Y. Shiloach and U. Vishkin. An o(n log n) parallel max-flow algorithm. J. Algorithms, 3:128--146, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....important when B is an order of magnitude smaller than Bj. Clearly we assume that arrays cannot be enlarged unless their contents is j=l copied into a larger part of memory. As for parallel algorithms, array allocation dominates in the literature, even for data structures like stacks (cf. [5], 3] Using array allocation in the example mentioned above, stack Sj can be maintained in 0 (K Bj) space while each simultaneous update of J stacks by at most K processors takes O (1ogK) time (cf. 5] The cooperation of processors leads to a term K in the space bound and a term log K in the ....

.... array allocation dominates in the literature, even for data structures like stacks (cf. 5] 3] Using array allocation in the example mentioned above, stack Sj can be maintained in 0 (K Bj) space while each simultaneous update of J stacks by at most K processors takes O (1ogK) time (cf. [5]) The cooperation of processors leads to a term K in the space bound and a term log K in the time bound (compared with the case of sequential algorithms) Thus the total memory bound is O (JK Bj) In this paper we will prove that a lower space bound of 0 (K ] j [ for each stack j is ....

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Shiloach, Y. and U. Vishkir, An O(n21ogn) parallel maxflow algorithm, Jml of Algo- rithms 3 (1982), pp. 128-146.

....algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [11] minimum weight spanning trees [32] shortest paths [32] global min cuts in weighted, undirected graphs [28] blocking flows and max flows [22, 43, 47], approximate computation of roots of polynomials [6, 37] sorting algorithms [32] and several problems in computational geometry [41] In contrast to boolean circuits where no lower bounds are known for unbounded depth circuits, our result gives a lower bound for a natural problem in a natural ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel MAX-FLOW algorithm. Journal of Algorithms, 3(2):128--146, June 1982.

....algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [6] minimum weight spanning trees [14] shortest paths [14] global min cuts in weighted, undirected graphs [13] blocking flows and max flows [9, 21], approximate computation of roots of polynomials [2, 18] sorting algorithms [14] and several problems in computational geometry [20] In constrast to Boolean circuits where no lower bounds are known for unbounded depth circuits, our result gives a lower bound for a natural problem in a natural ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel MAX-FLOW algorithm. Journal of Algorithms, 3(2):128--146, June 1982.

....algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [3] minimum weight spanning trees [14] shortest paths [14] global min cuts in weighted, undirected graphs [13] blocking flows and max flows [9, 21, 24], approximate computation of roots of polynomials [1, 17] sorting algorithms [14] and several problems in computational geometry [20] In constrast to Boolean circuits, where no lower bounds are known for unbounded depth circuits, our result gives a lower bound for a natural problem in a natural ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel MAX-FLOW algorithm. Journal of Algorithms, 3(2):128--146, June 1982.

....Goldberg and Tarjan s algorithm makes decisions locally. A node performs a change on the preflow, based on the knowledge it has about itself and its neighbors. A global vision of the network is not necessary. Therefore, the preflow push method naturally lends itself to parallel implementations [22, 26, 30]. By contrast, the parallel algorithm developed in this paper is based on the Ford Fulkerson method. This is due to the capabilities of the parallel model used, which can depict the structure of a graph in such a way that its global properties are easily extracted. 3 Models of Computation The two ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel MAX-FLOW algorithm. Journal of Algorithms, 3(2):128--146, 1982.

....this is successful and the mincut is situated at s or t then we have found a maxflow. 5. If the balancing of f has failed then find the maxflow of c with a polynomial worst case time algorithm. 2 The first step seems to be original. The concept of a preflow is used in a number of algorithms (cf. [5, 8, 14]) but only in the restricted sense that in each vertex the outgoing flow is at most equal to the incoming flow, i.e. a vertex may have a flow excess but never a flow deficit. In our algorithm we use preflows in which vertices may have flow deficits. Another important point is that step 4 ....

Shiloach, Y., U. Vishkin, An O(n 2 log n) parallel max-flow algorithm, J. of Alg., 3 (1982) 128-146. 16

....nodes in S. Step 2. For every matched arc (v; w) push excess flow from v to w. Step 3. Relabel nodes in T . Step 4. Push flow from active nodes in T along admissible arcs. Step 5. Relabel nodes in T . Step 6. Relabel nodes in S. Figure 5: The Match and Push procedure with n m processors [15, 30]. The bottleneck is Step 1, which takes O(log 3 n) time on a CRCW PRAM using n m processors [20] The next lemma, which bounds the number of executions of Match and Push, is the key to the analysis of the algorithm. Lemma 4.3 Procedure Match and Push with activity parameter l and distance ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) Parallel Max-Flow Algorithm. J. Algorithms, 3:128--146, 1982.

....to form what is called a enhanced protocol whose performance is bounded by O(h) 5 Conclusion In this paper a small subset of the actually existing algorithms has been presented. A parallel algorithm for finding the maximum flow in a weighted graph, has been proposed by Shiloach and Vishkin [8]. Detailed discussion on parallel graph algorithms can be found in [1] This literature also discusses about the sparse graph algorithms which have orthogonal issues to be considered, when compared with dense graph algorithms in the parallel domain. Readers can find distributed algorithms for ....

Y.Shiloach and U.Vishkin. An O(n 2 log n) Parallel Max-flow algorithm, Journal of Algorithms, 3:128-146, 1982. 32

....steps using k processors. The implementation given in this report assumes CREW capabilities of the target machine, since this simplifies somewhat in particular parallel searching. The parallel dictionary is used in [13] for reducing the space requirement of the parallel maximum flow algorithm of [12]. For this application a trivial extension of the data structure is needed for computing for each item in the dictionary (in a logarithmic number of operations) the sum of all items with smaller key currently in the dictionary. This extension has not been implemented. It is, however, indicated how ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel MAX-FLOW algorithm. Journal of Algorithms, 3:128--146, 1982.

.... Typically, researchers on efficient parallel algorithms have been interested in designing optimal parallel algorithms (even if their running time could not be upper bounded by O(log k n) where n is the input length and k is a constant, as demonstrated in a 1977 thesis by Eckstein, RC] [SV], VS] and [KRS] A notion of performance which is similar to the one of optimal algorithms, but not identical, is needed in the context of this paper. A serial emulation of a parallel algorithm is compared with the fastest serial algorithm for the same problem, where the serial emulation may ....

....processor allocation, which is an essential part of a PRAM algorithm, will typically not be such a serious issue for extracting the parallelism needed here. A starting point for a weaker model than the PRAM for describing parallelism might be Informal Data Parallelism, originally presented in [SV] using Brent s theorem) as per [Vi94] this has also been called the work time presentation paradigm for parallel algorithms, as elucidated by the author in [Ja] This paradigm suggests preceding a full PRAM description of a parallel algorithm by a high level description of work and time only. ....

Y. Shiloach and U. Vishkin. An O(nlog 2 n) parallel Max-Flow Algorithm. J. Algorithms, 3:128--146, 1982.

....in polynomial time) under logspace reductions [16] As a consequence, the existence of an NC algorithm for the maximum flow problem would imply that all problems in P have NC algorithms, a state of affairs that is considered highly unlikely. Thus parallel algorithms for the maximum flow problem [15, 22] fall far short of a polylogarithmic running time. However, NC algorithms are known for special cases. If the edge capacities are given in unary, then a reduction to perfect matching yields a randomized NC algorithm [20] An NC algorithm is known for the case of planar networks [19] and an ....

Y. Shiloach and U. Vishkin, An O(n 2 log n) parallel MAX-FLOW algorithm, J. Algorithms 3 (1982) 128--146.

....constructs. Both are also sequential, i.e. their evaluation could be serialized while preserving meaning. For each, I will use the cost measures of work and steps, informally known as the work time paradigm, which has been used to describe parallel algorithm complexity at an abstract level [11, 12, 29]. The work is the number of serial reductions. The step cost describes the number of (abstract) time units needed for evaluation, assuming an infinite number of processors and unit communication costs. Brent [6] showed how to relate the work and step complexities of an algorithm to a machine with ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. Journal of Algorithms, 3:128--146, 1982.

