Computing transitive closure on systolic arrays of fixed size

Forming the transitive closure of a binary relation (or directed graph) is an important part of many algorithms. When the relation is represented by a bit matrix, the transitive closure can be efficiently computed in parallel in a systolic array. Various such arrays for computing the transitive closure have been proposed. They all have in common, though, that the size of the array must be proportional to the number of nodes. Here we propose two ways of computing the transitive closure of an arbitrarily big graph on a systolic array of fixed size. The first method is a simple partitioning of a well-known systolic algorithm for computing the transitive closure. The second is a block-structured algorithm for computing the transitive closure. This algorithm is suitable for execution on a systolic array, that can multiply fixed size bit matrices and compute transitive closure of graphs with a fixed number of nodes. The algorithm is, however, not limited to systolic array implementations; it works on any parallel architecture that can form the transitive closure and product of fixed-size bit matrices efficiently. The shortest path problem, for directed graphs with weighted edges, can also be solved for arbitrarily large graphs on a fixed-size systolic array in the same manner, devised above, as the transitive closure is computed

Transitive Closure of Infinite Graphs and its Applications

Integer tuple relations can concisely summarize many types of information gathered from analysis of scientific codes. For example they can be used to precisely describe which iterations of a statement are data dependent of which other iterations. It is generally not possible to represent these tuple relations by enumerating the related pairs of tuples. For example, it is impossible to enumerate the related pairs of tuples in the relation {[i] -> [i+2] | 1 <= i <= n-2}. Even when it is possible to enumerate the related pairs of tuples, such as for the relation {[i,j] -> [i',j'] | 1 <= i,j,i',j' <= 100}, it is often not practical to do so. We instead use a closed form description by specifying a predicate consisting of affine constraints on the related pairs of tuples. As we just saw, these affine constraints can be parameterized, so what we are really describing are infinite families of relations (or graphs). Many of our applications of tuple relations rely heavily on an operation called transitive closure. Computing the transitive closure of these "infinite graphs" is very different from the traditional problem of computing the transitive closure of a graph whose edges can be enumerated. For example, the transitive closure of the first relation above is the relation {[i] -> [i'] | exists beta s.t. i'-i = 2beta and 1 <= i <= i' <= n}. As we will prove, this computation is not computable in the general case. We have developed algorithms that produce exact results in most commonly occurring cases and produce upper or lower bounds (as necessary) in the other cases. This paper will describe our algorithms for computing transitive closure and some of its applications such as determining which inter-processor synchronizations are redundant. (Also cross-referenced as UMIACS-TR-95-48

Efficiency of Automata in Semi-Commutation Verification Techniques

Computing the image of a regular language by the transitive closure of a relation is a central question in Regular Model Checking. In a recent paper Bouajjani, Muscholl and Touili proved that the class of APC regular languages is closed under all semi-commutation relations R. Moreover a recursive algorithm on the regular expression is given to compute the image of an APC language by the transitive closure of R. This paper provides a new approach, based on automata, of the same question. Our approach produces a simpler and more efficient algorithm which furthermore works for a larger class of regular languages closed under union, intersection, semi-commutation relations and conjugacy. The existence of this new class, PolC, answered the open question proposed in Bouajjani and al.'s paper

Regular hedge model checking

We extend the regular model checking framework so that it can handle systems with arbitrary width tree-like structures. Con gurations of a system are represented by trees of arbitrary arities, sets of con gurations are represented by regular hedge automata, and the dynamics of a system is modeled by a regular hedge transducer. We consider the problem of computing the transitive closure T + of a regular hedge transducer T. This construction is not possible in general. Therefore, we present a general acceleration technique for computing T+. Our method consists of enhancing the termination of the iterative computation of the different compositions Ti by merging the states of the hedge transducers according to an appropriate equivalence relation that preserves the traces of the transducers. We provide a methodology for effectively deriving equivalence relations that are appropriate. We have successfully applied our technique to compute transitive closures for some mutual exclusion protocols de ned on arbitrary width tree topologies, as well as for an XML application.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

A survey of parallel execution strategies for transitive closure and logic programs

