15 82 87

Page 1

Proc. of Int. Conf. on Recent Trends in Information, Telecommunication and Computing, ITC

On the Linear Time Construction of Minimum Spanning Tree Awadhesh Kumar Singh1, Ashish Negi2, Manish Kumar3, Vivek Rathee4 and Bharti Sharma5 1

1,2,3,4,5 National Institute of Technology Kurukshetra, Haryana, INDIA aksinreck@rediffmail.com, 2ashishnegi33@gmail.com, 3manishgl08@gmail.com, 4 rockonvivek@gmail.com, 5bharti_kanhiya@yahoo.co.in

Abstract— The article presents a simple algorithm to construct minimum spanning tree and to find shortest path between pair of vertices in a graph. Our illustration includes the proof of termination. The complexity analysis and simulation results have also been included. Index Terms— graph, spanning tree, ad hoc network

I. INTRODUCTION The graphical structures are popularly used to model computer networks. The spanning tree is one such structure. A spanning tree of a graph is the subgraph containing all the vertices and is a tree. The sum of its edge weights is called weight of the spanning tree. The smallest weight spanning tree, among all the possible spanning trees of a graph, is called minimum spanning tree (henceforth, MST). The ad hoc networks undergo topological changes due to the node movement or failure. Hence, the MST modeling such networks often needs to be reorganized. It is called the dynamic maintenance of MST. As, in the ad hoc network, nodes are energy constrained and the topological changes may be frequent, the MST computation method should be light weight and fast. The popular algorithms available in the literature, to compute MST, can be placed in two broad categories, namely, message efficient and time efficient. The message efficient algorithms, e.g. GHS algorithm [1], Chin-Ting [2], Gafni [3], and Awerbuch algorithm [4], exhibit linear or super linear time complexity; nevertheless, they are message optimal. On the other hand, the time efficient algorithms, e.g. Garay-Kutten-Peleg algorithm [5], Kutten-Peleg algorithm [6], and Elkin [7], exhibit sublinear time complexity; however, they are not message optimal. We present an algorithm to compute MST. Though, the algorithm is centralized, it exhibits message complexity better than algorithms [1–4] while keeping the time complexity order linear. II. THE ALGORITHM CONCEPT We consider a mobile ad hoc network (MANET) modeled as an undirected connected graph G = (V, E), where V is the set of vertices (nodes) and E is the set of edges (communication links) between them. Each edge eE has non-zero weight w. Each node has unique Id. Any two nodes are called neighbors if they are one hop away from each other and communicate directly. Also, we assume that despite multipath effect and varying channel conditions the message propagation between neighbor nodes is FIFO. We aim to collect the entire graph information at one or more nodes and use this information to create MST and to find the shortest paths between nodes. We present two methods for collecting the entire graph information. In the first method, all the graph information gets converged at a single node, which is not fixed a priori, called central DOI: 02.ITC.2014.5.15 © Association of Computer Electronics and Electrical Engineers, 2014


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.