Shortest Path Determination in a Wireless Packet Switch Network System in University of Calabar Using a Modified Dijkstra s Algorithm

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1 International Journal of Engineering and Technical Research (IJETR) ISSN: (O) (P), Volue-5, Issue-1, May 16 Shortest Path Deterination in a Wireless Packet Switch Network Syste in University of Calabar Using a Modified Dijkstra s Algorith Ofe O A, Edi E A, Ele S I, Abstract The proble of finding the shortest path between two odes is a well known proble in network analysis Optial routing has been widely studied for interconnection network This research considers the proble of finding the shortest path in a wireless packet switch network syste in the University of Calabar environent, Calabar, Nigeria Its theoretical approach, ipleentation and application First a historical background of sp algorith was given and basic concepts of network analysis in connection with congestion and delay issues were explored for the later use The report was concentrated on the odified Dijkstra s algorith and the open shortest path first (OSPF) algorith, use in finding single source shortest path in a one-to-all undirected wireless packet switch network syste in the University of Calabar environent, Calabar, Nigeria, with distance costs of links in order of O (n+) tie, where n is the nuber of nodes and is the nuber of links in the network The algorith was presented in the context of wireless packet switch network syste in the University of Calabar, Calabr, Nigeria The ai of the research is to designed a wireless packet switch network in the University of Calabar environent with the objective of deterining the shortest path, dij taken by a essage (packet) to traverse fro a source node (current) to a destination (sink) node considering four different routes in the network syste Index Ters Graph, transit distance (dij) wireless packet switch network adjacency list, adjacency atrix I INTRODUCTION In this research, our network odel is abstracted as a graph G (V, E) In graph theory, the shortest path proble is used to deterine the path between source vertices and destination vertices (or nodes) such that the su of the weights of its constituent edges is iniized A graph G is a collection of two sets V and E where V is the collection of vertices Vo, Vi, --- Vn-1also called nodes and E is the collection of edges e1, e, ----, en where an edge is an arc which connects two nodes (Delling, 8) This can be represented as: G = (V, E) V (G) = (Vo, V1, ----, Vn) or set of vertices E (G) = (e1, e, en) or set of edges In a graph structure, we have two ajor coponents naely: nodes and edges This we need to design the data Ofe O A, Departent of Coputer Science University of Calabar, Nigeria Edi E A, Departent of Coputer Science University of Calabar, Nigeria Ele S I, Departent of Coputer Science University of Calabar, Nigeria structure to keep these coponents in eory There are two ways of representing the graph in coputer eory First one is the sequential representation and second one is the linked list representation Adjacency atrix is the atrix, which keeps the inforation of adjacent nodes In order words we cannot say that this atrix keeps the inforation that whether this node is adjacent to any other node or not If the adjacency atrix of the graph is sparese then it is better to represent the graph through adjacency linked list Our network odel in the University of Calabar is a sparese wireless, undirected packet switch network thus we represent the graph using adjacency priority linked list In this adjacency link list representation of the graph, we will, aintain two lists First list will keep inforation of all nodes in the graph and second list will aintain a list of adjacency nodes for each node There are several cases in graph where we have a need to know the shortest path fro one node to another node In our wireless packet switch network, the switches or routers constitute the nodes, while the distances between any two switches constitute the edges of the network Our proble definition is to find the shortest path (dij) that a packet will take to traverse fro a source node to a destination node Our research scenario is a network routing concept There can be several paths for the packets to go fro one node to another node But the shortest path is that path in which the su of weights of the included edges is the iniu There are several algoriths to deterine or find the shortest path in a network graph Here we describe the odified dikjstra s algorith (Ahn, Raakrishna, Rang & Choi, 1) Fig 1 A wireless packet switch network odeled as a graph structure in the University of Calabar Fro the above graph structure, our objective is to find the shortest path that a packet can traverse fro the source node to the destination node The source node is node 1and the destination node is node 6 We label each node with transit distance or distance cost dij, 194 wwwerpublicationorg

2 Shortest Path Deterination in a Wireless Packet Switch Network Syste in University of Calabar Using a Modified Dijkstra s Algorith predecessor, and status Distance of node represents the shortest distance of that node fro the source node, and predecessor of node represents the node which precedes the given node in the shortest path fro the source Status of a node can be peranent or teporary, aking a node peranent eans that it has been included in the shortest path Teporary nodes can be relabeled if required but once a node is ade peranent it cannot be relabeled The procedure for our odified diskstra s algorith 1 Initially ake source node peranent by assigning to it a distance value of zero and ake it the current working node All other nodes are ade teporary, nodes that are directly reachable fro node 1 have their teporary label equal to dij while nodes that are not reachable fro node 1 have to infinity as their teporary labels Exaine all the teporary neighbours of the current working node and after checking the condition for iniu weight, relabel the required nodes 3 Fro all the teporary nodes, find out the node which has iniu value of distance, ake this node peranent and now this is our current working node 4 When there is a tie in steps and 3 choose any one but exactly once 5 Repeat steps,3 and 4 until destination node is ade peranent Advantages of our odified diskstra s algorith 1 With this algorith, we can iniize our cost while bulding a network This is because the odified dijkstra s will find the shortest path value fro a given source node to the destination node Therefore, we need not build uch of routers to build path fro a node to other With this algorith, we can also axiize the efficiency of the syste, since it will find out the iniu path value Recall that path weight is propagation delay for a syste II PROPOSED WORK There are several cases in graph where we need to know the shortest path fro one node to another node The following doains of application of our algorith follow this approach: Robot path planning, logistic distribution, link-state routing protocol, general electric supply syste, water distribution, railway track syste, driving direction on websites, like ap quest or Google ap, plant and facility layout arising in VLSI design, transportation, and the traveling salesan proble (Anjali, Datta & Joshi, 16) In our wireless packet switch network, our algorith is very useful in the network for routing concepts There are several paths for going fro one node to another node But the shortest path is the path in which the su of weights of the included edges is the iniu There are several algoriths to find out the shortest path Here we will describe the odified dijkstra s algorith We ipleent a total different concept to find out shortest path using the scaled ap of the University of Calabar The ap which we are using is scaled ap which return correct distance between two switches or routers in our network Here we will use link list to store and traverse N nodes or vertices We label each node (switch) with distance predecessor and status Distance of node represents the shortest distance of that node fro the source node, and predecessor of node represents the node which precedes the given node in the shortest path fro the source node, status of a node can be peranent or teporary III RESEARCH DESIGN PROBLEM AND FORMULATION Earlier researchers have perfored various operations on Dijkstra s algorith to deterine the shortest path between the nodes and have got the better results in their researches for the specified nuber of nodes But, the results were liited to the nuber of nodes fixed at the tie while declaring the size of the data structure Anjali, Datta & Joshi, (16) referring to Fuhao ZHANG, introduces the classical dijkstra s algorith in detail, and describes the useful process of ipleentation of the algorith It describes the adjacent node algorith which is a better optiization algorith based on Dijkstra s algorith This algorith akes correlation with each node in the different network topology and inforation, and avoids the use of co-related atrix that contains huge infinite