An Analysis of Least-Cost Routing using Bellman Ford and Dijkstra Algorithms in Wireless Routing Network

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1 An Analysis of Least-Cost Routing using Bellman Ford and Dijkstra Algorithms in Wireless Routing Network Centre for Telecommunication Research and Innovation (CeTRI), Faculty of Electronic and Computer Engineering, Universiti Teknikal Malaysia Melaka, Durian Tunggal, Melaka, Malaysia Abstract In this paper, two different shortest paths routing algorithms in respect of performance are discussed. The first one is Dijkstra algorithm in routing network. The second one is a method that can be used in negative cycles known as Bellman Ford Algorithm. The paper is about a network creation with 9 nodes routes using MATLAB environment. Both algorithms have been applied and their performances are compared in terms of attenuation versus distance and throughput versus traffic load, a fully comparison table has been created and discussed. Keywords: Djikstra algorithm, Bellman-Ford algorithm, performance, attenuation 1. Introduction The evolution of networking and computer information technology caused a wide attention on research about graphical theory was reported in the last few years. Besides, a variety of graphical structures and algorithms have been projected in many fields of science too. Nowadays, the shortest path algorithm has become one of the most hotspot researches in graph theory and its algorithms. In addition, shortest path algorithm is used to control traffic engineering in Internet Protocol networks and to enhance networking and communications systems [1]. A lot of protocols have been tested in the wireless sensor networks such as Public Key Matrix [2], Ant Colony Principle [3], Clustering Hierarchical Structure [4] and many others. Dijkstra's and Bellman-Ford are the most popular algorithms in computer science and networking. They are also widely used in operations research. They have many attractions for both computer architectures and networks. Unfortunately, a number of important features of these interesting algorithms are not as clear to the normal users as they should be. Specifically, one of the results of the history of the algorithm is that they are generally regarded as a computer science method rather than as an operation research method. One of the main reasons for the acceptance of Dijkstra's and Bellman Algorithms is that they are the most important and useful algorithms existing for producing (precise) optimum solutions to most classes of shortest path problems. The point being that these classes of problems are very important in theoretical and educational purposes [5]. Dijkstra s algorithm is the route algorithm prepared by the Dutch computer scientist Edsger Dijkstra in 1956 and published in It is a graph search algorithm that explains the singlesource shortest route problem for a graph with nonnegative valued edge route costs. Therefore, it produces the shortest path tree. This system is widely used in routing and also as a subroutine in other graph algorithms. The disadvantage of the algorithm is that it cannot handle maps with negative path weights. The Bellman-Ford algorithm computes single-source shortest paths in a weighted graph or digraph, where some of the edge weights may be negative. This algorithm is a modification of the one published in 1952 by Richard E. Bellman [6] and that by Lester Randolph Ford [7]. Many applications can be used using these algorithms such as routing in wireless network for disaster situation [8], intelligent fire evacuation system [9], defense system International Journal of Advancements in Computing Technology(IJACT) Volume 5, Number 10, June 2013 doi : /ijact.vol5.issue

