Transitive Closure

A graph is a collection of nodes and edges. Transitive closure matrix is a matrix formed by the reach-ability factor, which means if one node A of the graph is reachable from another node B, then there exists a positive reach-ability between A and B, negative reach-ability otherwise. This can be easily denoted by using binary denotation of 0 and 1. Graph powering is a technique in discrete mathematics and graph theory where our concern is to get the path between the nodes of a graph by using the powering principle. In simple words, if we take the rth power of any given graph G then that will give us another graph G(r) which has exactly the same vertices, but the number of edges will change. In the powered graph G(r) there will be a connection between any two nodes if there exists a path that has a length less than r between them. This small intuition can help us in finding the transitive closure of a graph in O(V^4) time complexity and O(V^2) space complexity. We can improve the time complexity of the above-mentioned algorithm by using Euler's Fast Powering Algorithm to O(V^3 logo).

ABSTRACT We describe two methods for analysing a complex network which focus on examining changes in the transitive closure of the network under directed and stochastic attack of its edges. Dynamic transitive closure analysis (DTCA) examines changes in critical transitive path for increasingly stringent edge length tolerance. Context specific DTCA is a fuzzy extension of DTCA, which examines changes in DTCA under randomly perturbed transitive contexts. We find that such strategies allow a researcher to identify highly connected elements within a network and uncover related sub-networks. Application to simulated random graphs demonstrates that these methods can be used to determine local network connections as well as quantify the overall transitive natures of the network. We have developed an adjustable resolution O(N ) parallel algorithm to carry out these analyses which scales nearly linearly with the number of nodes in the network.

The need for temporal reasoning is found throughout the engineering disciplines. James Allen introduced a representation for temporal reasoning based upon the concept of intervals. This approach provides a rich set of temporal relations for reasoning over events and changes in state. The full temporal algebra is Nscomplete however. The algorithm developed by Allen executes in 0(n3) time but only ensures consistency between any three intervals. This research presents an approach to representing interval relations as a bit-encoded form which captures the relationships between the end-points of the intervals. A bit-algebra is then defined which provides an algorithmic method for computing transitive relations without requiring the table lookup of Allen's algorithm. By reducing the set of ambiguous interval representations to the set of relationships which have unknown temporal extent, a robust subset of the full algebra is defined which maintains the direct computation of transitiv...

Catalunya, under grant SGR2001-0029, is also acknowledged. Abstract: The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position in the assignment matrix.

The property of transitivity is one of the most important for fuzzy binary relations, especially in the cases when they are used for the representation of real life similarity or ordering information. As far as the algorithmic part of the actual calculation of the transitive closure of such relations is concerned, works in the literature mainly focus on crisp symmetric relations, paying little attention to the case of general fuzzy binary relations. Most works that deal with the algorithmic part of the transitive closure of fuzzy relations only focus on the case of max-min transitivity, disregarding other types of transitivity. In this paper, after formalizing the notion of sparseness and providing a representation model for sparse relations that displays both computational and storage merits, we propose an algorithm for the incremental update of fuzzy sup-t transitive relations. The incremental transitive update (ITU) algorithm achieves the re-establishment of transitivity when an ...

We report on the performance evaluation of greedy parsing with asingle step lookahead (which we call flexible Parsing or FPas an alternative to the commonly used greedy parsing (withno-lookaheads) scheme. Greedy parsing is the basis of most popularcompression programs including UNIX compress andgzip, however it usually results in far from optimalparsing/compression with regard to the dictionary constructionscheme in use. Flexible parsing, however, is optimal [MS99], i.e.partitions any given input to the smallest number of phrasespossible, for dictionary construction schemes which satisfy theprefix property throughout their execution. We focus on the application of FP in the context of theLZW variant of the Lempel-Ziv'78 dictionary construction method[Wel84, ZL78], which is of considerable practical interest. Weimplement two compression algorithms which use (1) FP withLZW dictionary (LZW-FP), and (2) FP with analternative flexible dictionary (FPA as introduced in [Hor95]). Ourimp...

This paper focuses on the optimization of the navigation through voluminous subsumption hierarchies of topics employed by portal catalogs like Netscape Open Directory (ODP). We advocate for the use of labeling schemes for modeling these hierarchies in order to efficiently answer ...

We study queries to spatial databases, where spatial data are modeled as semi-algebraic sets, using the relational calculus with polynomial inequalities as a basic query language. We work with the extension of the relational calculus with terminating transitive closures. The main result is that this language can express the linearization of semialgebraic databases. We also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases. As a consequence of these results, we obtain a ...

Recursive queries are required for many database applications. Among them we can mention Bill-Of-Material (BOM), various kinds of networks (transportation, telecommunication, etc.), workflows, processing semi-structured data (XML, RDF), and others. The support for recursive queries in current query languages is limited. In particular, this concerns the corresponding extensions of SQL in Oracle and DB2. In this paper we present recursive

In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that Ag is nonnegative, A and Ag have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices. AMS subject classifications. 15A18, 15A48, 15A21

A hypergraph formalism is introduced to represent database schemata. In particular, a database schema B , described by one full join dependency and a set of functional dependencies, is represented by a (database) hypergraph H , containing both undirected and directed hyperedges. Undirected hyperedges correspond to the relations in the join dependency, and directed hyperedges correspond to the functional dependencies. In addition, two classes of database hypergraphs are defined: e -acyclic hypergraphs and e -independent hypergraphs. A hypergraph is e -acyclic if it is equivalent to some acyclic hypergraph; it is e -independent if it is equivalent to some independent (i.e., cover-embedding) hypergraph. Furthermore, the closure of a database hypergraph is defined as the extension of the transitive closure of a graph. By using a lower bound and an upper bound of the hypergraph closure (called L-closure and U-closure, respectively), it is proved that two e -acyclic ( e -independent) hype...

ABSTRACT In this paper, we investigate an extension of the description logic SHIQ\mathcal{SHIQ}–a knowledge representation formalism used for the Semantic Web–with transitive closure of roles occurring not only in concept inclusion axioms but also in role inclusion axioms. It was proved that adding transitive closure of roles to SHIQ\mathcal{SHIQ} without restriction on role hierarchies may lead to undecidability. We have identified a kind of role inclusion axioms that is responsible for this undecidability and we propose a restriction on these axioms to obtain decidability. Next, we present a tableaux-based algorithm that decides satisfiability of concepts in the new logic.

Abstract. Transformation rules are a convenient means to specify partially defined and nondeterministic computations. We describe a rulebased programming system which has primitive operators for defining elementary rules and for computing with unions, ...