Unfaithful Glitch Propagation in Existing Binary Circuit Models

We show that no existing continuous-time, binary value-domain model for digital circuits is able to correctly capture glitch propagation. Prominent examples of such models are based on pure delay channels (P), inertial delay channels (I), or the elaborate PID channels proposed by Bellido-D\'iaz et al. We accomplish our goal by considering the solvability/non-solvability border of a simple problem called Short-Pulse Filtration (SPF), which is closely related to arbitration and synchronization. On one hand, we prove that SPF is solvable in bounded time in any such model that provides channels with non-constant delay, like I and PID. This is in opposition to the impossibility of solving bounded SPF in real (physical) circuit models. On the other hand, for binary circuit models with constant-delay channels, we prove that SPF cannot be solved even in unbounded time; again in opposition to physical circuit models. Consequently, indeed none of the binary value-domain models proposed so far (and that we are aware of) faithfully captures glitch propagation of real circuits. We finally show that these modeling mismatches do not hold for the weaker eventual SPF problem.Comment: 23 pages, 15 figure

New transience bounds for long walks in weighted digraphs

International audienceWe consider the sequence of maximal weights of walks of lengt n between two fixed nodes in a weighted digraph. It is known that these sequences show a periodic behavior after an initial transient. We identify relevant graph parameters and propose a modular strategy to derive new upper bounds on the transient. To the best of our knowledge, our bounds are the first that are both asymptotically tight and potentially subquadratic. In particular, the new bounds show that the transient is linear in the number of nodes in bi-directional trees. Besides, our results enable a fine-grained performance analysis and give guidelines for the design of distributed systems based on max-plus recursions

Diffusive clock synchronization in highly dynamic networks

International audienceThis paper studies the clock synchronization problem in highly dynamic networks. We show that diffusive synchronization algorithms are well adapted to environments in which the network topology may change unpredictably. In a diffusive algorithm, each node repeatedly (i) estimates the clock difference to its neighbors via broadcast of zero-bit messages, and (ii) updates its local clock according to a weighted average of the estimated differences. The system model allows for drifting local clocks, running at possibly different frequencies. We show that having a rooted spanning tree in the network at every time instance suffices to solve clock synchronization. We do not require any stability of the spanning tree, nor do we impose that the links of the spanning tree be known to the nodes. Explicit bounds on the convergence speed are obtained. In particular, our results settle an open question posed by Simeone and Spagnolini to reach clock synchronization in dynamic networks in the presence of nonzero clock drift. We also identify certain reasonable assumptions that allow for a significant higher convergence speed, e.g., bidirectional networks or random graph models

Transience Bounds for Distributed Algorithms

International audienceA large variety of distributed systems, like some classical synchronizers, routers, or schedulers, have been shown to have a periodic behavior after an initial transient phase (Malka and Rajsbaum, WDAG 1991). In fact, each of these systems satisfies recurrence relations that turn out to be linear as soon as we consider max-plus or min-plus algebra. In this paper, we give a new proof that such systems are eventually periodic and a new upper bound on the length of the initial transient phase. Interestingly, this is the first asymptotically tight bound that is linear in the system size for various classes of systems. Another significant benefit of our approach lies in the straightforwardness of arguments: The proof is based on an easy convolution lemma borrowed from Nachtigall (Math. Method. Oper. Res. 46) instead of purely graph-theoretic arguments and involved path reductions found in all previous proofs