Efficient parallel recognition algorithms of cographs and distance hereditary graphs

On an extension of distance hereditary graphs

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

In the literature, there are quite a few sequential and parallel algorithms for solving problems on distance-hereditary graphs. With an n-vertex and m-edge distance-hereditary graph G, we show that the efficient domination problem on G can be solved in O(log/sup 2/ n) time using O(n + m) processors on a CREW PRAM. Moreover, if a binary tree representation of G is given, the problem can be optimally solved in O(log n) time using O(n/log n) processors on an EREW PRAM.

On polygon numbers of circle graphs and distance hereditary graphs

The Hamiltonian problem on distance-hereditary graphs

We present a parallel recognition algorithm for bipartite-permutation graphs. The algorithm can be executed in O(log n) time on the CRCW PRAM if O(n/sup 3//log n) processors are used, or O(log/sup 2/ n) time on the CREW PRAM if O(n/sup 3//log/sup 2/n) processors are used. Chen and Yesha (1993) have presented another CRCW PRAM algorithm that takes O(log/sup 2/n) time if O(n/sup 3/) processors are used. Compared with Chen and Yesha's algorithm, our algorithm requires either less time and fewer processors on the same machine model, or fewer processors on a weaker machine model. Our algorithm can also be applied to determine if two bipartite-permutation graphs are isomorphic.

In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x, y of G d(x, y) ≤ r(x) + r(y) + 1 holds where r : V → N is a given vertex function. Furthermore, we show that for homogeneously orderable graphs with r-dominating cliques the cardinality of a maximum strict r-packing set equals the cardinality of a minimum r-dominating clique provided the last parameter is not two. Finally, we present two efficient algorithms: The first one decides whether a given homogeneously orderable graph has an r-dominating clique and, if so, computes both a minimum r-dominating clique and a maximum strict r-packing set of the graph. The second one computes a minimum connected r-dominating set in a homogeneously orderable graph. © 1997 John Wiley & Sons, Inc. Networks 30: 121–131, 1997

We provide efficient parallel algorithms for the minimum separation, offset range, and optimal offset problems for single-layer channel routing. We consider all the variations of these problems that have linear-time sequential solutions rather than limiting attention to the “river-routing” context, where single-sided connections are disallowed. For the minimum separation problem, we obtain O(lg N) time on a CREW PRAM or O(lg N/lg lg N) time on a CRCW PRAM, both with optimal work (processor-time product) of O(N), where N is the number of terminals. For the offset range problem, we obtain the same time and processor bounds as long as only one side of the channel contains single-sided nets. For the optimal offset problem with single-sided nets on one side of the channel, we obtain time O(lg N lg lg N) on a CREW PRAM or O(lg N) time on a CRCW PRAM with O(N lg lg N) work. Not only does this improve on previous results for river routing, but we can obtain an even better time of O((lg lg N)2) on the CRCW PRAM in the river routing context.KeywordsParallel AlgorithmMinimum SeparationChannel BoundaryConvex CornerEfficient Parallel AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Laminar structure of ptolemaic graphs with applications

A simple paradigm for graph recognition: application to cographs and distance hereditary graphs

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

Let G be a connected graph and $S \subseteq V( G )$. Then the Steiner distance of S in G, denoted by $d_G ( S )$, is the smallest number of edges in a connected subgraph of G that contains S. A connected graph G is k-Steiner distance-hereditary, $k \geq 2$, if, for every $S \subseteq V( G )$ such that $|S| = k$ and every connected induced subgraph H of G containing $S,d_H ( S ) = d_G ( S )$. It is shown that if G is 2-Steiner distance-hereditary, then G is k-Steiner distance-hereditary for all $k \geq 2$. Furthermore, it is shown that if G is k-Steiner distance-hereditary $( k \geq 3 )$, then G need not be $( k - 1 )$-Steiner distance-hereditary. An efficient algorithm for determining the Steiner distance of a set of k vertices in a k-Steiner distance-hereditary graph is discussed, and a characterization of 2-Steiner distance-hereditary graphs that leads to an efficient algorithm for testing whether a graph is 2-Steiner distance-hereditary is given.

Homogeneously orderable graphs

An efficient implementation of a consistent labeling technique can lead to a fast solution in solving all kinds of tree search problems by reducing the search space from a potentially exponential size to a much smaller size. In this paper a scalable parallel algorithm is presented in achieving labeling consistency. This algorithm is optimal in execution time as well as in processor-time product. It is suitable for implementation on most distributed memory multiprocessor system.

B+-trees algorithm is a high efficient and common-used data organization algorithm in database system. But B+-trees algorithm has a disadvantage of low storage utilization. This disadvantage has a bad effect especially on embedded database systems, which have limit storage. To overcome this drawback, this paper presents an improved B+-trees algorithm, and applies it to an embedded database-Berkeley DB. Experimental results show that the improved B+-trees algorithm can attain higher storage utilization at the expense of more execution time than the conventional B+-trees algorithm.