Beeping a Maximal Independent Set

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.Comment: arXiv admin note: substantial text overlap with arXiv:1108.192

Graph Relabelling Systems A Tool for Encoding, Proving, Studying and Visualizing Distributed Algorithms

International audienc

About randomised distributed graph colouring and graph partition algorithms

AbstractWe present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O(logn) with probability 1-o(n-1). Each message containing 1 bit, its bit complexity per channel is O(logn).From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree Δ and size n that uses at most Δ+1 colours and halts in time O(logn) with probability 1-o(n-1).We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O(logn) with probability 1-o(n-1). We study some parameters such as the number, the size and the radius of trees of the spanning forest

Jump-diffusion unravelling of a non Markovian generalized Lindblad master equation

The "correlated-projection technique" has been successfully applied to derive a large class of highly non Markovian dynamics, the so called non Markovian generalized Lindblad type equations or Lindblad rate equations. In this article, general unravellings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unravelling can be interpreted in terms of measurements continuous in time, but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and we discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.Comment: 23 page

Visualization of Distributed Algorithms Based on Graph Relabelling Systems1 1This work has been supported by the European TMR research network GETGRATS, and by the “Conseil Régional d' Aquitane”.

AbstractIn this paper, we present a uniform approach to simulate and visualize distributed algorithms encoded by graph relabelling systems. In particular, we use the distributed applications of local relabelling rules to automatically display the execution of the whole distributed algorithm. We have developed a Java prototype tool for implementing and visualizing distributed algorithms. We illustrate the different aspects of our framework using various distributed algorithms including election and spanning trees

Sigmoid stricture associated with diverticular disease should be an indication for elective surgery with lymph node clearance

BACKGROUND: The literature concerning stricture secondary to diverticulitis is poor. Stricture in this setting should be an indication for surgery because (a) of the potential risk of cancer and (b) morbidity is not increased compared to other indications for colectomy. The goal of this report is to study the post-surgical morbidity and the quality of life in patients after sigmoidectomy for sigmoid stricture associated with diverticular disease. METHOD: This is a monocenter retrospective observational study including patients with a preoperative diagnosis of sigmoid stricture associated with diverticular disease undergoing operation between Jan 1, 2007 and Dec 31, 2013. The GastroIntestinal Quality of Life Index was used to assess patient satisfaction. RESULTS: Sixteen patients were included of which nine were female. Median age was 69.5 (46-84) and the median body mass index was 23.55kg/m(2) (17.2-28.4). Elective sigmoidectomy was performed in all 16 patients. Overall, complications occurred in five patients (31.2%) (4 minor complications and 1 major complication according to the Dindo and Clavien Classification); none resulted in death. Pathology identified two adenocarcinomas (12.5%). The mean GastroIntestinal Quality of Life Index was 122 (67-144) and 10/11 patients were satisfied with their surgical intervention. CONCLUSION: Sigmoid stricture prevents endoscopic exploration of the entire colon and thus it may prove difficult to rule out a malignancy. Surgery does not impair the quality of life since morbidity is similar to other indications for sigmoidectomy. For these reasons, we recommend that stricture associated with diverticular disease should be an indication for sigmoidectomy including lymph node clearance

Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying H\"ormander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity