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22
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
SublinearTime Parallel Algorithms for Matching and Related Problems
, 1988
"... This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial struc ..."
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Cited by 33 (6 self)
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This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial structure of the above problems, which leads to new algorithmic techniques. In particular, we show how to use maximal matching to extend, in parallel, a current set of nodedisjoint paths and how to take advantage of the parallelism that arises when a large number of nodes are "active" during an execution of a pushrelabel network flow algorithm. We also show how to apply our techniques to design parallel algorithms for the weighted versions of the above problems. In particular, we present sublineartime deterministic parallel algorithms for finding a minimumweight bipartite matching and for finding a minimumcost flow in a network with zeroone capacities, if the weights are polynomially ...
Parallel Algorithmic Techniques for Combinatorial Computation
 Ann. Rev. Comput. Sci
, 1988
"... this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165. ..."
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Cited by 29 (3 self)
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this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165.
Efficient parallel algorithms for chordal graphs
"... We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing ..."
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Cited by 26 (0 self)
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We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to our results is an efficient parallel algorithm for finding a perfect elimination ordering.
On Identifying Strongly Connected Components in Parallel
, 2000
"... . The standard serial algorithm for strongly connected components is based on depth first search, which is difficult to parallelize. We describe a divideandconquer algorithm for this problem which has significantly greater potential for parallelization. For a graph with n vertices in which deg ..."
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Cited by 23 (4 self)
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. The standard serial algorithm for strongly connected components is based on depth first search, which is difficult to parallelize. We describe a divideandconquer algorithm for this problem which has significantly greater potential for parallelization. For a graph with n vertices in which degrees are bounded by a constant, we show the expected serial running time of our algorithm to be O(n log n). 1 Introduction A strongly connected component of a directed graph is a maximal subset of vertices containing a directed path from each vertex to all others in the subset. The vertices of any directed graph can be partitioned into a set of disjoint strongly connected components. This decomposition is a fundamental tool in graph theory with applications in compiler analysis, data mining, scientific computing and other areas. The definitive serial algorithm for identifying strongly connected components is due to Tarjan [15] and is built on a depth first search of the graph. For a grap...
Designing Checkers for Programs that Run in Parallel
 Algorithmica
, 1994
"... Program correctness for parallel programs is an even more problematic issue than for serial programs. We extend the theory of program result checking to parallel programs, and find general techniques for designing such result checkers that work for many basic problems in parallel computation. These ..."
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Cited by 13 (2 self)
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Program correctness for parallel programs is an even more problematic issue than for serial programs. We extend the theory of program result checking to parallel programs, and find general techniques for designing such result checkers that work for many basic problems in parallel computation. These result checkers are simple to program and are more efficient than the actual computation of the result. For example, sorting, multiplication, parity, the all pairs shortest path problem and majority all have constant depth result checkers, and the result checkers for all but the last problem use a linear number of processors. We show that there are Pcomplete problems (evaluating straightline programs, linear programming) that have very fast, even constant depth, result checkers. 1 Introduction Verifying a program to see if it is correct is a problem that every programmer has encountered. Even the seemingly simplest of programs can be full of hidden bugs, and in the age of massive software...
Towards overcoming the transitiveclosure bottleneck: efficient parallel algorithms for planar digraphs
 J. Comput. System Sci
, 1993
"... Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms ..."
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Cited by 11 (1 self)
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Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms for digraph reachability. To pinpoint the nature of the bottleneck, we de* velop a collection of polylogtime reductions among reachability problems. These reductions use only linear processors and work for general graphs. Furthermore, for planar digraphs, we give polylogtime algorithms for the following problems: (1) directed ear decomposition, (2) topological ordering, (3) digraph reachability, (4) descendent counting, and (5) depthfirst search. These algorithms use only linear processors and therefore reduce the complexity to within a polylog factor of optimal.
An Efficient Parallel Algorithm for the Minimal Elimination Ordering (MEO) of an Arbitrary Graph
, 1989
"... . We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O(nm) processors on a CREW PRAM, for an nvertex, medge graph, and is optimal up to a polylogarithmic factor with resp ..."
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Cited by 11 (5 self)
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. We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O(nm) processors on a CREW PRAM, for an nvertex, medge graph, and is optimal up to a polylogarithmic factor with respect to the best sequential algorithm of Rose, Tarjan and Lueker ([RTL 76]). The MEO problem for arbitrary graphs arises in a number of combinatorial optimization problems, as well as in database applications, scheduling problems, and the An Extended Abstract has appeared in [DK 89]. y present address: Basser Department of Computer Science, University of Sydney, NSW 2006, Australia z Research partially supported by the Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/41, and by the SERC Grant GRE 68297. sparse Gaussian elimination on symmetric matrices. It was believed before to be inherently sequential, and strongly resisting sublinear parallel time (subli...
An Efficient Parallel Algorithm That Finds Independent Sets Of Guaranteed Size
, 1990
"... . Every graph with n vertices and m edges has an independent set containing at least n 2 =(2m +n) vertices. We present a parallel algorithm that nds an independent set of this size and runs in O(log 3 n) time on a CRCW PRAM with O((m + n)(m; n)= log 2 n) processors, where (n; m) is a functiona ..."
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Cited by 10 (0 self)
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. Every graph with n vertices and m edges has an independent set containing at least n 2 =(2m +n) vertices. We present a parallel algorithm that nds an independent set of this size and runs in O(log 3 n) time on a CRCW PRAM with O((m + n)(m; n)= log 2 n) processors, where (n; m) is a functional inverse of Ackerman's function. The ideas used in the design of this algorithm are also used to design an algorithm that, with the same resources, nds a vertex coloring satisfying certain minimality conditions. Key words. Turan's theorem, independent set, NC, graph, parallel computation, deterministic AMS(MOS) subject classications. 68Q22, 68R10, 68R05 1. Introduction. This paper presents a fast parallel algorithm that, given a graph G, nds an independent set of G whose size is bounded from below. The bound depends on the number n of vertices and number m of edges of G, and cannot be improved in these terms. Since constructing a maximum independent set is NPhard, it cannot be so...