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.

....C. It is easily checked that for any connected component C of S j i V j , the set of neighbors of C outside C is a cut. Q.E.D. Note that the ordered partition (V 1 ; Vn ; fvn g) can be determined in O(n m) time and by a CRCW PRAM in logarithmic time with a linear processor number [18]. Now we consider only the subsequence of nonempty partion elements and denote it by V 1 ; V k , i.e. V k = fvn g. Nevertheless, V i is the set of neighbors of a vertex v i 2 V i 1 [ V k that are not in V i 1 [ V k . v 1 ; v k Gamma1 is also called the generating ....

Y. Shiloach, U. Vishkin, An O(log n) Parallel Connectivity Algorithm, Journal of Algorithms 3 (1982), S. 57-67. 9

....running time using the minimal possible number of processors. The algorithm we obtain is randomised. Relatively simple CRCW PRAM algorithms that find the connected components of a graph G = V; E) deterministically in O(log n) time using m n processors were obtained by Shiloach and Vishkin [SV82] and by Awerbuch and Shiloch [AS87] More complicated deterministic CRCW PRAM connectivity algorithms that run in O(log n) time using 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 ....

Y. Shiloach and U. Vishkin. An O(log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57--67, 1982. 34

.... in O(log 3 n) time with O(m) processors, using the maximal matching algorithm of Israeli and Shiloach [20] Each iteration of the second stage of the algorithm is essentially a connectivity computation, which can be computed in O(log n) time and O(m) processors for the undirected case [29], and O(log 2 n) time and BFS(n; m) processors for the directed case [27] This leads to the following theorem. Theorem 3.3 1. On undirected graphs, the Maximal Paths algorithm runs in O( p n log 3 n) time using O(n m) processors. 2. On directed graphs, the Maximal Paths algorithm runs ....

....modification is that the stage terminates when the total amount of excess at active nodes is less than l = m 2=3 , where l is the activity parameter. The key part of this stage is the Push and Relabel procedure. This procedure can be implemented by using either techniques of Shiloach and Vishkin [29] or 15 procedure Zero One(V; E; s; t) first stage] for all v 2 V Gamma fsg do d(v) 0; d(s) n; for all (v; w) 2 E do f(v; w) 0; for all v 2 V do e f (v) 0; for all v 2 V such that (s; v) 2 E do begin f(s; v) 1; e f (v) e f (v) 1; end; while the total amount of excess at ....

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

....to the maximality condition of for j. 2(Lemma) Note that all steps of the algorithm BFC with the exception of the first step can be executed by an EREW PRAM in O(log n) time using O(n m) log n processors. The first step can be done by a CRCW PRAM in O(log n) time with O(n m) processors [33]. Therefore we get the following result. Theorem 4 Suppose G is a graph not containing the house or a cycle of length greater than 4 as an induced subgraph. Then a breadth first search tree can be computed in the same time and processor bound as a spanning tree, i.e. in O(log 2 n) time with O(n ....

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

....are unlike the regular arrays used in numerical code for dense matrix computations. As a result, we need to understand how to program, so that more than one thread can work simultaneously on the same data structure. Availability of Theoretical Results A number of algorithms (eg connected component[48], maximum flow[23] have well understood theoretical complexity. The question is whether these algorithms can be implemented efficiently in an actual parallel environment. Using the performance statistics of these implementations, we can further fine tune the runtime and compiler in the following ....

....an obvious optimization might be to avoid creating new threads in the 1 processor case. 4 The alternative is static optimizations. The compiler may explicitly re structure the code to reduce thread switching and synchronization. For example, an implementation of the connected component algorithm[48] is such that for each vertex of the graph, we create a new thread. However, each iteration of the algorithm has several points where all the threads have to synchronize. This requires a large amount of thread switching, and the performance deteriorates rapidly with large number of vertices. There ....

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

....algorithm for the same problem but running it for only O(log p) rounds and then nishing the computation with some other O(log p) rounds CGM algorithm. See also J aJ a s accelerated cascading technique for the PRAM [23] Steps 1 and 2 of Algorithm 4 simulate Shiloch and Vishkin s PRAM algorithm [34], but for log p phases only. Each vertex v has a pointer to a vertex parent(v) such that the parent(v) pointers always form trees. The trees are also referred to as a supervertices. A tree of height one is called a star. An edge (u; v) is live if parent(u) 6= parent(v) Shiloch and Vishkin s PRAM ....