value, and aking it ore reliable and suitable analysis of the networks for ass data () It is prove that this algorith can save a lot of eory space and is ore reliable to the network with huge nodes, but it was found that as node grew larger this approach gets slow in searching nodes In this research, our design proble is to design a wireless network syste in the University of Calabar, Calabar, Nigeria, with the objective of deterining the shortest path fro the source, (current) node to the destination (sink) node in an undirected network syste Thus our odel posses the following attributes: 1 It is a single source shortest path proble in a one-to-all network The edge lengths are non-negative 3 The graph is an undirected one 4 The shortest path is coputed using just coparison and addition odel The attribute of a node in the network are: 1 Its distance value, and Its status label Fro our network design odel in fig 1, the odel of research is designed using the linked list priority queue data structure The siulation is done at different levels of priorities The priority levels are as follows; At level 1: Node 1 Reachable nodes fro node 1 are nodes, 3 and 4 The teporary labels for these nodes are their direct distances fro node 1 to the node in question No teporary label distance is updated here fro upper boundary to a lower boundary (Goldberg, Kaplan, & Werneck, 8) Nodes that cannot be reached directly fro node 1 have as their teporary label In this odel, the researchers used full line for edges that are reachable fro the source nodes and broken lines for edges that cannot be reached directly fro the source 195 wwwerpublicationorg

3 International Journal of Engineering and Technical Research (IJETR) ISSN: (O) (P), Volue-5, Issue-1, May 16 node Thus, the node odel at this level is: At level : Copare the teporary labels of n, n 3 and n4 and ake the sallest of the the current peranent label No teporary label is updated at this stage Thus, if n < n3 and n < n4 then pl = n else pl n : Since 3 < 7 and 3 < 4 n 3 Thus, the logical structure of the odel at level becoes: At level 4: Update teporary label for node 5: TL(5) = in {, 4 + 3} = 7 Copare TLS for n3, n5 and n6 and ake the sallest a peranent label If n3 < n5 and n3 < n 6 then pl = n3 Else pl n3 since 5 < 7 and 5 < 1 n 3 5 Thus, the logical structure of the odel at level 3 becoes: At level 3: Update the teporary labels for the nodes that are directly reachable fro node ; copare their TLs and ake the sallest TL a peranent label The new TL for node 3 = in {, 3 + } = 5 in {, + 7} = 7 : new TL for node 3 = 5 and new TL for node 6 = in {, 3 + 9} = 1 now copare the TLs and ake the sallest of the a peranent label If n3 < n4 and n3 < n6 then pl = n3 otherwise pl n3 : Since the IF stateent = False Another coparison was tested with n4 IF n4 < n3 and n4 < n6 then pl = n4 Else pl n4 Since 4 < 5 and 4 < 1 then At level 5: Update teporary label for node 5 and 6 n5: in{, 5 + 3} = 8 in {, 4 + 3} = 7 TL for n5 = 7 n6: in {, 3 + 9} = 1 in {, 5 + 6} = 11 TL for n6 = 11 Now copare TLS for node 5 and 6 and ake the sallest a pl IF n5 < n 6 then n 5 7 Thus, the logical structure of the odel at level 5 becoes: n 4 4 Thus, the logical structure of the odel at level 3 becoes: 196 wwwerpublicationorg

4 Shortest Path Deterination in a Wireless Packet Switch Network Syste in University of Calabar Using a Modified Dijkstra s Algorith At level 6: Node 6 is the last node to be visited Update the teporary label of node 6 and ake it peranent n6: in {, 5 + 6} = 11 in {, 7 + 3} = n1 1 n 3 13 n3 1n n nn It is required to represent G in a coputer eory with an adjacency atrix of size N<n Solution Consider a graph G having n = 6 nodes or vertices as shown Thus, node 6 has a new TL of 1 and it is autoatically ade peranent The algorith converges at this point The shortest path dij of the network ode is 1 and the path length or link whose dij = 1 is the shortest path of the network Fro the logical siulation of the wireless packet switch network odel in the University of Calabar, the shortest path for the packets to traverse fro the source node to the destination is as follows: n 1 n 4 n 5 n n 1 n 4 n 5 n 6 IF TL1 < TL < < T < K Then pl = TL1 otherwise pl TL1 subject to further coparison and selection IV RESEARCH FORMULATION To resolve this proble and to give the best solution for increasing nuber of nodes we used the priority queue link list having the capacity to store N nodes with soe constraints containing the value predecessor, status and distance (Kiran & Ranjit, 1) Proble: Given a