The Expressive Power and Complexity of Dynamic Process Graphs

Abstract

A model for parallel and distributed programs, the dynamic process graph, is investigated under graph-theoretic and complexity aspects. Such graphs are capable of representing all possible executions of a parallel or distributed program in a very compact way. The size of this representation is small — in many cases only logarithmic with respect to the size of any execution of the program. An important feature of this model is that the encoded executions are directed acyclic graphs with a regular structure, which is typical of parallel programs, and that it embeds constructors for parallel programs, synchronization mechanisms as well as conditional branches.

In a previous paper we have analysed the expressive power of the general model and various restrictions. Furthermore, from an algorithmic point of view it is important to decide whether a given dynamic process graph can be executed correctly and to estimate the minimal deadline given enough parallelism. Our model takes into account communication delays between processors when exchanging data.

In this paper we study a variant with output restriction. It is appropriate in many situations, but its expressive power has not been known exactly. First, we investigate structural properties of the executions of such dynamic process graph s $\mathcal{G}$. A natural graph-theoretic conjecture that executions must always split into components isomorphic to subgraphs of $\mathcal{G}$ turns out to be wrong. We are able to establish a weaker property. This implies a quadratic bound on the maximal deadline in contrast to the general case, where the execution time may be exponential. However, we show that the problem to determine the minimal deadline is still intractable, namely this problem is $\mathcal{N}$ EXPTIME-complete as is the general case. The lower bound is obtained by showing that this kind of dynamic process graph s can represent certain Boolean formulas in a highly succint way.