....v of V 0 let parent 0 (v) be the smallest label parent(w) of a vertex w 2 V 0 which is in the same connected component with respect to G 00 = V 0 ; E 0 ) For each vertex u 2 V stored at processor P i set parent(u) parent 0 (parent(u) Note that parent(u) 2 V 0 . Lemma 10 [34] The number of di erent trees after iteration k of Step 2 is bounded by ( 2 3 ) k n. We obtain Theorem 5 Algorithm 4 computes the connected components and spanning forest of a graph G = V; E) with n vertices and m edges on a CGM with p processors and O( n m p ) local memory per ....

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Shiloch, Y. and Vishkin, U. \An O(log n) parallel connectivity algorithm,\ Journal of Algorithms, 3(1), pp. 57-67, 1983.

....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 ....

....relations. In this case, edges are the axioms of the relation while vertices form the range relation. Hence, using these algorithms the cost of computing a transitive closure is affected by the size of the range. Based on the same hooking technique, another PRAM algorithm for CCug is in [27, 1]. In this algorithm, the time is reduced to O(log(v) and the processors required are 2e v. However, the algorithm is for the strongest CRCW model which could suggest an additional time factor log(v e) in order to simulate it on the weak CREW model [29, 21] According to this simulation, for ....

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Shiloach Y. and Vishkin, U. An O(log n) Parallel Connectivity Algorithm. J. Algorithms 3, 1 (1982), 57-67.

....for list ranking and connected components, problems which have little intrinsic locality, and can therefore be considered as very hard cases. Context. List ranking and connected components have been intensively studied in the parallel domain. Theoretically, the problems have been solved optimally [1, 2] (see also [3] Practically, direct implementations of these algorithms do not lead to good performance. Several later papers are dealing with this problem [4, 5, 6, 7] On a powerful parallel computer, the maximum speedup that may be reached with P processing units, PUs, is about P=3. We address ....

....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 ....

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

....vertices is denoted Y in the proof) In the CRCW PRAM model this test can also be done in constant time (it is a boolean and operation for each v = 2 X) If v 2 Y , then we also put M(v) T rue. Clearly, X [ Y is a dominating set. Using well known NC algorithms for nding components (such as [26]) we can compute the components of the subgraph of G induced by X [ Y in O(log n) parallel time on a CRCW PRAM. If there is only one component, then we are nished. Otherwise, we proceed as follows: We compute distances joining all pairs of vertices of G (namely, the n n matrix A of the ....

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

....of C and the sets C x are given. Proof: This follows immediately from last proposition and the fact that connected components can be computed in the time bounds as mentioned in the corollary (for the sequential case, see any textbook on algorithms, e.g. 9] and for the parallel case, see [30]) 2 (corollary) 9 4.2 The tree structure of overlap components First we show now the following lemma that has been mentioned before. Lemma 9 (Overlap(C) ae) is a tree like ordering. For each overlap component C 1 of C, there is a unique overlap component C 2 , such that C 1 ae C 2 and for no ....

....the problem to determine the lowest common ancestor. We have to determine a linear number of lowest common ancestors (of min(c) and max(c) and we have a tree of linear size. This allow us to determine Max(c) for all c simultaneously by an EREW PRAM in logarithmic time with a logarithmic workload [30], and therefore sequentially in linear time. Theorem 2 Max can be determined by an EREW PRAM in logarithmic time with a linear workload with respect to the the size of the input (V; C) The overall result of this section is therefore the following. Theorem 3 The overlap components can be ....

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

....are the child clusters of c that contain u and v respectively. For each cluster c, we compute a spanning tree T c of E c . Then the minimum spanning tree of G is just the union of all T c . This can be parallelized, and we get a time bound of O(log n) and a processor bound of O(n) on a CRCW PRAM [16] Next we root TS to the vertex x where x; y are the two vertices of the root graph. Now consider any nonterminal edge g = uv that is finally replaced by H g . We distinguish between 10 Introverted Nonterminals: The parent of u or the parent of v is in H g and Social Nonterminals: The parents ....

....the edge from f to the parent of f is of the weight of f and the parent of f has a weight at least of that of f . This procedure can be done in linear time and in logarithmic time with a linear processor number, because we can determine connected components in the same bounds ( in parallel see [16]) 5.1.8 Extension of the Algorithm Connected Graphs in General We build up the following tree T that consists of the set C of 2 connected components and the set A of articulation vertices of G. An articulation vertex a is joint by an edge with a component c 2 C if a 2 c. We root T to a ....

