Hamamache Kheddouci

A survey on self-stabilizing algorithms for independence, domination, coloring, and matching in graphs

Dijkstra defined a distributed system to be self-stabilizing if, regardless of the initial state, the system is guaranteed to reach a legitimate (correct) state in a finite time. Even though the concept of self-stabilization received little attention when it was introduced, it has become one of the most popular fault tolerance approaches. On the other hand, graph algorithms form the basis of many network protocols. They are used in routing, clustering, multicasting and many other tasks. The objective of this paper is to survey the self-stabilizing algorithms for dominating and independent set problems, colorings, and matchings. These graph theoretic problems are well studied in the context of self-stabilization and a large number of algorithms have been proposed for them.

ECGK: An efficient clustering scheme for group key management in MANETs

Mobile Ad hoc NETworks (or MANETs) are flexible networks that are expected to support emerging group applications such as spontaneous collaborative activities and rescue operations. In order to provide secrecy to these applications, a common encryption key has to be established between group members of the application. This task is critical in MANETs because these networks have no fixed infrastructure, frequent node and link failures and a dynamic topology. The proposed approaches to cope with these characteristics aim to avoid centralized solutions and organize the network into clusters. However, the clustering criteria used in the literature are not always adequate for key management and security. In, this paper, we propose, a group key management framework based on a trust oriented clustering scheme. We show that trust is a relevant clustering criterion for group key management in MANETs. Trust information enforce authentication and is disseminated by the mobility of nodes. Furthermore, it helps to evict malicious nodes from the multicast session even if they are authorized members of the group. Simulation results show that our solution is efficient and typically adapted to mobility of nodes.

Exact values for the b-chromatic number of a power complete k-ary tree

Abstract Let G be a graph on vertices x1, x2,…, xn . The b-chromatic number of G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i ≤ k, has at least one representant xi adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we give the exact value for the b-chromatic number of power graphs of a complete binary tree.

A new clustering approach for symbolic data and its validation: application to the healthcare data

Graph coloring is used to characterize some properties of graphs. A b-coloring of a graph G (using colors 1,2,...,k) is a coloring of the vertices of G such that (i) two neighbors have different colors (proper coloring) and (ii) for each color class there exists a dominating vertex which is adjacent to all other k-1 color classes. In this paper, based on a b-coloring of a graph, we propose a new clustering technique. Additionally, we provide a cluster validation algorithm. This algorithm aims at finding the optimal number of clusters by evaluating the property of color dominating vertex. We adopt this clustering technique for discovering a new typology of hospital stays in the French healthcare system.

A cluster based mobility prediction scheme for ad hoc networks

The ad hoc networks are completely autonomous wireless networks where all the users are mobile. These networks do not work on any infrastructure and the mobiles communicate either directly or via other nodes of the network by establishing routes. These routes are prone to frequent ruptures because of nodes mobility. If the future movement of the mobile can be predicted in a precise way, the resources reservation can be made before be asked, which enables the network to provide a better QoS. In this aim, we propose a virtual dynamic topology, which on one hand, will organize the network as well as possible and decreases the impact of mobility, and on the other hand, is oriented user mobility prediction. Our prediction scheme uses the evidence theory of Dempster-Shafer in order to predict the future position of the mobile by basing itself on relevant criteria. These ones are related to mobility and network operation optimisation. The proposed scheme is flexible and can be extended to a general framework. To show the relevance of our scheme, we combine it with a routing protocol. Then, we implemented the prediction-oriented topology and the prediction scheme which performs on it. We implemented also a mobility prediction based routing protocol. Simulations are made according to a set of elaborate scenarios.

Efficient distributed lifetime optimization algorithm for sensor networks

One of the main design challenges in Wireless Sensor Networks (WSN) is to prolong the system lifetime, while achieving acceptable quality of service for applications. In WSN, each sensor node is battery powered and it is not convenient to recharge or replace the batteries in many cases, especially in remote and hostile environments. Due to the limited capabilities of sensor nodes, it is usually desirable that a WSN should be deployed with high density and thus redundancy can be exploited to increase the networks lifetime. In this paper, we introduce an efficient lifetime optimization and self-stabilizing algorithm to enhance the lifetime of wireless sensor networks especially when the reliabilities of sensor nodes are expected to decrease due to use and wear-out effects. Our algorithm seeks to build resiliency by maintaining a necessary set of working nodes and replacing failed ones when needed. We provide some theoretical and simulation results, that fully demonstrate the usefulness of the proposed algorithm.

A Distance Measure for Large Graphs based on Prime Graphs

Abstract Graphs are universal modeling tools. They are used to represent objects and their relationships in almost all domains: they are used to represent DNA, images, videos, social networks, XML documents, etc. When objects are represented by graphs, the problem of their comparison is a problem of comparing graphs. Comparing objects is a key task in our daily life. It is the core of a search engine, the backbone of a mining tool, etc. Nowadays, comparing objects faces the challenge of the large amount of data that this task must deal with. Moreover, when graphs are used to model these objects, it is known that graph comparison is very complex and computationally hard especially for large graphs. So, research on simplifying graph comparison gainedan interest and several solutions are proposed. In this paper, we explore and evaluate a new solution for the comparison of large graphs. Our approach relies on a compact encoding of graphs called prime graphs. Prime graphs are smaller and simpler than the original ones but they retain the structure and properties of the encoded graphs. We propose to approximate the similarity between two graphs by comparing the corresponding prime graphs. Simulations results show that this approach is effective for large graphs.

Grundy number of graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 <= i <= k, is adjacent to (i-1) vertices colored with each color j, 1 <= j <= i-1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an algorithm to generate all graphs for a given Grundy number.

A strict strong coloring of trees

In this paper we introduce and study a new coloring parameter of a graph called the strict strong coloring (short SSColoring). A SSColoring of a graph G is a vertex proper coloring of G such that each vertex of G is adjacent to at least one not empty color class. The minimum number of colors among all SSColorings is called strict strong chromatic (short SSChromatic) number, denoted by @gss(G). In this paper we prove the NP-completeness of the problem, we discuss the @gss(G) number of trees by giving some bounds. Finally, we give an optimal polynomial algorithm for SSColoring of trees.

Component-cardinality-constrained critical node problem in graphs

An effective way to analyze and apprehend the structural properties of networks is to find their most critical nodes. This makes them easier to control, whether the purpose is to keep or to delete them. Given a graph, Critical Node Detection problem (CNDP) consists in finding a set of nodes, deletion of which satisfies some given connectivity metrics in the induced graph. In this paper, we propose and study a new variant of this problem, called Component-Cardinality-Constrained Critical Node Problem (3C-CNP). In this variant, we seek to find a minimal set of nodes, removal of which constrains the size of each connected component in the induced graph to a given bound. We prove the NP-hardness of this problem on a graph of maximum degree Δ = 4 , through which we deduce the NP-hardness of CNP (Arulselvan etźal., 2009) on the same class of graphs. Also, we study 3C-CNP on trees for different cases depending on node weights and connection costs. For the case where node weights and connection costs have non-negative values, we prove its NP-completeness. While, for the case where node weights (or connection costs) have unit values, we present a polynomial algorithm. Also, we study 3C-CNP on chordal graphs, where we show that it is NP-complete on split graphs, and polynomially solvable on proper interval graphs.