Design of Self-Stabilizing Approximation Algorithms via a Primal-Dual Approach

Self-stabilization is an important concept in the realm of fault-tolerant distributed computing. In this paper, we propose a new approach that relies on the properties of linear programming duality to obtain self-stabilizing approximation algorithms for distributed graph optimization problems. The power of this new approach is demonstrated by the following results: - A self-stabilizing 2(1+?)-approximation algorithm for minimum weight vertex cover that converges in O(log? /(?log log ?)) synchronous rounds. - A self-stabilizing ?-approximation algorithm for maximum weight independent set that converges in O(?+log^* n) synchronous rounds. - A self-stabilizing ((2?+1)(1+?))-approximation algorithm for minimum weight dominating set in ?-arboricity graphs that converges in O((log?)/?) synchronous rounds. In all of the above, ? denotes the maximum degree. Our technique improves upon previous results in terms of time complexity while incurring only an additive O(log n) overhead to the message size. In addition, to the best of our knowledge, we provide the first self-stabilizing algorithms for the weighted versions of minimum vertex cover and maximum independent set

Improving Connectionist Energy Minimization

Symmetric networks designed for energy minimization such as Boltzman machines and Hopfield nets are frequently investigated for use in optimization, constraint satisfaction and approximation of NP-hard problems. Nevertheless, finding a global solution (i.e., a global minimum for the energy function) is not guaranteed and even a local solution may take an exponential number of steps. We propose an improvement to the standard local activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm, called activate, is uniform and does not assume that the network is tree-like. It can identify tree-like subnetworks even in cyclic topologies (arbitrary networks) and avoid local minima along these trees. For acyclic networks, the algorithm is guaranteed to converge to a global minimum from any initial state of the system (self-stabilization) and remains correct under various types of schedulers. On the negative side, we show that in the presence of cycles, no uniform algorithm exists that guarantees optimality even under a sequential asynchronous scheduler. An asynchronous scheduler can activate only one unit at a time while a synchronous scheduler can activate any number of units in a single time step. In addition, no uniform algorithm exists to optimize even acyclic networks when the scheduler is synchronous. Finally, we show how the algorithm can be improved using the cycle-cutset scheme. The general algorithm, called activate-with-cutset, improves over activate and has some performance guarantees that are related to the size of the network's cycle-cutset.Comment: See http://www.jair.org/ for any accompanying file

Self-stabilizing k-clustering in mobile ad hoc networks

In this thesis, two silent self-stabilizing asynchronous distributed algorithms are given for constructing a k-clustering of a connected network of processes. These are the first self-stabilizing solutions to this problem. One algorithm, FLOOD, takes O( k) time and uses O(k log n) space per process, while the second algorithm, BFS-MIS-CLSTR, takes O(n) time and uses O(log n) space; where n is the size of the network. Processes have unique IDs, and there is no designated leader. BFS-MIS-CLSTR solves three problems; it elects a leader and constructs a BFS tree for the network, constructs a minimal independent set, and finally a k-clustering. Finding a minimal k-clustering is known to be NP -hard. If the network is a unit disk graph in a plane, BFS-MIS-CLSTR is within a factor of O(7.2552k) of choosing the minimal number of clusters; A lower bound is given, showing that any comparison-based algorithm for the k-clustering problem that takes o( diam) rounds has very bad worst case performance; Keywords: BFS tree construction, K-clustering, leader election, MIS construction, self-stabilization, unit disk graph

Making Self-Stabilizing Algorithms for Any Locally Greedy Problem

Self-stabilizing algorithms are a way to deal with network dynamicity, as it will update itself after a network change (addition or removal of nodes or edges), as long as changes are not frequent. We propose an automatic transformation of synchronous distributed algorithms that solve locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks. Mendable problems are a generalization of greedy problems where any partial solution may be transformed -instead of completed- into a global solution: every time we extend the partial solution, we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it. In order to do this, we propose the first explicit self-stabilizing algorithm computing a (k,k-1)-ruling set (i.e. a "maximal independent set at distance k"). By combining this technique multiple times, we compute a distance-K coloring of the graph. With this coloring we can finally simulate Local model algorithms running in a constant number of rounds, using the colors as unique identifiers. Our algorithms work under the Gouda daemon, similar to the probabilistic daemon: if an event should eventually happen, it will occur

The (k,l) Coredian tree for Ad Hoc Networks

In this paper, we present a new efficient strategy for constructing a wireless tree network containing n nodes of diameter ∆ while satisfying the QoS requirements such as bandwidth and delay. Given a tree network T , a coredian path is a path in T that minimizes the centdian function, a k-coredian tree is a subtree of T with k leaves that minimizes the centdian function, and a (k, l)-coredian tree is a subtree of T with k leaves and diameter l at most that minimizes the centdian function. The (k, l)-coredian tree can serve as a backbone for a network, where the internal nodes belong to the backbone and the leaves serve as the heads of the clusters covering the rest of the network. We show that a coredian path can be constructed at O(∆) time with O(n) messages and a k-coredian tree can be constructed at O(k∆) time with O(kn) messages. We provide an O(n 2 ) time construction algorithm for the (k, l)-coredian tree that requires O(n 2 ) messages. We also give upper and lower bounds for a number of nodes covered by the k cluster heads in random geometric graph using critical transmission range of connected network. Finally, simulation is presented for various values of n and k

Optimization in a Self-Stabilizing Service Discovery Framework for Large Scale Systems

Ability to find and get services is a key requirement in the development of large-scale distributed sys- tems. We consider dynamic and unstable environments, namely Peer-to-Peer (P2P) systems. In previous work, we designed a service discovery solution called Distributed Lexicographic Placement Table (DLPT), based on a hierar- chical overlay structure. A self-stabilizing version was given using the Propagation of Information with Feedback (PIF) paradigm. In this paper, we introduce the self-stabilizing COPIF (for Collaborative PIF) scheme. An algo- rithm is provided with its correctness proof. We use this approach to improve a distributed P2P framework designed for the services discovery. Significantly efficient experimental results are presented