight incident edges, and in ensuring that no cycles are formed by the contraction. It would take too long to fully carry out the contractions in each iteration, so we use an idea of Shiloach and Vishkin ([20]; see also [1] and represent the sets of merged nodes using parent pointers. A processor is associated with each edge and each node. For each node v, the algorithm maintains a parent pointer p(v) The parent pointers form a structure consisting of trees where each root points to itself. A tree ....

Y. Shiloach and U. Vishkin, \An O(logn) parallel connectivity algorithm," J. Algorithms 3, 1982, pp. 57-67.

....then the set of nodes B form a clique, by Lemma 1.3. The procedure assumes the existence of edges between nodes in B without ever checking for their presence. Speci cally, in computing connected components of a graph involving nodes of B, the procedure uses the algorithm of Shiloach and Vishkin [43], suitably modi ed to take into account the fact that any two nodes of B are adjacent. Depending on the nodes in B, the procedure Stratify(G ; C) calls one of three subprocedures, in which most of the work is done. In each procedure, we make use of parallel pre x computation, due to Ladner ....

....of an induced subgraph of G containing nodes of C and nodes of B. We want to implement this step in a way that the actual edges between nodes of B are not involved. To carry out the connected components computation (and to nd spanning trees of the components) we use the connectivity algorithm of [43], suitably modi ed to take into account our assumption that every two blue nodes are adjacent: we start by constructing a tree containing all the blue nodes in H D (and another tree containing all the blue nodes in D. We then execute the algorithm of [43] using these arti cial edges of these ....

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Y. Shiloach and U. Vishkin, \An O(log n) parallel connectivity algorithm," J. Algorithms 3 (1982), pp. 57-67.

....and Chen [5] who achieve a work optimal algorithm for dense graphs. Nath and Maheshwari [30] provided an algorithm which require O(log 2 n) time using O(n 2 ) processors on the weakest EREW PRAM model. On the powerful CRCW PRAM model, O(log n) time deterministic algorithms exist (e.g. see [40]) In particular, Cole and Vishkin [7] attain a nearly optimal processor bound, i.e. O( n m) log (3) n= log n) processors, on the STRONG CRCW PRAM model. Awerbuch and Shiloach [1] provided a PRIORITY CRCW PRAM algorithm requiring O(n m) processors. Johnson and Metaxas [23] provided the first ....

Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57--67, 1982.

....one, and then solving the problem on the resulting undirected graph, is possible. If E 0 = O(n) it even leads to ecient solutions, since ecient solutions for undirected graphs are possible. One of them of them, pointed to us recently, that is time optimal (for undirected graphs) appears in [SV82] for CRCW PRAM, and can be translated into the model of [HLL99] using O(jE 0 j log n) messages. Two others, one message optimal, and one communication optimal, we present in the full version of our current paper. These algorithms, too, can be translated into the directed model, with the ....

Yossi Shiloach and Uzi vishkin. An O(log n) Parallel connectivity algorithm. Journal of Algorithms 3, pp. 57-67, 1982.

....have dimension 2 z Theta 2 z 1 , B has dimension 2 z 1 Theta 2 z 1 . The algorithm needs n processors and takes time O(n 1=2 ) The access patterns of this algorithm only depend on the dimensions of the matrices. The connected components algorithm was adapted from Shiloach and Vishkin [19]. For a given undirected graph with n = max(2 Delta #edges; #nodes) the connected components are computed. The algorithm needs n processors and takes time O(log n) The graph is represented by two arrays HEAD;TAIL. For a given edge e, HEAD[e] and TAIL[e] give the nodes to which e is adjacent. ....

Y. Shiloach and Uzi Vishkin. An O(log n) parallel connectivity algorithm. Journal of Algorithms, 3:57--67, 1982.

....and edges, we assign a processor to each set of vertices of size k. Thus, the step 1 1 can be performed in a constant time. The step 1 2 can be carried out using an algorithm that computes connected components. The parallel algorithm for this task runs in O(log n) time and uses O(n m) processors [23]. Next we count the number of processors. The number of the set of vertices of size k is 0 n k 1 = O(n k ) For each set we check the reachability that requires O(n m) processors. Hence, we need at most a total of O(n k 1 n k m) processors. Now we turn to the recognition problem of ....

Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. Journal of Algorithms, 3:57--67, 1982.

....problem with spanning trees and section 3 offers conclusions and topics for further research. 3 The Connected Component Problem and Spanning Trees Shiloach and Vishkin offer an optimal parallel connected component algorithm for the parallel random access memory (PRAM) model of computation in [26]. This algorithm requires O(logn) time and n 2m processors. We take inspiration from this algorithm and generalize it for an LARGB with P processors. We begin with the 2m edges input 2m P edges per processing element. First each processing element conducts a breadth first search on the edges ....

