Efficient self-stabilizing construction of disjoint MDSs in distance-2 model

We study the deterministic silent self-stabilizing construction of two disjoint minimal dominating sets (MDSs) in anonymous networks. We focus on algorithms where nodes share only their status (i.e. the name of their MDS to which they belong, if they belong to a MDS). We prove that such an algorithm cannot be designed in distance-1 model under a central daemon; therefore, we study this problem in the distance-2 model under a central daemon. We present an algorithm building two disjoint minimal dominating sets such that one of them is also a maximal independent set (MIS). Any execution of this algorithm converges in 5n moves. Our approach to compute this value is novel: the number of moves is not computed per node. We propose a second algorithm faster than the first one at the expense of the independence property of one of the constructed sets. A node executes at most 2 moves. If the network is not anonymous, the presented algorithms can be translated into a silent self-stabilizing algorithms converging in O($n$•$m$) moves in the distance-1 model under the distributed daemon where m is the number of edges and n the number of nodes. This improves the complexity of O($n$.$m$) moves of proposed algorithms with the same assumptions

Client–server and cost effective sets in graphs

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Stabilization Time in Weighted Minority Processes

A minority process in a weighted graph is a dynamically changing coloring. Each node repeatedly changes its color in order to minimize the sum of weighted conflicts with its neighbors. We study the number of steps until such a process stabilizes. Our main contribution is an exponential lower bound on stabilization time. We first present a construction showing this bound in the adversarial sequential model, and then we show how to extend the construction to establish the same bound in the benevolent sequential model, as well as in any reasonable concurrent model. Furthermore, we show that the stabilization time of our construction remains exponential even for very strict switching conditions, namely, if a node only changes color when almost all (i.e., any specific fraction) of its neighbors have the same color. Our lower bound works in a wide range of settings, both for node-weighted and edge-weighted graphs, or if we restrict minority processes to the class of sparse graphs

A General Stabilization Bound for Influence Propagation in Graphs

We study the stabilization time of a wide class of processes on graphs, in which each node can only switch its state if it is motivated to do so by at least a $\frac{1+\lambda}{2}$ fraction of its neighbors, for some $0 < \lambda < 1$. Two examples of such processes are well-studied dynamically changing colorings in graphs: in majority processes, nodes switch to the most frequent color in their neighborhood, while in minority processes, nodes switch to the least frequent color in their neighborhood. We describe a non-elementary function $f(\lambda)$, and we show that in the sequential model, the worst-case stabilization time of these processes can completely be characterized by $f(\lambda)$. More precisely, we prove that for any $\epsilon>0$, $O(n^{1+f(\lambda)+\epsilon})$ is an upper bound on the stabilization time of any proportional majority/minority process, and we also show that there are graph constructions where stabilization indeed takes $\Omega(n^{1+f(\lambda)-\epsilon})$ steps

Stabilization Bounds for Influence Propagation from a Random Initial State

We study the stabilization time of two common types of influence propagation. In majority processes, nodes in a graph want to switch to the most frequent state in their neighborhood, while in minority processes, nodes want to switch to the least frequent state in their neighborhood. We consider the sequential model of these processes, and assume that every node starts out from a uniform random state. We first show that if nodes change their state for any small improvement in the process, then stabilization can last for up to $\Theta(n^2)$ steps in both cases. Furthermore, we also study the proportional switching case, when nodes only decide to change their state if they are in conflict with a $\frac{1+\lambda}{2}$ fraction of their neighbors, for some parameter $\lambda \in (0,1)$. In this case, we show that if $\lambda < \frac{1}{3}$, then there is a construction where stabilization can indeed last for $\Omega(n^{1+c})$ steps for some constant $c>0$. On the other hand, if $\lambda > \frac{1}{2}$, we prove that the stabilization time of the processes is upper-bounded by $O(n \cdot \log{n})$

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Probabilistic methods for distributed information dissemination

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 457-484).The ever-increasing growth of modern networks comes with a paradigm shift in network operation. Networks can no longer be abstracted as deterministic, centrally controlled systems with static topologies but need to be understood as highly distributed, dynamic systems with inherent unreliabilities. This makes many communication, coordination and computation tasks challenging and in many scenarios communication becomes a crucial bottleneck. In this thesis, we develop new algorithms and techniques to address these challenges. In particular we concentrate on broadcast and information dissemination tasks and introduce novel ideas on how randomization can lead to powerful, simple and practical communication primitives suitable for these modern networks. In this endeavor we combine and further develop tools from different disciplines trying to simultaneously addresses the distributed, information theoretic and algorithmic aspects of network communication. The two main probabilistic techniques developed to disseminate information in a network are gossip and random linear network coding. Gossip is an alternative to classical flooding approaches: Instead of nodes repeatedly forwarding information to all their neighbors, gossiping nodes forward information only to a small number of (random) neighbors. We show that, when done right, gossip disperses information almost as quickly as flooding, albeit with a drastically reduced communication overhead. Random linear network coding (RLNC) applies when a large amount of information or many messages are to be disseminated. Instead of routing messages through intermediate nodes, that is, following a classical store-and-forward approach, RLNC mixes messages together by forwarding random linear combinations of messages. The simplicity and topology-obliviousness of this approach makes RLNC particularly interesting for the distributed settings considered in this thesis. Unfortunately the performance of RLNC was not well understood even for the simplest such settings. We introduce a simple yet powerful analysis technique that allows us to prove optimal performance guarantees for all settings considered in the literature and many more that were not analyzable so far. Specifically, we give many new results for RLNC gossip algorithms, RLNC algorithms for dynamic networks, and RLNC with correlated data. We also provide a novel highly efficient distributed implementation of RLNC that achieves these performance guarantees while buffering only a minimal amount of information at intermediate nodes. We then apply our techniques to improve communication primitives in multi-hop radio networks. While radio networks inherently support broadcast communications, e.g., from one node to all surrounding nodes, interference of simultaneous transmissions makes multihop broadcast communication an interesting challenge. We show that, again, randomization holds the key for obtaining simple, efficient and distributed information dissemination protocols. In particular, using random back-off strategies to coordinate access to the shared medium leads to optimal gossip-like communications and applying RLNC achieves the first throughput-optimal multi-message communication primitives. Lastly we apply our probabilistic approach for analyzing simple, distributed propagation protocols in a broader context by studying algorithms for the Lovász Local Lemma. These algorithms find solutions to certain local constraint satisfaction problems by randomly fixing and propagating violations locally. Our two main results show that, firstly, there are also efficient deterministic propagation strategies achieving the same and, secondly, using the random fixing strategy has the advantage of producing not just an arbitrary solution but an approximately uniformly random one. Both results lead to simple, constructions for a many locally consistent structures of interest that were not known to be efficiently constructable before.by Bernhard Haeupler.Ph.D