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

....C. It is easily checked that for any connected component C of S j i V j , the set of neighbors of C outside C is a cut. Q.E.D. Note that the ordered partition (V 1 ; Vn ; fvn g) can be determined in O(n m) time and by a CRCW PRAM in logarithmic time with a linear processor number [18]. Now we consider only the subsequence of nonempty partion elements and denote it by V 1 ; V k , i.e. V k = fvn g. Nevertheless, V i is the set of neighbors of a vertex v i 2 V i 1 [ V k that are not in V i 1 [ V k . v 1 ; v k Gamma1 is also called the generating ....

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

....more a practical aspect. One does not get a lower time bound in the order. Another aspect that might be discussed is the parallelization. The components of the tree split procedure are O(n) computations of connected components and reorganization of the tree. First can be parallelized very easily [16]. The parallelization of the second component of the tree split procedure might be a topic for a masters or honors thesis. The improved RTL algorithm might be replaced by a variation of the algorithm of [8] ....

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

....p numbers in O(log p) time using p processors on EREW PRAM. Sum, products and conjunction of p elements can be implemented on EREW PRAM with parallel prefix computation in O(log p) time using p log p processors. The connected components of a graph with p vertices and q edges can be computed [SV82] on CRCW PRAM in O(log p) time using p q processors. The complexities of our algorithms follow easily from the complexities of these routines. In the following algorithm, we will need the degree d(u) of each vertex u, it can be computed with a prefix sum on its adjacency list. Algorithm 1. ....

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

....dimension of the feature space, the required operations for each feature space unit can be carried out independent of the other units. Thus, using parallel processing we can speed up transforming the feature space. The connected component analysis can also be speeded up using parallel processing [NS80, SV82]. Parallel processing algorithms will be specially useful when the number of units m or the number of dimensions d is high. For large number of dimensions we may have N K = m d . For such cases, we can also perform principle component analysis [Sch92] to find the most important features and ....

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

....[24] 3. In Step 3, we solve n 2D convex hull problems using the method by Amato and Preparata. Note that, the total size of the n 2D convex hull problems is O(n) and, thus, the total time is O(log n) using n processors. 4. For Step 4, we apply any optimal NC 1 connected component algorithm [27][39]. ....

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

....of both matrices. This example comes very close to the worst case described in section 9.1 and therefore R approximately matches the upper bound. The second example B1 is computing the connected components of an undirected graph with v nodes and e edges. For the PRAM we use an algorithm of [45] in a form presented in [20] Its runtime is O(log v) steps on a PRAM with 2e (virtual) processors. The formal explanation and the proofs for correctness and runtime can be found in [45] On a PRAM with n v physical processors we have t D1 = 300 e n 108 v n ) log v as analyzed in appendix ....

....the connected components of an undirected graph with v nodes and e edges. For the PRAM we use an algorithm of [45] in a form presented in [20] Its runtime is O(log v) steps on a PRAM with 2e (virtual) processors. The formal explanation and the proofs for correctness and runtime can be found in [45]. On a PRAM with n v physical processors we have t D1 = 300 e n 108 v n ) log v as analyzed in appendix A and c D1 as computed in equation 7.1. For the distributed memory machine we could use an algorithm from [4] that runs on a hypercube. Its runtime is O(log 2 v) on v 3 processors. ....

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

....problem. Hirschberg et. al [Hir76, HCS79, CLC82] showed an O(n 2 ) work algorithm for CREW PRAMs. This algorithm starts with the adjacency matrix of the graph and shrinks the graph based on local connectiviety information at each step, recomputing the new matrix each time. Shiloach et. al [SV82, AS87] use a similar idea for sparse graphs. This CRCW PRAM algorithm starts with a list of edges, forming trees of connected vertices and grafting smaller trees to form larger trees till all vertices of a component are in the same tree. The work complexity of this algorithm is O( m n) log n) m ....

....Suppose i and j lie in the same component but P(i) 6= P(j) Consider the path from i to j. There must exist two consecutive nodes i 0 and j 0 on this path such that the edge (i 0 ; j 0 ) 2 G, and P(i 0 ) 6= P(j 0 ) But then 2 Contrast this with the grafting operation of [SV82] 4 KUMAR, GODDARD, AND PRINS p C i j k Figure 1. Chain switching the algorithm could not have terminated since either P(i 0 ) P(j 0 ) and P(i 0 ) must change or vice versa. 2.2. Complexity: Let us define a chain as follows: ffl i and P(i) lie on a chain. Succ(i) P(i) and Pred(P(i) ....

