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Efficient Parallel Algorithms for Tree Accumulations
 Science of Computer Programming
, 1994
"... Accumulations are higherorder operations on structured objects; they leave the shape of an object unchanged, but replace elements of that object with accumulated information about other elements. Upwards and downwards accumulations on trees are two such operations; they form the basis of many tree ..."
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Accumulations are higherorder operations on structured objects; they leave the shape of an object unchanged, but replace elements of that object with accumulated information about other elements. Upwards and downwards accumulations on trees are two such operations; they form the basis of many tree algorithms. We present two Erew Pram algorithms for computing accumulations on trees taking O(log n) time on O(n= log n) processors, which is optimal.
RealTime Minimum Vertex Cover For TwoTerminal SeriesParallel Graphs
 Proceedings of the Thirteenth Conference on Parallel and Distributed Computing and Systems
, 2000
"... Tree contraction is a powerful technique for solving a large number of graph problems on families of recursively definable graphs. The method is based on processing the parse tree associated with a member of such a family of graphs in a bottomup fashion, such that the solution to the problem is ..."
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Tree contraction is a powerful technique for solving a large number of graph problems on families of recursively definable graphs. The method is based on processing the parse tree associated with a member of such a family of graphs in a bottomup fashion, such that the solution to the problem is obtained at the root of the tree. Sequentially, this can be done in linear time with respect to the size of the input graph. In parallel, efficient and even cost optimal tree contraction algorithms have also been developed. In this paper we show how the method can be applied to compute the cardinality of the minimum vertex cover of a twoterminal seriesparallel graph. We then construct a realtime paradigm for this problem and show that in the new computational environment, a parallel algorithm is superior to the best possible sequential algorithm, in terms of the accuracy of the solution computed. Specifically, there are cases in which the solution produced by a parallel algorithm ...
An Optimal Parallel Matching Algorithm for Cographs
 Journal of Parallel and Distributed Computing
, 1994
"... The class of cographs, or complementreducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$ver ..."
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The class of cographs, or complementreducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$vertex cograph $G$ represented by its parse tree as input, our algorithm finds a maximum matching in $G$ in O($logn$) time using O($n0$) processors in the EREWPRAM model. Key Words: list ranking, tree contraction, matching, parenthesis matching, scheduling, operating systems, cographs, parallel algorithms, EREWPRAM. 1. Introduction A wellknown class of graphs arising in a wide spectrum of practical applications [1,2,7] is the class of cographs, or complementreducible graphs. The cographs are defined recursively as follows: . a singlevertex graph is a cograph; . if $G$ is a cograph, then its complement $G bar$ is also a cograph; . if $G$ and $H$ are cographs, then their union is also a cog...
Efficient Implementation of Tree Skeletons on DistributedMemory Parallel Computers
, 2006
"... The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electron ..."
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The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. These works may not be reposted without the explicit permission of the copyright holder.
Experimentation, Theory
"... Tree contraction algorithms, whose idea was first proposed by Miller and Reif, are important parallel algorithms to implement efficient parallel programs manipulating trees. Despite their efficiency, the tree contraction algorithms have not been widely used due to the difficulties in deriving the tr ..."
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Tree contraction algorithms, whose idea was first proposed by Miller and Reif, are important parallel algorithms to implement efficient parallel programs manipulating trees. Despite their efficiency, the tree contraction algorithms have not been widely used due to the difficulties in deriving the tree contracting operations. In particular, the derivation of the tree contracting operations is much difficult when multiple values are referred and updated in each step of the contractions. Such computations often appear in dynamic programming problems on trees. In this paper, we propose an algebraic approach to deriving tree contraction programs from recursive tree programs, by focusing on the properties of commutative semirings. We formalize a new condition for implementing tree reductions with the tree contraction algorithms, and give a systematic derivation of the tree contracting operations. Based on it, we implemented a code generator for tree reductions, which has an optimization mechanism that can remove unnecessary computations in the derived parallel programs. As far as we are aware, this is the first step towards an automatic parallelization system for the development of efficient tree programs.
CHARACTERIZATION OF EFFICIENTLY PARALLEL SOLVABLE PROBLEMS ON DISTANCEHEREDITARY GRAPHS ∗
"... Abstract. In this paper, we sketch common properties ofa class ofsocalled subgraph optimization problems that can be systematically solved on distancehereditary graphs. Based on the found properties, we then develop a general problemsolving paradigm that solves these problems efficiently in paral ..."
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Abstract. In this paper, we sketch common properties ofa class ofsocalled subgraph optimization problems that can be systematically solved on distancehereditary graphs. Based on the found properties, we then develop a general problemsolving paradigm that solves these problems efficiently in parallel. As a byproduct, we also obtain new lineartime algorithms by a sequential simulation ofour parallel algorithms. Let Td(V , E) and Pd(V , E) denote the time and processor complexities, respectively, required to construct a decomposition tree ofa distancehereditary graph G =(V,E) on a PRAM model Md. Based on the proposed paradigm, we show that the maximum independent set problem, the maximum clique problem, the vertex connectivity problem, the domination problem, and the independent domination problem can be sequentially solved in O(V  + E) time, and solved in parallel in O(Td(V , E) + log V ) time using O(Pd(V , E)+V  / log V ) processors on Md. By constructing a decomposition tree under a CREW PRAM, we also show that
Optimal Tree Ranking is in NC
"... This paper places the optimal tree ranking problem in NC. A ranking is a labeling of the nodes with natural numbers such that if nodes u and v have the same label then there exists another node with a greater label on the path between them. An optimal ranking is a ranking in which the largest label ..."
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This paper places the optimal tree ranking problem in NC. A ranking is a labeling of the nodes with natural numbers such that if nodes u and v have the same label then there exists another node with a greater label on the path between them. An optimal ranking is a ranking in which the largest label assigned to any node is as small as possible among all rankings. An O(n) sequential algorithm is known. Researchers have speculated that this problem is P complete. We show that for an nnode tree, one can compute an optimal ranking in O(log n) time using n 2 = log n CREW PRAM processors. In fact, our ranking is super critical in that the label assigned to each node is absolutely as small as possible. We achieve these results by showing that a more general problem, which we call the super critical numbering problem, is in NC. No NC algorithm for the super critical tree ranking problem, approximate or otherwise, was previously known; the only known NC algorithm for optimal tree ranking ...
rDomination problems on trees and their homogeneous extensions
, 1995
"... . A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of so ..."
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. A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of some results on homogeneous sets we present a linear time algorithm computing the vertex sets of the connected components of the complement of an arbitrary graph. Moreover we present efficient parallel algorithms solving the problems rdominating set, r dominating clique and connected rdominating set (and thus the Steiner tree problem) on trees by reducing these problems to algebraic tree computations. Using these algorithms we can compute minimum rdominating cliques and minimum connected rdominating sets in homogeneous extensions of trees in linear sequential and logarithmic parallel time using a linear number of processors. Finally, we give a more involved sequential algorithm solv...