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The coefficient matrix of a fairly large system of equations is generally very sparse. Thus in the computer-aided analysis of such a system of equations, of prime importance are the sparsity-oriented solution and storage techniques. Motivated by this problem, in the present paper we first define a bipartite network associated with a given system of equations, and then discuss the problem of sparsity-preserving pivot ordering with the use of the bipartite network.to row I\ and column I, and then deleting the row and the column. If through this operation the following relations hold between the x i j of X(A) and . Y: ". ' ) of X(A#"~")For any N~, # 0 of A, let where V and A are Boolean addition and multiplication, respectively, then we designate this operation as an s-pirofing with p i w r ukl.Henceforth, let A*"-" denote a matrix obtained from A by applying an s-pivoting with pivot u k i .

This paper studies the problem of scheduling jobs on a multiprocessor system with a common memory to minimize \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {w_i \left({f_i } \right)} $\end{document}, where fi denotes flow‐times of jobs i, and wi(fi) denotes monotonically nondecreasing functions of fi. Arriving jobs are assigned to available processors according to an appropriate scheduling rule. Once a job is placed on a processor, it retains the processor during its processing and repeats local computations and accesses to the common memory. First, the scheduling problem is formulated into a zero‐one integer programming problem. Next, a procedure is presented for computing a lower bound on the optimal cost. Finally, the performance of the proposed procedure is evaluated using simulation experiments for the problem of minimizing mean flow‐time.

A b s t r a c t -T h i s paper treats mutual exclusion in distributed autonomous environments. The most important property of the autonomous network treated in this paper is its membership variability, that is, frequent occurence of entries of new nodes and exits of old nodes. Thus, when the network is large-scale, it is not possible for each node to keep up the information of all other nodes. In [4], we have designed a mutual exclusion algorithm of the single shared resource case for distributed environments of autonomous nodes, based on Chandy-Misra protocol for Dining Philosopher(diners) problems [l]. We in this paper consider mutual exclusion of multiple shared resource cases. We give the condition that no more than k nodes can use the shared resource. Since the acyclic graph model used in our model can be represented by a marked graph model, this model is used in analysis.