An experimental validation of the PRO model for parallel and distributed computation

National audienceThe Parallel Resource-Optimal (PRO) computation model was introduced by Gebremedhin et al. [2002] as a framework for the design and analysis of efficient parallel algorithms. The key features of the PRO model that distinguish it from previous parallel computation models are the full integration of resource-optimality into the design process and the use of a {granularity function as a parameter for measuring quality. In this paper we present experimental results on parallel algorithms, designed using the PRO model, for two representative problems: list ranking and sorting. The algorithms are implemented using SSCRAP, our environment for developing coarse-grained algorithms. The experimental performance results observed agree well with analytical predictions using the PRO model. Moreover, by using different platforms to run our experiments, we have been able to provide an integrated view of the modeling of an underlying architecture and the design and implementation of scalable parallel algorithms

Incrementally Maintaining the Number of l-cliques

The main contribution of this paper is an incremental algorithm to update the number of $l$-cliques, for $l \geq 3$, in which each node of a graph is contained, after the deletion of an arbitrary node. The initialization cost is $O(n^{\omega p+q})$, where $n$ is the number of nodes, $p=\lfloor \frac{l}{3} \rfloor$, $q=l \pmod{3}$, and $\omega=\omega(1,1,1)$ is the exponent of the multiplication of two $n x n$ matrices. The amortized updating cost is $O(n^{q}T(n,p,\epsilon))$ for any $\epsilon \in [0,1]$, where $T(n,p,\epsilon)=\min\{n^{p-1}(n^{p(1+\epsilon)}+n^{p(\omega(1,\epsilon,1)-\epsilon)}),n^{p \omega(1,\frac{p-1}{p},1)}\}$ and $\omega(1,r,1)$ is the exponent of the multiplication of an $n x n^{r}$ matrix by an $n^{r} x n$ matrix. The current best bounds on $\omega(1,r,1)$ imply an $O(n^{2.376p+q})$ initialization cost, an $O(n^{2.575p+q-1})$ updating cost for $3 \leq l \leq 8$, and an $O(n^{2.376p+q-0.532})$ updating cost for $l \geq 9$. An interesting application to constraint programming is also considered

Cutting planes and the elementary closure in fixed dimension

The elementary closure $P'$ of a polyhedron $P$ is the intersection of $P$ with all its Gomory-Chvátal cutting planes. $P'$ is a rational polyhedron provided that $P$ is rational. The known bounds for the number of inequalities defining $P'$ are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. If $P$ is a simplicial cone, we construct a polytope $Q$, whose integral elements correspond to cutting planes of $P$. The vertices of the integer hull $Q_I$ include the facets of $P'$. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of $Q_I$

Performance Implications by the Hierarchical Design of Clusters

Colloque avec actes et comité de lecture. internationale.International audienceWe present experimental results for the evaluation of PC clusters that differ on several aspect of their architecture, such as being mono or biprocessors and having a high bandwidth interconnection network or not. These experiments confirm that a good equilibrium between the speed (throughput) of the different components is crucial for a satisfactory performance of such a machine. On the other hand they also show that the latency of the underlying infrastructure is of less importance and can be hidden by applying techniques from coarse grained parallelism

A deductive model checking approach for hybrid systems

In this paper we propose a verification method for hybrid systems that is based on a successive elimination of the various system locations involved. Briefly, with each such elimination we compute a weakest precondition (strongest postcondition) on the predecessor (successor) locations such that the property to be proved cannot be violated. This is done by representing a given verification problem as a second-order predicate logic formula which is to be solved (proved valid) with the help of a second-order quantifier elimination method. In contrast to many ``standard'' model checking approaches the method as described in this paper does not perform a forward or backward reachability analysis. Experiments show that this approach is particularly interesting in cases where a standard reachability analysis would require to travel often through some of the given system locations. In addition, the approach offers possibilities to proceed where ``standard'' reachability analysis approaches do not terminate

Superposition and chaining for totally ordered divisible abelian groups

We present a calculus for first-order theorem proving in the presence of the axioms of totally ordered divisible abelian groups. The calculus extends previous superposition or chaining calculi for divisible torsion-free abelian groups and dense total orderings without endpoints. As its predecessors, it is refutationally complete and requires neither explicit inferences with the theory axioms nor variable overlaps. It offers thus an efficient way of treating equalities and inequalities between additive terms over, e.g., the rational numbers within a first-order theorem prover

Ultimate parallel list ranking?

Two improved list-ranking algorithms are presented. The ``peeling-off'' algorithm leads to an optimal PRAM algorithm, but was designed with application on a real parallel machine in mind. It is simpler than earlier algorithms, and in a range of problem sizes, where previously several algorithms where required for the best performance, now this single algorithm suffices. If the problem size is much larger than the number of available processors, then the ``sparse-ruling-sets'' algorithm is even better. In previous versions this algorithm had very restricted practical application because of the large number of communication rounds it was performing. This main weakness of this algorithm is overcome by adding two new ideas, each of which reduces the number of communication rounds by a factor of two