Stefan D. Bruda

Improving A Solution's Quality Through Parallel Processing

The primary purpose of parallel computation is the fast execution of computational tasks that are too slow to perform sequentially. However, it was shown recently that a second equally important motivation for using parallel computers exists: Within the paradigm of real-time computation, some classes of problems have the property that a solution to a problem in the class computed in parallel is better than the one obtained on a sequential computer. What represents a better solution depends on the problem under consideration. Thus, for optimization problems, ‘better’ means ‘closer to optimal’. Similarly, for numerical problems, a solution is ‘better’ than another one if it is ‘more accurate’. The present paper continues this line of inquiry by exploring another class enjoying the aforementioned property, namely, cryptographic problems in a real-time setting. In this class, ‘better’ means ‘more secure’. A real-time cryptographic problem is presented for which the parallel solution is provably, considerably, and consistently better than a sequential one.It is important to note that the purpose of this paper is not to demonstrate merely that a parallel computer can obtain a better solution to a computational problem than one derived sequentially. The latter is an interesting (and often surprising) observation in its own right, but we wish to go further. It is shown here that the improvement in quality can be arbitrarily high (and certainly superlinear in the number of processors used by the parallel computer). This result is akin to superlinear speedup—a phenomenon itself originally thought to be impossible.

The characterization of data-accumulating algorithms

Abstract. Data-accumulating algorithms (d-algorithms for short), extensively studied in [12], work on an input considered as a virtually endless stream. The computation terminates when all the currently arrived data have been processed before another datum arrives. In this paper a finer characterization of the class of d-algorithms is given, and it is shown that this class is identical to the class of on-line algorithms under a proper definition of the latter. The parallel implementation of d-algorithms is then investigated. It is found that, in general, the speedup achieved through parallelism can be made arbitrarily large for almost any such algorithm. On the other hand, we prove that for d-algorithms whose static counterparts manifest only unitary speedup, no improvement is possible through parallel implementation.

A Case Study in Real-Time Parallel Computation

A correcting algorithm is one that receives an endless stream of correc- tions to its initial input data and terminates when all the corrections received have been taken into account. We give a characterization of correcting algorithms based on the theory of data-accumulating algorithms. In particular, it is shown that any correcting algorithm exhibits superunitary behavior in a parallel computation setting if and only if the static counterpart of that correcting algorithm manifests a strictly superunitary speedup. Since both classes of correcting and data-accumulating algorithms are included in the more general class of real-time algorithms, we show in fact that many problems from this class manifest superunitary behavior. Moreover, we give an example of a real-time parallel computation that pertains to neither of the two classes studied (namely, correcting and data-accumulating algorithms), but still manifests superunitary behavior. Because of the aforementioned results, the usual measures of performance for parallel algorithms (that is, speedup and efficiency) lose much of their ability to convey effectively the nature of the phenomenon taking place in the real-time case. We propose therefore a more expressive measure that captures all the relevant parameters of the computation. Our proposal is made in terms of a graphical representation. We state as an open problem the investigation of such a measure, including finding an analytical form for it.

Real-time computation: a formal definition and its applications

The concept of real time has different meanings in the systems and theory communities. Thus, the existing formal real-time models do not capture all the practically relevant aspects of such computations. This article proposes a new definition that, we believe, allows a unified treatment of all practically meaningful variants of realtime computations. We use the developed formalism to mode two important features of real-time algorithms, namely the presence of deadlines and the real-time arrival of input data. We also emphasize the expressive power of our model by using it to formalize aspects from the areas of real-time database systems and ad hoc networks. We offer formulations of the recognition problem for real-time database systems and of the routing problem in ad hoc networks. Finally, we suggest a variant of our formalism that is suited for modelling parallel distributed real-time algorithms. We believe that the proposed formalism is a first step towards a unified and realistic complexity theory for real-time parallel computations.

Real-time computation: A formal definition and its applications

Abstract The concept of real time has different meanings in the systems and theory communities. Thus, the existing formal real-time models do not capture all the practically relevant aspects of such computations. This paper proposes a new definition that, we believe, allows a unified treatment of all practically meaningful variants of real-time computations. We use the developed formalism to mode two important features of real-time algorithms, namely the presence of deadlines and the real-time arrival of input data. We also emphasize the expressive power of our model by using it to formalize aspects from the areas of real-time database systems and ad hoc networks. We offer formulations of the recognition problem for real-time database systems and of the routing problem in ad hoc networks. Finally, we suggest a variant of our formalism that is suited for modelling parallel distributed real-time algorithms. We believe that the proposed formalism is a first step towards a unified and realistic complexity theory for real-time parallel computations.

Parallel real-time numerical computation: beyond speedup. III

For pt. II see Technical Report No. 99-423, Dept. of Comput. and Inf. Sci., Queens University, Kingston, Ontario, May 1999. Parallel computers can do more than simply speed up sequential computations. They are capable of finding solutions that are far better in quality than those obtained by sequential computers. This fact is demonstrated by analyzing sequential and parallel solutions to numerical problems in a real-time paradigm. In this setting, numerical data required to solve a problem are received as input by a computer system, at regular intervals. The computer must process its inputs as soon as they arrive. It must also produce its outputs at regular intervals, as soon as they are available. We show that for some real-time numerical problems a parallel computer can deliver a solution that is significantly more accurate than that computed by a sequential computer. Similar results were derived recently in the areas of real-time optimization and real-time cryptography.

Pursuit and Evasion on a Ring: An Infinite Hierarchy for Parallel Real-Time Systems

Abstract. We show that, for any positive integer n , there exists at least one timed ω -language Ln which is accepted by a 2n -processor real-time algorithm using arbitrarily slow processors, but cannot be accepted by a (2n-1) -processor real-time algorithm. It follows therefore that real-time algorithms form an infinite hierarchy with respect to the number of processors used. Furthermore, such a result holds for any model of parallel computation.

The characterization of parallel real-time optimization problems

We identify the class of optimization problem expressible as independence systems that can be solved in real time using a parallel machine with polynomially bounded resources as being exactly the class of matroid for which the size of the optimal solution can be computed in parallel real time. We also extend previous results, showing that the solution obtained by a parallel algorithm is arbitrarily better than the solution reported by a sequential one not only for the real-time minimum-weight spanning tree (as previously known). Indeed, we show that, for all practical purposes, such a property does in fact hold for any optimization problem that falls into the aforementioned class.