scholarly journals LEARNING AUTOMATA-BASED ALGORITHMS FOR FINDING MINIMUM WEAKLY CONNECTED DOMINATING SET IN STOCHASTIC GRAPHS

2010 ◽  
Vol 18 (06) ◽  
pp. 721-758 ◽  
Author(s):  
JAVAD AKBARI TORKESTANI ◽  
MOHAMMAD REZA MEYBODI
Keyword(s):  
Dominating Set ◽  
Learning Automata ◽  
Sampling Rate ◽  
Connected Dominating Set ◽  
Stochastic Graphs ◽  
Vertex Set ◽  
Stochastic Graph ◽  
Standard Sampling ◽  
Definition Of ◽  
Weakly Connected Dominating Set

A weakly connected dominating set (WCDS) of graph G is a subset of G so that the vertex set of the given subset and all vertices with at least one endpoint in the subset induce a connected sub-graph of G. The minimum WCDS (MWCDS) problem is known to be NP-hard, and several approximation algorithms have been proposed for solving MWCDS in deterministic graphs. However, to the best of our knowledge no work has been done on finding the WCDS in stochastic graphs. In this paper, a definition of the MWCDS problem in a stochastic graph is first presented and then several learning automata-based algorithms are proposed for solving the stochastic MWCDS problem where the probability distribution function of the weight associated with the graph vertices is unknown. The proposed algorithms significantly reduce the number of samples needs to be taken from the vertices of the stochastic graph. It is shown that by a proper choice of the parameters of the proposed algorithms, the probability of finding the MWCDS is as close to unity as possible. Experimental results show the major superiority of the proposed algorithms over the standard sampling method in terms of the sampling rate.

scholarly journals Finding Maximum Clique in Stochastic Graphs Using Distributed Learning Automata

2015 ◽  
Vol 23 (01) ◽  
pp. 1-31 ◽  
Author(s):  
Alireza Rezvanian ◽  
Mohammad Reza Meybodi
Keyword(s):  
Community Structure ◽  
Dominating Set ◽  
Learning Automata ◽  
Random Variables ◽  
Graph Model ◽  
Maximum Clique ◽  
Distribution Functions ◽  
Maximum Clique Problem ◽  
Stochastic Graphs ◽  
Stochastic Graph

Because of unpredictable, uncertain and time-varying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, every feature of the graph such as path, clique, spanning tree and dominating set, to mention a few, should be treated as a stochastic feature. For example, choosing stochastic graph as the graph model of an online social network and defining community structure in terms of clique, and the associations among the individuals within the community as random variables, the concept of stochastic clique may be used to study community structure properties. In this paper maximum clique in stochastic graph is first defined and then several learning automata-based algorithms are proposed for solving maximum clique problem in stochastic graph where the probability distribution functions of the weights associated with the edges of the graph are unknown. It is shown that by a proper choice of the parameters of the proposed algorithms, one can make the probability of finding maximum clique in stochastic graph as close to unity as possible. Experimental results show that the proposed algorithms significantly reduce the number of samples needed to be taken from the edges of the stochastic graph as compared to the number of samples needed by standard sampling method at a given confidence level.

New Self-Stabilizing Algorithms for Minimal Weakly Connected Dominating Sets

2015 ◽  
Vol 26 (02) ◽  
pp. 229-240
Author(s):  
Yihua Ding ◽  
James Z. Wang ◽  
Pradip K. Srimani
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Graph Algorithm ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Time And Space ◽  
Stabilization Time ◽  
Connected Dominating Sets ◽  
Space Requirement ◽  
Weakly Connected Dominating Set

In this paper, we propose two new self-stabilizing algorithms, MWCDS-C and MWCDS-D, for minimal weakly connected dominating sets in an arbitrary connected graph. Algorithm MWCDS-C stabilizes in O(n4) steps using an unfair central daemon and space requirement at each node is O(log n) bits at each node for an arbitrary connected graph with n nodes; it uses a designated node while other nodes are identical and anonymous. Algorithm MWCDS-D stabilizes using an unfair distributed daemon with identical time and space complexities, but it assumes unique node IDs. In the literature, the best reported stabilization time for a minimal weakly connected dominating set algorithm is O(nmA) under a distributed daemon [1], where m is the number of edges and A is the number of moves to construct a breadth-first tree.

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