Jean Duprat

Some results about on-line computation of functions

Complexity results that allow the exact determination or bounding of the online delay of most common arithmetic and elementary functions are presented. These results show that many classical online operators presented in the literature are optimal in delay (but not necessarily in period). The authors propose a way to conserve, for large numbers of manipulations, the main advantage of online arithmetic (the capability of digit-level pipelining) by presenting sparse online arithmetic.<<ETX>>

The complexity of searching in X + Y and other multisets

Abstract Bounds on sorting and selection in a matrix A = X + Y have been studied for the case where X and Y are two sorted n -vectors of reals. In this paper we construct a special family of such matrices to prove a lower bound of Ω( n ) for the problem of searching in X + Y . As an O( n ) algorithm for searching exists in the literature, we conclude that the complexity of searching in X + Y is Θ( n ). We also address searching in Σ X i as well.

New redundant representations of complex numbers and vectors

A new redundant representation for complex numbers, called polygonal representation, is presented. This representation enables fast carry-free addition (in a way quite similar to the carry-free addition in signed-digits number systems), and is convenient for multiplication. In addition, the technique is extended to handle n-dimensional vectors.<<ETX>>

Complexity of selection in X + Y

Let X and Y be two sorted n-vectors and A = X + Y be an n×n matrix with sorted rows and columns such that aij=xi+yj. Let 1⩽k<n2. Vyskoc (1987) claimed that selecting the (k+1)st element of A could be done in O(logn) time if the kth element is known. In this note we prove that this result is not exact by showing that O(n) is a lower bound for the problem under Vyskocs hypothesis. We also describe an O(n) algorithm and conclude by showing how the same algorithm can be used for searching on such matrices.

Towards a High Precision Massively Parallel Computer

Here, we deal with a new fine grain parallel-pipelined architecture made up of heterogeneous digit online arithmetic units (A Us). We present some main issues of such an architecture, including the model of computation, new scheduling heuristics and examples of linear algebra computations. Using parallel discrete-event simulations and computation visualization on a massively parallel computer, we present some measures of its performance.

On the Simulation of Pipelining of Fully Digit On-Line Floating-Point Adder Networks on Massively Parallel Computers

This paper deals with the simulation of on-line floating-point adders networks on parallel machines. We present the first results of the simulations of pipelining at digit-level on MasPar, a massively parallel SIMD computer.

Systolic triangularization over finite fields

Abstract We propose a systolic architecture for the triangularization of large dense n × n matrices over GF(p), where p is a prime number. The solution of large dense linear systems over GF(p) is the major computational step in various algorithms issuing from arithmetic number theory and computer algebra. The proposed architecture implements the triangularization via a new algorithm as robust as Gaussian elimination with partial pivoting, and the operation of the array remains purely systolic. The algorithm triangularizes a dense n × n matrix in time 2n on a triangular array of n (n + 1) 2 elementary processors, which is (to our knowledge) the best area-time performance.

About Constructive vectors

We explore several ways of constructively implementing vectors in proof assistants and we discuss their advantages and their drawbacks.

Parallel simulation of heterogeneous arithmetic units networks and high precision dot products

In this paper we deal with a new high precision computation of the dot product. The key idea is to use hundreds of digit-serial arithmetic units or operators that allow a massive digit-level pipelining. Parallel discrete-event simulations performed on a memory-distributed massively parallel computer show that with a limited number of arithmetic units, the computation of dot product when performed using a classical algorithmic technique (i.e. serial cumulative multiplications) is almost as fast as the case where an optimal divide-and-conquer algorithmic technique is used. Interconnection networks for both algorithmic techniques are considered.<<ETX>>

LAIOS: a parallel execution of PROLOG by data copies

Abstract LAIOS (Lattice for Artificial Intelligence Oriented System) is a project of a multiprocessor architecture oriented to artificial intelligence applications. The execution model is the AND/OR tree; the control is realized by automata and authorizes OR parallelism and AND pipelining. PROLOG programs are compiled into frame nodes and execution dynamically creates process nodes. The data flows along the branches of the execution tree. In each process node, a computation is realized by a universal operator. The data are physically copied step by step, making parallelism and nondeterministic implementations easier. An architecture is defined, supporting such a model. It uses self-defined data, structured in blocks. The lattice is composed of modules connected by two topologies corresponding to the two levels of communications. A hexagonal network where neighbours communicate by dual-ported memories allows the data circulation. A centralized network distributes the program, all the modules are connected by a shared bus to a mass storage containing the knowledge base. First simulations have validated the model and the architecture. The performance (without limitations of the number of nodes) is still modest, about tens of KLIPS depending on the program. These simulations permit one to understand the rules and the parameters that govern the parallel execution of PROLOG programs on this lattice and using this model.