graph G = {V, E} where V is the set of nodes (or vertices) in G and E the set of edges in G, if n = V is the nuber of vertices in G, then the size of the adjacency atrix required to store G in a coputer eory is n x n (or n): G requires a 6 x 6 (or 36) adjacency atrix to represent it in a coputer eory This representation is shown below Notice that this adjacency atrix is a syetric atrix that is MT = M thus, ij = ij ij {1,,3,4,5,6} and wij = i =j Hence, if all duplicate is reove, starting fro i = j downward, as follows; Thus, the size of this new atrix is the su of each row given as = 15 In general, n n - + n n - (n - 1) = n (n - 1) Hence, the size of atrix N is N = n (n - 1) Where n = nuber of nodes Clearly N < n in fact 197 wwwerpublicationorg

5 Theore: N is less than 5% of n that is, N < 5% n Proof: Suppose N = X% n, then N(n 1) = xn 1, n n = xn 1 ==> 1 (n n) = xn ==> x = 5(n - n) = 5 n(n 1) n n ==> 5 5 = x (1) n Since x depends on n, we can take liit as n : li x (n) = 5 and hence, the proof: n eg in the above atrix, n = 6, thus, x (6) = 5 5/6 = x(6) = 4167% ==> 15 is 4167% of 36 Therefore, percentage reduction = = 5833% Instead of creating a ultidiensional array M for storing the graph G, we can rearrange the reaining eleents of the atrix M as follow; Observe that each eleent have been isolated with a faint line while each row separated by thick line as shown above But atrix M is a -D array ij, hence, we need to provide an interface for this new one diensional array so as to ake it appear ulti-diensional Row R and Colun Cij -Diensional array is divided into rows and coluns In the case of atrix ij, i = row and j = colun nubers Hence, we need to find a forula R for the row and Cij for the colun C r 1= 1 1 st row begins at eleent 3 r = n 1 +1 nd row begins at eleent => r 3 = n 1 + n +1 3 rd row beings at eleent 1 => r 4 = n 1 + n + n th row begins at eleent 3 => r = n-1+n-+n-3+ +n-(-1)+1 r = ( 1) n - ( 1) International Journal of Engineering and Technical Research (IJETR) ISSN: (O) (P), Volue-5, Issue-1, May 16 n ( -1) ( 1) + Thus, r = ( 1)(n ) + 1 () Equation () is the required forular for the row nuber If = I, we have r i = (i 1)(n i) + 1 V COLUMN Given a pair of vertices i, j, where i<j, then the colun where the weight of the graph G is stored is given by Cij = j i 1 (3) i<j Therefore, given two vertices (or nodes) i and j, the edge fro i to j or j to i is given by Aij = ri + Cij (4) Equation (4) is the forular that turns A (a single diension array) into Aij (a ultidiensional array) Proof: We ust show that Mij = Aji ij {1,,3,4,5,6} M1 = 3, A1 = r1 + C1 ((1 1)((6) 1))/ ( 1 1) A1 = = 1 check the index 1 in A M35 = 3, A35 = r3 + C35 ((3 1)((6) 1))/ (5 3 1) A35 = = 13 check the index 3 in A VI CONCLUSION Our odified Dijkstra s algorith is an iproved version of the traditional Dijkstra s algorith Currently, our algorith used priority queue with link list with soe constraints, plus one odified Dijkstra call This odified Dijkstra s algorith should iprove the running tie of our algorith by about 5833% and the shortest path link is node 1 node 4 nodes 5 node 6 where node 1 is the source (current) node while node 6 is the destination (sink) node in our wireless packet switch network syste in the University of Calabar, Calabar, Nigeria REFERENCES [1] Anjali, J, Datta, U and Joshi, K K (16) Modification in Dijkstra s Algorith to Find the Shortest Path for N Nodes with Constraint International Journal of Scientific Engineering and Applied Science, (1) [] Ahn, C W, Raakrishna, R S, Rang, C G & Choi, I C (1) Shortest path routing algorith using hopfield neural network, Electron Lett, 37(19): [3] Delling, D & Nannicini, G (8) Bidirectional Core-Based Routing in Dynaic Tie-Dependent Road Networks In S-H Hong, H Nagaochi, and T Fukunaga, editors, Proceedings of the 19th International Syposiuon Algoriths and Coputation (ISAAC 8), volue 5369 of Lecture Notes in Coputer Science, pp [4] Delling, D (8) Tie-Dependent SHARC-Routing In Proceedings of the 16th Annual European Syposiuon Algoriths (ESA 8), volue 5193 of Lecture Notes in Coputer Science, pp [5] Goldberg, A V Kaplan, H & Werneck, R F (8) Reach for A*: Shortest Path Algoriths with Preprocessing In C Deetrescu, A V Goldberg, and D S Johnson, editors, Shortest Paths: Ninth DIMACS Ipleentation Challenge, DIMACS Book Aerican Matheatical Society [6] [7] Kiran, Y & Ranjit B, (1) An Approach to Find Kth Shortest Path using Fuzzy Logic International Journal of Coputational Cognition, 8(1) 198 wwwerpublicationorg

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