A Formal Model For Real-Time Parallel Computation

The imposition of real-time constraints on a parallel computing environment- specifically high-performance, cluster-computing systems- introduces a variety of challenges with respect to the formal verification of the system's timing properties. In this paper, we briefly motivate the need for such a system, and we introduce an automaton-based method for performing such formal verification. We define the concept of a consistent parallel timing system: a hybrid system consisting of a set of timed automata (specifically, timed Buchi automata as well as a timed variant of standard finite automata), intended to model the timing properties of a well-behaved real-time parallel system. Finally, we give a brief case study to demonstrate the concepts in the paper: a parallel matrix multiplication kernel which operates within provable upper time bounds. We give the algorithm used, a corresponding consistent parallel timing system, and empirical results showing that the system operates under the specified timing constraints.Comment: In Proceedings FTSCS 2012, arXiv:1212.657

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log m\})$ space to process this stream. For each $p \ge 1$, we give a very simple deterministic algorithm that makes $p$ passes over the input stream and returns an appropriately certified $(p+1)n^{1/(p+1)}$-approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than $0.99\,n^{1/(p+1)}/(p+1)^2$ is unachievable for a $p$-pass semi-streaming algorithm, even allowing randomisation. In particular, this implies that achieving a $\Theta(\log n)$-approximation requires $\Omega(\log n/\log\log n)$ passes, which is tight up to the $\log\log n$ factor. These results extend to a relaxation of the set cover problem where we are allowed to leave an $\varepsilon$ fraction of the universe uncovered: the tight bounds on the best approximation factor achievable in $p$ passes turn out to be $\Theta_p(\min\{n^{1/(p+1)}, \varepsilon^{-1/p}\})$. Our lower bounds are based on a construction of a family of high-rank incidence geometries, which may be thought of as vast generalisations of affine planes. This construction, based on algebraic techniques, appears flexible enough to find other applications and is therefore interesting in its own right.Comment: 20 page

Energy Efficiency and Renewable Energy Management with Multi-State Power-Down Systems

A power-down system has an on-state, an off-state, and a finite or infinite number of intermediate states. In the off-state, the system uses no energy and in the on-state energy it is used fully. Intermediate states consume only some fraction of energy but switching back to the on-state comes at a cost. Previous work has mainly focused on asymptotic results for systems with a large number of states. In contrast, the authors study problems with a few states as well as systems with one continuous state. Such systems play a role in energy-efficiency for information technology but are especially important in the management of renewable energy. The authors analyze power-down problems in the framework of online competitive analysis as to obtain performance guarantees in the absence of reliable forecasting. In a discrete case, the authors give detailed results for the case of three and five states, which corresponds to a system with on-off states and three additional intermediate states “power save”, “suspend”, and “hibernate”. The authors use a novel balancing technique to obtain optimally competitive solutions. With this, the authors show that the overall best competitive ratio for three-state systems is 95 role= presentation style= box-sizing: border-box; max-height: none; display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e95 and the authors obtain optimal ratios for various five state systems. For the continuous case, the authors develop various strategies, namely linear, optimal-following, progressive and exponential. The authors show that the best competitive strategies are those that follow the offline schedule in an accelerated manner. Strategy “progressive” consistently produces competitive ratios significantly better than 2

When--and how--can a cellular automaton be rewritten as a lattice gas?

Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the term `cellular automaton' or `lattice gas' for the dynamic system itself as well as for its presentation. The two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari (and by Durand--Lose for more than two dimensions) that any invertible CA can be rewritten as an LG (with a possibly much more complex ``unit cell''). But until now it was not known whether this is possible in general for noninvertible CA--which comprise ``almost all'' CA and represent the bulk of examples in theory and applications. Even circumstantial evidence--whether in favor or against--was lacking. Here, for noninvertible CA, (a) we prove that an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the rest--noninvertible and nonsurjective--which comprise all the typical ones, including Conway's `Game of Life'. For these (b) we prove by explicit construction that all the one-dimensional ones are representable as LG, and (c) we present and motivate the conjecture that this result extends to any number of dimensions. The tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale.Comment: 16 page