.... depth (D) complexities have been calculated in this way, the formula T = O(W=P D lg P ) 3) places an upper bound on the asymptotic running time of an algorithm on the CRCW PRAM model (P is the number of processors) This formula can be derived from Brent s scheduling principle [14] as shown in [41, 7, 30]. The lg P term shows up because of the cost of allocating tasks to processors, and the cost of implementing the sum and scan operations. On the scan PRAM [6] where it is assumed that the scan operations are no more expensive than references to the shared memory (they both require O(lg P ) on a ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

....S) 1) and the bound on a PRAM without scan primitives is t = O(W=P S lg n) 2) where P is the number of processors. This formula can be derived from Brent s scheduling principle [9] as shown in [4] and is similar to the use of Brent s principle as used informally to aid algorithm analysis in [24, 16]. Nesl is a strongly typed side effect free language. It runs within an interactive environment, has a Lisp like syntax 2 , and allows the user to run the environment, including the compiler, on a local workstation while executing interactive calls to Nesl programs on the CRAY Y MP or CM 2 (or ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

....the running time if the algorithm is executed on a sequential processor. The depth represents the best possible running time, assuming an ideal machine with an unlimited number of processors. Work and depth have been used informally for many years to describe the performance of parallel algorithms [23], especially when teaching them [17, 16] The claim is that it is easier to describe, think about, and analyze algorithms in terms of work and depth rather than in terms of running time on a processor based model (a model based on P processors) Furthermore, work and depth together tell us a lot ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

.... step (S) complexities have been calculated in this way, the formula T = O(W=P S lg P ) 3) places an upper bound on the asymptotic running time of an algorithm on the CRCW PRAM model (P is the number of processors) This formula can be derived from Brent s scheduling principle [10] as shown in [34, 5, 23]. The lg P term shows up because of the cost of allocating tasks to processors, and the cost of implementing the sum and scan operations. On the scan PRAM [4] where it is assumed that the scan operations are no more expensive than references to the shared memory (they both require O(lg P ) on a ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

....for Advanced Computer Studies, and was partially supported by NSF grants CCR 9111348 and CCR 8906949. E mail address: matias research.att.com. 1 Introduction Many parallel algorithms are designed using the well known work time methodology. See [13, Chapter 1. 3] for a detailed discussion and [19] for an early use in a pram algorithm. Guided by Brent s theorem [3] the methodology suggests to first describe a meta algorithm in terms of a sequence of rounds; each round may include any number of independent constant time operations. Second, the meta algorithm is implemented on a p processor ....

Y. Shiloach and U. Vishkin. An O(n 2 lg n) parallel Max-Flow algorithm. Journal of Algorithms, 3:128--146, 1982.

....In fact, the lack of random access seems to be a bigger problem than the lack of parallelism. This research was sponsored in part by the Wright Laboratory, Aeronautical Systems Center, Air Force Materiel Command, USAF, and the Advanced Research Projects Agency (ARPA) under grant number F33615 93 1 1330 and contract number F19628 91 C 0168. It was also supported in part by an NSF Young Investigator Award and by Finmeccanica. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies or ....

....is sparse. Jones [18] related the time augmented semantics of simple while loop language to that of an equivalent machine language in order to study the effect of constant factors in time complexity. The work step paradigm has been used for many years for informally describing parallel algorithms [36, 19] It was first included in a formal model by Blelloch in the VRAM [5] The NESL language [6] a data parallel functional language, includes complexity measures based on work and steps and has been used for describing and teaching parallel algorithms. Skillicorn [37] also introduced cost ....