....conducts a breadth first search on the edges it stores locally. The result being a spanning forest for those edges [2] Before proceeding we will breifly review the functions used to complete the connected component processing. For further details of these functions the reader is referred to [26]. Conditional Hooking: if the parent of i is a root, j is adjacent to i in the original graph, and parent[j] parent[i] then hook j to i. Star Hooking: If i belongs to a star, j is adjacent to i in the original graph, and j is not in i s star then hook j to i. Shortcutting: if i is not a root, ....

Yossi Shiloach and Uzi Vishkin. An O(log n) Parallel Connectivity Algorithm. Journal of Algorithms, 3:57--67, 1982.

....in case there are long learning chains. For instance, it may happen for three nodes v; u; w, that while v assigns f(u) to be its leader, u may assign f(w) to be its own leader. The solution to this difficulty is to use log n 1 rounds of leader updates, a process called pointer jumping [SV82]. In our context, this is done as follows. Phase k pointer jumping: The pointer jumping part of the phase consists of log n 1 indirection resolutions (abbreviated IR henceforth) The idea in an IR is that each node records locally, for each of its neighbors, the previous leader ID it announced. ....

Y. Shiloach and U. Vishkin, An O(log n) parallel connectivity algorithm, J. of Algorithms Vol. 3, (1982), 57--67.

....SIMD CM 2, multiprocessors such as MasPar, CM5 and even on a network of workstations communicating via PVM. NESL is discussed in more detail in section 4.2. 2.2 Previous Work 2.2. 1 CRCW PRAM Algorithms There are several simple deterministic O(lg n) time CRCW algorithms for this problem, 1] [21], most of these algorithms are however not optimal. A near optimal deterministic O(lg n) time algorithm using (n m)ff(n; m) lg n processors was presented by Cole and Vishkin [8] this algorithm is considered complex. Recently, simpler, near optimal, O(lg n) time deterministic algorithms using ....

....This is known as nested parallelism. An example of a nested sequence of integers is [ 1, 3] 0, 2, 4, 6] 5, 7, 9] If, using apply to each, the built in function sum( is applied to this nested sequence sum(v) v in [ 1, 3 ] 0, 2, 4, 6 ] 5, 7, 9] the result would be the sequence [4, 12, 21], where the sum( function has been applied to each subsequence in parallel. 4.2.6 Divide and Conquer algorithms Nested parallelism is very useful in the implementation of divide and conquer algorithms. A divide and conquer algorithm breaks the original data into smaller parts, applies the same ....

Y. Shiloach and U. Vishkin. A O(lg n) Parallel Connectivity Algorithm. Journal Of Algorithms 3; p57--67, 1982.

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Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. J. Algorithms, 3:57--67, 1982.

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Y. Shiloch and U. Vishkin, An O(log n) Parallel Connectivity Algorithm, J. Algorithms, 3(1) (1983), 57--67. F. Dehne, A. Ferreira, E. Caceres, S. W. Song, and A. Roncato

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Y. Shiloch and U. Vishkin, An O(log n) Parallel Connectivity Algorithm, J. Algorithms, 3(1) (1983), 57--67. F. Dehne, A. Ferreira, E. Caceres, S. W. Song, and A. Roncato

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Y. Shiloch and U. Vishkin, An O(log n) Parallel Connectivity Algorithm, J. Algorithms, 3(1) (1983), 57--67. F. Dehne, A. Ferreira, E. Caceres, S. W. Song, and A. Roncato

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Shiloach, Y. and Vishkin, U. An O(log n) parallel connectivity algorithm. J. Algorithms 3, 1(Mar. 1982), pp. 57-67. 20

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Y. Shiloach and U. Vishkin. An o(log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57-67, 1983.

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Y. Shiloach and U. Vishkin, An O(log n) parallel connectivity algorithm, J. Algorithms 3 (1982) 57-67. 6

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Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57--67, 1982.

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Y. Shiloch and U. Vishkin, An O(log n) Parallel Connectivity Algorithm, J. Algorithms, 3(1) (1983), 57--67. F. Dehne, A. Ferreira, E. Caceres, S. W. Song, and A. Roncato

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Y. Shiloach and U. Vishkin. An O(log n) Parallel connectivity algorithm. Journal of Algorithms 3, pp. 57-67, 1982.

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Y. Shiloach and U. Vishkin. An O(log n) Parallel Connectivity Algorithm. J. Algorithms 3, (1982), 57--67.

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Y. Shiloach and U. Vishkin. An O#log n# parallel connectivity algorithm. Journal of Algorithms 3 #1981#, pp. 57#67.

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Y. Shiloach, U. Vishkin, An O(log n) parallel connectivity algorithm, J. Algorithms, 3 (1982), pp. 57-63.

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