Split: a flexible and efficient algorithm to vector-descriptor product

Abstract

Many Markovian stochastic structured modeling formalisms like Petri nets, automata networks and process algebra represent the infinitesimal generator of the underlying Markov chain as a descriptor instead of a traditional sparse matrix. A descriptor is a compact and structured storage based on a sum of tensor (Kronecker) products of small matrices that can be handled by many algorithms allowing affordable stationary and transient solutions even for very large Markovian models. One of the most efficient algorithms used to compute iterative solutions of descriptors is the Shuffle algorithm which is used to perform the multiplication by a probability vector. In this paper we propose an alternative algorithm called Split, since it offers a flexible solution between the pure sparse matrix approach and the Shuffle algorithm using a hybrid solution. The Split algorithm puts the Shuffle approach in perspective by presenting a faster execution time for many cases and at least the same efficiency for the worst cases. The Split algorithm is applied to solve two SAN models based on real problems showing the practical contribution of this paper.

Kronecker descriptor partitioning for parallel algorithms

“…However, Kronecker product terms can be exploited in different ways. The Split algorithm [3] allows each tensor product to be partitioned into smaller work units called AUNFs that can be distributed among processors. Note that the step (c) suffers a great impact in different partitioning approaches, i.e., by Kronecker product terms itself and by AUNFs quantities, mainly because tensor product terms can strongly vary matrices types, dimensions, sparsity, and all these characteristics determine the cut-parameter and consequently the total number of AUNFs.…”

“…To efficiently perform VDP, specialized algorithms are proposed throughout the years, namely the traditional Shuffle Algorithm and the flexible Split Algorithm [3]. The main differentiation between the algorithms concerns the additional memory requirements and the computational cost in terms of needed multiplications.…”

“…The main differentiation between the algorithms concerns the additional memory requirements and the computational cost in terms of needed multiplications. Figure 1 graphically presents the Split idea [3] when applied to the multiplication of a vector by a generic tensor term. The algorithm affects all Kronecker product terms inside a descriptor, at each iteration of a numerical method [7] (e.g.…”

Efficient vector-descriptor product exploiting time-memory trade-offs

“…• to use the flexible hybrid vector-descriptor algorithm called Split [11,32], which is a rather recent approach currently applied to a subset of the SAN formalism where the interaction between components is limited to synchronized events, i.e., there are no functional rates or probabilities in the model. Once again, as in the canonical matrix diagrams approach, the efficiency of the Split algorithm depends on many internal choices.…”

OpenMP-based Parallel Algorithms for Solving Kronecker Descriptors

“…Each term corresponds to a set of small matrices and tensor product operators. Specialized numerical algorithms have been proposed throughout the years, namely the traditional Shuffle algorithm [11,12] and the flexible Split algorithm [10]. The main difference between them concerns the additional memory requirements and the computational cost in terms of floating-point multiplications.…”