Yossi Shiloach and Uzi Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

....of some of these algorithms. Though many parallel versions of maximum flow algorithms exist, there is no implementation that runs in polylogarithmic time on a polynomial number of processors. The most efficient parallel maximum flow algorithm runs in O(n 2 log n) time using O(n) processors [14, 18]. Thus the maximum flow problem is not known to be in the class NC. The Goldberg Tarjan algorithm [14] is based on a method called the preflow push method. Preflowpush algorithms work in a localized manner as compared to the Ford Fulkerson or the Edmonds Karp algorithms. Thus, distributed ....

Y. Shiloach and U. Vishkin, An O(n 2 log n) parallel max-flow algorithm, Journal of Algorithms, 3 (1982), pp. 128--146.

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Y. Shiloach and U. Vishkin. An o(n log n) parallel max-flow algorithm. J. Algorithms, 3:128--146, 1982.

....model, such a description mode permits having a different (flexible, or elastic ) number of operations in different parallel rounds. This relates to the well known work time methodology for describing parallel algorithms (and the VRAM model in [Bl90] Since its introduction and original use in [SV82] and [Vi81] the work time methodology has become a standard framework for describing parallel algorithms (see, e.g. Ja92] Section 11 notes that much of the empirical work on studying the limits of ILP extractable from code, by Lam Wilson and others, appear to fit a more abstract version of ....

....code provides is already in an imperative form. This last contribution resembles the contribution of the PRAM model in providing imperative algorithms; see for instance the contributions of [SV81] and [BH85] in turning Valiant s [Va75] decision tree algorithms imperative, and the contribution of [SV82] in proposing to use Brent s non imperative theorem in the work time methodology for presenting (imperative) PRAM algorithms where all overheads are taken into account . The paper [Vi97] presented the following theoretical insight. Using prefix sum as the only addition to a standard ....

Y. Shiloach and U. Vishkin. An O(n 2 logn) parallel max-flow algorithm. J. of Algorithms, 3 (1982), 128--146.

....available machines such as the LogP model of [CKP 93] We give an additional justification for considering the range m p for the size of the read write shared memory m. One of the most useful methodologies for describing PRAM algorithms is the data parallel style, as suggested in [SV 82] A similar style was dubbed data parallel in [HS86] and has been used extensively in [J 92] as one example. A parallel algorithm is described in terms of a sequence of steps, and for each step the set of operations to be performed (concurrently) in that step is given. Using an informal ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel max-flow algorithm. J. Algorithms, 3(2):128-146, 1982.

.... full PRAM design is guided by a theorem due to [Bre74] the problem, however, is that the theorem holds only for a non standard model of parallel computation, where assignment of processors to jobs can be done free of charge; the methodology was first used for the design of a PRAM algorithm in [SV82b] and is elucidated in [Vis90] and [J aJ91] who call it the work time framework; typical applications of this methodology solve the processor assignment problem in an ad hoc manner; however, sometimes proper processor assignment can be achieved using general methods for balancing loads among ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128-- 146, 1982.

....of phase i into an array of length N 0 i prior to the phase. Consider the first phase i 0 for which N 0 i 0 n=log n. The compacted array of size N 0 i 0 , will be used for all subsequent phases. Our presentation of the parallel algorithm follows a methodology that was introduced in [SV82b] and became standard since then, see [KR88] First, the algorithm is described in terms of work and time only, as was done above. Second, a theorem by Brent [Bre74] is used to guide actual assignment of processors to jobs. The theorem formally holds only in some abstract model of parallel ....

Y. Shiloach and U. Vishkin. An O(n 2 log n) parallel Max-Flow algorithm. J. Algorithms, 3:128--146, 1982.

....general, 1 We note that using the total number of operations and the number of parallel cycles as the key measure for evaluating parallel algorithms is rather common. It is used by the so called work time methodology for describing parallel algorithms. This methodology was first presented in [23] (relying on a 1974 theorem by Brent for an abstract model of parallel computation) Later the book [13] used the methodology as a framework for describing most of the parallel algorithms presented there. For brevity, we avoid elaborating on this issue here. where one may need to compare two ....

Y. Shiloach and U. Vishkin. An O(n 2 logn) parallel max-flow algorithm. Journal of Algorithms, 3(2):128--146, 1982.

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