2 for mission planning approach using UAV [10], and also in automatic indoor navigation system [11]. 2. Djikstra and Bellman-Ford Algorithm Theory For a specified source vertex (node), the algorithm calculates all paths and finds out the path with lowest cost (i.e. the shortest distance) between that node and every other node in the network. It can also be used in calculating costs of shortest paths from a single node source to a single destination node by discontinuing the algorithm once the shortest path to the destination node has been reached. For example, if the node of the graph represents hosts and edge path costs represent path distances between pairs of nodes connected by a direct link, Dijkstra s algorithm can be used to find the minimum cost or shortest link path between one host and all other hosts in the network. Moreover, Dijkstra algorithm was highly motivated by Bellman's Principle of Optimality and that both theoretically and optimality related to the mathematical optimization method known as dynamic programming. One of the immediate implications of this perspective is that this widely used algorithm can be combined in the dynamic programming. The Bellman Ford algorithm sometimes denoted to as the Label Correcting Algorithm LCA, computes single-source shortest paths in a weighted routes (where some of the links weights may be negative). On the other hand and unlike Dijkstra, Bellman-Ford algorithm does not choose optimal set of nodes to scan: it simply iterates at all routes that may face in any directions. For each node, the algorithm may set as a "predecessor". At the end of the calculated route, the path from the distention to the source point can be traced by tracking the node, which was named as a predecessor in the path. Bellman-Ford algorithm goes over edges, not over nodes (vertices) of the graph, and gives all edges a directional value (thus route from node A to B can have different values comparing to the route from node B to A). A couple of reversed edges must be counted for every connection if the node relations in the task are not directional. The purpose of both algorithms describes the way to find the nodes shortest path problem. The primary difference in these algorithms is the fact that Dijkstra carries the overall information of the network in every node. On the other hand, Bellman-Ford algorithm doesn t care about the overall network costs where each node needs to know the cost or distances with immediately neighbor nodes. Another difference is when a negative cost is available in the routes. In such case the Dijkstra algorithm cannot be useful. However, if native values are in the network, the Bellman-Ford method can handle the process to calculate the minimum route [12] Djikstra s Algorithm Dijkstra's algorithm finds the shortest path length between two nodes in a graph. The algorithm works as follows; select the source node to be initial point. Then set the outline a set S of nodes, and set it to empty points. As the algorithm progresses, the set S will save those nodes to which a shortest path has been found. Later on, label the source node with 0, and add it into S. Next step we need to consider each node not in S connected by a route from the newly inserted node. Label the node not in S with the label of the newly inserted node and store the length of the route. But if the node not in S was already labeled, its new label will be min (label of newly inserted node + length of route, old label). Then pick a vertex not in S with the lowest label, and add it to S. Repeat the previous step, until the destination node is in S or there are no labeled nodes not in S. If the target is labeled; its label will be the path link distance from source to destination. If it is not labeled, there is no path. An upper bound of the running time of Dijkstra and Bellman algorithm on a graph with routes M and nodes N can be expressed as a function of M and N using the Big-O notation. Bellman complexity can be represented as O (mn), and Dijkstra O (n 2 ). 110

3 Figure 1. Dijkstra s algorithm [12] The instruction is theoretically as follows: to begin, mark the distance or cost to every intersection on the map with infinity. Even it is not infinity distance but to start the assumption and to note that node has not yet been visited; some alternatives of this technique simply leave the intersection unmarked. For the first iteration the current node will be the starting point and the distance to it will be labeled as zero. For the following iterations (after the first), the current intersection will be the closest unvisited node to the starting point. From the current point, update the distance to every unvisited node that is immediately neighbor to the first node or directly connected to it. This is ended by determining the sum of the distance between an unvisited intersection and the value of the current intersection, and relabeling the unvisited nodes with this value if it is less than its current value [12] Bellman-Ford Algorithm Bellman-Ford algorithm describes the shortest path in the general case in which routes of a certain network can have negative weight as long as the loop contains no negative cycles. For example it uses d[u] as an upper bound on the distance d [u,v] from u to v. The algorithm progressively reduces an estimate d[v] on the cost of the shortest path from the source node u to each node v in V until it reach the actual shortest-path. The algorithm yields Boolean True if the given digraph contains no negative cycles that are reachable from source node s otherwise it returns Boolean FALSE. Figure 2. Bellman Ford algorithm [13] 111

