Parallel recognition of the consecutive ones property with applications
TL;DR: This paper gives the first NC algorithm for recognizing the consecutive 1's property for rows of a (0, 1)-matrix, and shows that the maximum matching problem for arbitrary convex bipartite graphs can be solved within the same complexity bounds.
About: This article is published in Journal of Algorithms.The article was published on 1991-09-01. It has received 33 citations till now. The article focuses on the topics: Matching (graph theory) & Complete bipartite graph.
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TL;DR: A logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs, which yields a canonicallabel of convex graphs and isomorphism and automorphism problems for these graph classes are solvable in logspace.
Abstract: We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.
38 citations
TL;DR: The minimum fill-in problem for bipartite permutation graphs can be solved efficiently by a randomized parallel algorithm and the first efficient parallel algorithms for several problems on bipartitespermutation graphs are given.
Abstract: In this paper, we further study the properties of bipartite permutation graphs. We give first efficient parallel algorithms for several problems on bipartite permutation graphs. These problems include transforming a bipartite graph into a strongly ordered one if it is also a permutation graph; testing isomorphism; finding a Hamiltonian path/cycle; solving a variant of the crossing number problem; and others. All these problems can be solved in O(log2n) time with O(n3) processors on a Common CRCW PRAM. We also show that the minimum fill-in problem for bipartite permutation graphs can be solved efficiently by a randomized parallel algorithm. © 1993 by John Wiley & Sons, Inc.
34 citations
TL;DR: The result presented here substantially broadens the class of graphs for which there are known efficient parallel isomorphism testing algorithms, and includes, among others, proper interval graphs and doubly convex bipartite graphs.
Abstract: In this paper, we explore some properties of identification matrices and exhibit some uses of identification matrices in studying the graph isomorphism problem, a famous open problem. We show that, given two graphs in the form of a certain identification matrix, isomorphism can be tested efficiently in parallel if at least one matrix satisfies the circular 1s property, and more efficiently in parallel if at least one matrix satisfies the consecutive 1s property. Graphs which have identification matrices satisfying the consecutive 1s property include, among others, proper interval graphs and doubly convex bipartite graphs. The result presented here substantially broadens the class of graphs for which there are known efficient parallel isomorphism testing algorithms.
33 citations
TL;DR: An accurate proof of the characterization of proper circular arc graphs is presented and the first efficient parallel algorithm which not only recognizes proper circular arcs graphs but also constructs proper circularArc representations is obtained.
Abstract: Based on Tucker's work, we present an accurate proof of the characterization of proper circular arc graphs and obtain the first efficient parallel algorithm which not only recognizes proper circular arc graphs but also constructs proper circular arc representations. The algorithm runs inO(log2
n) time withO(n
3) processors on a Common CRCW PRAM. The sequential algorithm can be implemented to run inO(n
2) time and is optimal if the input graph is given as an adjacency matrix, so to speak.
19 citations
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TL;DR: In this paper, a parallel algorithm is proposed to recognize a doubly convex-bipartite graph and it is proposed that the algorithm runs in O(log n) time using O( n 3 log n ) processors on the CRCW PRAM, or O( log2 n)Time using O 3 log 2 n processor on the CREWPRAM.
16 citations
References
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Book•
01 Jan 1980
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
Abstract: Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition. New edition of the "Classic" book on the topic Wonderful introduction to a rich research area Leading author in the field of algorithmic graph theory Beautifully written for the new mathematician or computer scientist Comprehensive treatment
4,090 citations
TL;DR: The consecutive ones test for the consecutive ones property in matrices and for graph planarity is extended to a test for interval graphs using a recently discovered fast recognition algorithm for chordal graphs.
1,622 citations
TL;DR: In this article, the problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, 1)-matrices: when can the rows of such a matrix be permuted so as to make the 1's in each colum appear consecutively.
Abstract: : According to present genetic theory, the fine structure of genes consists of linearly ordered elements. A mutant gene is obtained by alteration of some connected portion of this structure. By examining data obtained from suitable experiments, it can be determined whether or not the blemished portions of two mutant genes intersect or not, and thus intersection data for a large number of mutants can be represented as an undirected graph. If this graph is an interval graph, then the observed data is consistent with a linear model of the gene. The problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, 1)-matrices: When can the rows of such a matrix be permuted so as to make the 1's in each colum appear consecutively. A complete theory is obtained for this latter problem, culminating in a decomposition theorem which leads to a rapid algorithm for deciding the question, and for constructing the desired permutation when one exists.
1,329 citations
TL;DR: A recurstve construction is used to obtain a product circuit for solving the prefix problem and a Boolean clrcmt which has depth 2[Iog2n] + 2 and size bounded by 14n is obtained for n-bit binary addmon.
Abstract: The prefix problem is to compute all the products x t o x2 . . . . o xk for i ~ k .~ n, where o is an associative operation A recurstve construction IS used to obtain a product circuit for solving the prefix problem which has depth exactly [log:n] and size bounded by 4n An application yields fast, small Boolean ctrcmts to simulate fimte-state transducers. By simulating a sequentml adder, a Boolean clrcmt which has depth 2[Iog2n] + 2 and size bounded by 14n Is obtained for n-bit binary addmon The size can be decreased significantly by permitting the depth to increase by an addmve constant
1,159 citations
Book•
25 Aug 2011
TL;DR: A new algorithm for finding the blocks (biconnected components) of an undirected graph and a general algorithmic technique that simplifies and improves computation of various functions on trees is introduced.
Abstract: In this paper we propose a new algorithm for finding the blocks (biconnected components) of an undirected graph. A serial implementation runs in $O(n + m)$ time and space on a graph of n vertices and m edges. A parallel implementation runs in $O(\log n)$ time and $O(n + m)$ space using $O(n + m)$ processors on a concurrent-read, concurrent-write parallel RAM. An alternative implementation runs in $O(n^2 /p)$ time and $O(n^2 )$ space using any number $p \leqq n^2 /\log ^2 n$ of processors, on a concurrent-read, exclusive-write parallel RAM. The last algorithm has optimal speedup, assuming an adjacency matrix representation of the input. A general algorithmic technique that simplifies and improves computation of various functions on trees is introduced. This technique typically requires $O(\log n)$ time using processors and $O(n)$ space on an exclusive-read exclusive-write parallel RAM.
501 citations