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Y. Shiloach and U. Vishkin. An o(log n) parallel connectivity algorithm. Journal of Algorithms, 3(1):57--67, 1982. 14 KUMAR, GODDARD, AND PRINS

....write conflict resolution rules have been proposed according to which the CRCW and ERCW models can be divided into submodels. We will explicitly discuss only Arbitrary CRCW PRAM model, but the results shown in this paper are valid for other CRCW and ERCW models, too (see [15, 13] Arbitrary ([19]) If two or more processors write into a memory location in a given step, then one of the values is selected arbitrarily to become the new value. Nothing is known about the selection. From now on, N denotes the number of PRAM processors, P the number of real processors, and Q the number of nodes ....

Y. Shiloach and U. Vishkin. A log n Parallel Connectivity Algorithm. Journal of Algorithms, 3:57--63, 1982.

.... of several sequential MST algorithms, see [Tar83, Chapter 6] In parallel models, the previous results for the MST problem were O(log 2 n) using n 2 = log 2 n CREW PRAM [HCS79, CLC82] or n 2 EREW PRAM processors [NM82] and O(log n) time using n m PRIORITY CRCW PRAM processors [AS87, SV82], or (n m) log log log n= log n STRONG CRCW PRAM processors [CV86] using very elaborate techniques. Other parallel algorithms are reported in [KRS90, KR84, Ben80, SJ81] Recently, CL93] have improved the running time of [JM91] to O(log n log log n) mainly by providing a recursive version of the ....

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

....of both matrices. This example comes very close to the worst case described in section 6.1 and therefore R approximately matches the upper bound. The second example B1 is computing the connected components of an undirected graph with v nodes and e edges. For the PRAM we use an algorithm of [29] in a form presented in [14] Its runtime is O(log v) steps on a PRAM with 2e (virtual) processors. The formal explanation and the proofs for correctness and runtime can be found in [29] On a PRAM with n v physical processors we have t D1 = 300 e n 108 v n ) log v as analyzed in [3] and c ....

....the connected components of an undirected graph with v nodes and e edges. For the PRAM we use an algorithm of [29] in a form presented in [14] Its runtime is O(log v) steps on a PRAM with 2e (virtual) processors. The formal explanation and the proofs for correctness and runtime can be found in [29]. On a PRAM with n v physical processors we have t D1 = 300 e n 108 v n ) log v as analyzed in [3] and c D1 as computed in equation 1. For the distributed memory machine we could use an algorithm from [5] that runs on a hypercube. Its runtime is O(log 2 v) on v 3 processors. For a ....

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

....although the depth has increased. 4.3.3 Improved Versions of Connected Components There are many improvements to the two basic connected component algorithms we described. Here we mention some of them. The deterministic algorithm can be improved to run in O(log n) depth with the same work bounds [10, 68]. The basic idea is to interleave the hooking steps with the shortcutting steps. The one tricky aspect is that we must always hook in the same direction (i.e. from smaller to larger) so as not to create cycles. Our previous technique to solve the star graph problem therefore does not work. ....

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

....As a matter of fact, the great teacher was very impressed by my program. However, he also had reason to feel embarrassed when I demonstrated that one of his carefully elaborated algorithms (computing connected components of an undirected graph, a variant of the algorithm by Shiloach and Vishkin [27]) was less correct than proved (it never terminated) but that added only to my FUN. So I would like to conclude from this second example my two claims from the end of the previous section, and I would like to add my third claim : 3. Implementing algorithms helps to avoid mistakes. 4 FUN with ....

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

....in parallel if u is in a rooted star and p(u) 6= p(v) then 2.1 p(p(u) p(v) 3. Pointer jumping. perform a pointer jumping operation on all vertices until there is no change in the current set of tree loops; Algorithm 2. An algorithm for finding connected components by Shiloach and Vishkin [50] as described in Chapter 5.1.3 of J aJ a [25] parent of its parent) For each vertex whose grandparent is different from its parent, we mark (concurrent write) a flag f for its grandparent. Any vertex that is marked is not in a rooted star. Every unmarked vertex reads the flag f from its ....

....Any vertex that is marked is not in a rooted star. Every unmarked vertex reads the flag f from its grandparent. Unmarked vertices whose grandparents are marked are also not in rooted stars. 2.2. Shiloach and Vishkin. The next algorithm we implemented (Algorithm 2) is by Shiloach and Vishkin [50] and also appears in Chapter 5.1.3 of J aJ a [25] In each of the hooking and pointer jumping iteration of this algorithm, two hooks are performed as in Algorithm 1. The first hook, which is called conditional hooking, is similar to conditional star hooking as described in Algorithm 1, except that ....

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

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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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Y. Shiloach and U. Vishkin, An O(log n) parallel connectivity algorithm, J. Algorithms 3 (1982), pp. 56-67.

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