Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from $n$ states to $2^n$ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with $n$ states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model, each state has a fixed probability to be final. We prove that for any $d\geq 1$, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than $n^d$ states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on $n$, as long as it is not too close to $0$ and $1$, at distance at least $\Omega(\frac1{\sqrt{n}})$ to be precise, therefore allowing models with a sublinear number of final states in expectation.

Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from n states to 2ⁿ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with n states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model each state has a fixed probability to be final. We prove that for any d ≥ 1, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than n^d states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on n, as long as it is not too close to 0 and 1, at distance at least Ω(1/√n) to be precise, therefore allowing models with a sublinear number of final states in expectation.

We consider the distributed Cholesky factorization on homogeneous nodes. Inspired by recent progress on asymptotic lower bounds on the total communication volume required to perform Cholesky factorization, we present an original data distribution, Symmetric Block Cyclic (SBC), designed to take advantage of the symmetry of the matrix. We prove that SBC reduces the overall communication volume between nodes by a factor of square root of 2 compared to the standard 2D block-cyclic distribution. SBC can easily be implemented within the paradigm of task-based runtime systems. Experiments using the Chameleon library over the StarPU runtime system demonstrate that the SBC distribution reduces the communication volume as expected, and also achieves better performance and scalability than the classical 2D block-cyclic allocation scheme in all configurations. We also propose a 2.5D variant of SBC and prove that it further improves the communication and performance benefits.

We give a probabilistic analysis of parameters related to alpha-gapped repeats and palindromes in random words, under both uniform and memoryless distributions (where letters have different probabilities, but are drawn independently). More precisely, we study the expected number of maximal alpha-gapped patterns, as well as the expected length of the longest alpha-gapped pattern in a random word.

We consider allocation problems that arise in the context of service allocation in Clouds. More specifically, we assume on the one part that each computing resource is associated with a capacity, that can be chosen using the Dynamic Voltage and Frequency Scaling (DVFS) method, and with a probability of failure. On the other hand, we assume that the services run as a set of independent instances of identical Virtual Machines (VMs). Moreover, there exists a Service Level Agreement (SLA) between the Cloud provider and the client that can be expressed as follows: the client comes with a minimal number of service instances that must be alive at anytime, and the Cloud provider offers a list of pairs (price, compensation), the compensation having to be paid by the Cloud provider if it fails to keep alive the required number of services. On the Cloud provider side, each pair actually corresponds to a guaranteed reliability of fulfilling the constraint on the minimal number of instances. In this context, given a minimal number of instances and a probability of success, the question for the Cloud provider is to find the number of necessary resources, their clock frequency and an allocation of the instances (possibly using replication) onto machines. This solution should satisfy all types of constraints (both capacity and reliability constraints). Moreover, it should remain valid during a time period (with a given reliability in presence of failures) while minimizing the energy consumption of used resources. We assume in this paper that this time period, that typically takes place between two redistributions, is fixed and known in advance. We prove deterministic approximation ratios on the consumed energy for algorithms that provide guaranteed reliability and we provide an extensive set of simulations that prove that homogeneous solutions are close to optimal.

Peer-to-peer overlay networks have proven their efficiency for storing and retrieving data at large scale, but new services are required to take the actual performances of resources into account. In this paper, we describe a fully decentralized algorithm, called "dating service" meant to organize communications in a fully heterogeneous network, that ensures that communication capabilities of the nodes are not exceeded. We prove that with high probability, this service ensures that a constant fraction of all possible communications is organized. Interestingly enough, this property holds true even if a node is not able to choose another node uniformly at random. In particular, the dating service can be implemented over existing DHT-based systems. In order to illustrate the expressiveness and the usefulness of proposed service, we also present a possible practical application of the dating service. As an illustration, we propose an algorithm for rumor spreading that enables to broadcast a unit-size message to all the nodes of a P2P system in logarithmic number of steps with high probability.

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We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-O(1/n). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node u and any radius r the number of nodes within distance 2r from u is at most a constant times the number of nodes within distance r, our augmentation process proceeds with high probability in O(log n log D) communication rounds, with O(log n log D) messages of size O(log n) bits sent per node and requiring only O(log n log D) bit space in each node, where n is the number of nodes, and D the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most O(log2 n log2 D) with probability 1 - O(1/n), and of expected length O(log n log2 D). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinberg's model as a plausible explanation for the small world phenomenon in large real interaction networks.