An important feature of database technology of the nineties is the use of parallelism for speeding up the execution of complex queries. This technology is being tested in several experimental database architectures and a few commercial systems for conventional select-project-join queries. In particular, hash-based fragmentation is used to distribute data to disks under the control of different processors in order to perform selections and joins in parallel. With the development of new query languages, and in particular with the definition of transitive closure queries and of more general logic programming queries, the new dimension of recursion has been added to query processing. Recursive queries are complex; at the same time, their regular structure is particularly suited for parallel execution, and parallelism may give a high efficiency gain. We survey the approaches to parallel execution of recursive queries that have been presented in the recent literature. We observe that research on parallel execution of recursive queries is separated into two distinct subareas, one focused on the transitive closure of Relational Algebra expressions, the other one focused on optimization of more general Datalog queries. Though the subareas seem radically different because of the approach and formalism used, they have many common features. This is not surprising, because most typical Datalog queries can be solved by means of the transitive closure of simple algebraic expressions. We first analyze the relationship between the transitive closure of expressions in Relational Algebra and Datalog programs. We then review sequential methods for evaluating transitive closure, distinguishing iterative and direct methods. We address the parallelization of these methods, by discussing various forms of parallelization. Data fragmentation plays an important role in obtaining parallel execution; we describe hash-based and semantic fragmentation. Finally, we consider Datalog queries, and present general methods for parallel rule execution; we recognize the similarities between these methods and the methods reviewed previously, when the former are applied to linear Datalog queries. We also provide a quantitative analysis that shows the impact of the initial data distribution on the performance of methods

Context-Free Path Querying by Matrix Multiplication

Graph data models are widely used in many areas, for example, bioinformatics, graph databases. In these areas, it is often required to process queries for large graphs. Some of the most common graph queries are navigational queries. The result of query evaluation is a set of implicit relations between nodes of the graph, i.e. paths in the graph. A natural way to specify these relations is by specifying paths using formal grammars over the alphabet of edge labels. An answer to a context-free path query in this approach is usually a set of triples (A, m, n) such that there is a path from the node m to the node n, whose labeling is derived from a non-terminal A of the given context-free grammar. This type of queries is evaluated using the relational query semantics. Another example of path query semantics is the single-path query semantics which requires presenting a single path from the node m to the node n, whose labeling is derived from a non-terminal A for all triples (A, m, n) evaluated using the relational query semantics. There is a number of algorithms for query evaluation which use these semantics but all of them perform poorly on large graphs. One of the most common technique for efficient big data processing is the use of a graphics processing unit (GPU) to perform computations, but these algorithms do not allow to use this technique efficiently. In this paper, we show how the context-free path query evaluation using these query semantics can be reduced to the calculation of the matrix transitive closure. Also, we propose an algorithm for context-free path query evaluation which uses relational query semantics and is based on matrix operations that make it possible to speed up computations by using a GPU.Comment: 9 pages, 11 figures, 2 table

Data fragmentation for parallel transitive closure strategies

Addresses the problem of fragmenting a relation to make the parallel computation of the transitive closure efficient, based on the disconnection set approach. To better understand this design problem, the authors focus on transportation networks. These are characterized by loosely interconnected clusters of nodes with a high internal connectivity rate. Three requirements that have to be fulfilled by a fragmentation are formulated, and three different fragmentation strategies are presented, each emphasizing one of these requirements. Some test results are presented to show the performance of the various fragmentation strategie

Matching Dependencies with Arbitrary Attribute Values: Semantics, Query Answering and Integrity Constraints

Matching dependencies (MDs) were introduced to specify the identification or matching of certain attribute values in pairs of database tuples when some similarity conditions are satisfied. Their enforcement can be seen as a natural generalization of entity resolution. In what we call the "pure case" of MDs, any value from the underlying data domain can be used for the value in common that does the matching. We investigate the semantics and properties of data cleaning through the enforcement of matching dependencies for the pure case. We characterize the intended clean instances and also the "clean answers" to queries as those that are invariant under the cleaning process. The complexity of computing clean instances and clean answers to queries is investigated. Tractable and intractable cases depending on the MDs and queries are identified. Finally, we establish connections with database "repairs" under integrity constraints.Comment: 13 pages, double column, 2 figure