4 The algorithm is related to Dijkstra algorithm and very similar to, where the only difference is that when working with negative costs since it can t work with a greedy algorithm. A distributed variant of the Bellman Ford is used as the algorithm for dynamic Distance Vector Routing (DVR) protocols, for example the algorithm is statistically distributed in the Routing Information Protocol (RIP), because it includes a number of nodes (routers) within the system, a collection of IP networks typically owned by an Internet Service Provider. One way to measure the performance of network is by measuring attenuation versus distance. The strength of a signal falls off with increasing distance due to attenuation that raises with proportional distances increments. As distance increases many factor can be introduced to the system that affects the signals strength. Attenuation is a term used to explain the drop of the link signal strength that occurs on the connection links over distance and is measured in db. The further you are away from the exchange, the higher your attenuation figure will be resulting in incremental signal loss. There isn't much you can do regarding the path loss due to attenuation. This is because that is largely dependent upon link length, and the amount of copper in/or joints on your line. So the only technique that may be used to minimize communication loss is by choosing the shortest distance to cover the entire route. If the algorithm used has a slower response and takes longer time to stabilize then it will be more affected to incremental in distance resulting in more attenuation. The throughput is a degree of how fast we can actually transmit data through a network. Although, at first look, bandwidth in bits per second and throughput seem the same, but actually they are different. A link may have a bandwidth of B bit per second, but we can only send T bit per second through this link with T always less than B. In general, the bandwidth is a potential dimension of a link. Throughput can be measured as an individual device (such as a router) or a part of the network. Performance management monitors the throughput to make sure that it is not reduced to undesirable levels. Throughput can be represented as the number of bits passing through a point in a second. It is well known that definition from bits to packets and from a point to a network. Throughput can be defined in a network as the number of packets passing through the network during a unit of time. When the load is below the maximum capacity of the network, then the throughput rises proportionally with the load. It is expected that throughput remains constant after the load reaches the maximum capacity. In the paper we fix the number of nodes to 9 then we make the transmitting rate variable [14]. 3. Results and discussion From the designed algorithm of Dijkstra with 9 randomly created nodes the network will be as shown in the next figure. The designed algorithm calculates the distance for the all nodes, it returns the minimum distance and highlighted with red color the shortest route. Reffering to Figure 3, in order to send data from source node 1 to destination node 9 the data will take the shortest route. The shortest path is calculated as d = and the flow of the route goes as follows p = [ ]. On the other hand Bellman Ford has been designed with the same specification as Dijkstra and randomly generated nodes as show in the next figure. The calculated path in Figure 4 gives d = and the nodes flow from source to destination will be p =[ ]. Thus result in shorter route can be found using Dijksra algorithm. The second test is on algorithm performance and based on attenuation versus distance as shown in the next graph. The Dijkestra algorithm in Figure 5 below is highlighted with blue color performs better with distance increments comparing to Bellman Ford that appears with red color. This is due to faster response in the Dijkstra algorithm, which required less convergence time. The next part is the comparison in term of throughput versus traffic loads. 112

5 Figure 3. Dijkstra Shortest distance route Figure 4. Bellman shortest distance route

6 Figure 5. Attenuation versus distance in Dijkstra and Bellman Ford algorithms 10 9 Diji 8 7 Throughput /Mbit/s traffic load/mbit/s Figure 6. Dijkstra throughput versus traffic loads. 114

7 10 9 Bell 8 7 Throughput /Mbit/s traffic load/mbit/s Figure 7. Bellman throughput versus traffic load From figure 6 and 7, the throughput increases proportional to the increase in traffic load. Thus results of the performance test in terms of throughput versus traffic load the Dijkstra algorithm performs slightly well than Bellman algorithm. Based on the result analysis and literature review, evaluation of the relative advantages of the two algorithms should consider two things; the processing time needed and the amount of information that must be gathered from other vertices in the network or Internet. The assessment will be depending on the implementation methodology and the precise implementation. If the network implemented with negative routes the Bellman can be used. On the other hand, if higher response is needed and lower load desirable, then Dijkstra will be better choice. 4. Conclusion There are a lot of proposals of determining a shortest path routing in networks. One of the most widely used methods is the Dijksta algorithm. Usually, if the classical methods are not able to find the shortest route or have some problems, other alternatives can be used. Bellman Ford algorithm can find the shortest route even if the network contains negative cycles. A comparison between the two algorithms has been performed. This paper also shows the discovery of the performance of each algorithm in terms of attenuation versus distance and throughput versus traffic load. Dijkstra performs better in terms of attenuation since it performs faster calculation with lower convergence time. Dijkstra algorithm also proves a lower load generation comparing to Bellman Ford algorithm. 5. Acknowledgment Authors would like to thank Universiti Teknikal Malaysia Melaka for sponsoring this project. This project was funded by university short-term grant PJP/2011/FKEKK(38C)/S Deep appreciations are also dedicated to anyone who directly or indirectly involved in this project. 6. References [1] Liu Xiao-Yan, Chen Yan-Li, Application of Dijkstra Algorithm in Logistics Distribution Lines, In the Proceeding of Third International Symposium on Computer Science and Computational Technology, P. R. China, pp , August [2] Jie Huang, Bei Huang, A Security Routing Protocol Based on Public Key Matrix for Wireless Sensor Networks, Journal of Convergence Information Technology, Vol. 7, No. 3, pp ,

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