Shortest Path Algorithm in Computer Networks

In computer networks, the shortest path algorithms aim to find the optimal paths between the network nodes so that routing cost is minimized. They are direct applications of the shortest path algorithms proposed in graph theory.

Explanation

Consider that a network comprises of N vertices (nodes or network devices) that are connected by M edges (transmission lines). Each edge is associated with a weight, representing the physical distance or the transmission delay of the transmission line. The target of shortest path algorithms is to find a route between any pair of vertices along the edges, so the sum of weights of edges is minimum. If the edges are of equal weights, the shortest path algorithm aims to find a route having minimum number of hops.

Common Shortest Path Algorithms

Some common shortest path algorithms are

• Bellman Fords Algorithm

• Dijkstras Algorithm

• Floyd Warshalls Algorithm

The following sections describes each of these algorithms.

Bellman Ford Algorithm

Input A graph representing the network; and a source node, s

Output Shortest path from s to all other nodes.

• Initialize distances from s to all nodes as infinite (); distance to itself as 0; an array dist[] of size |V| (number of nodes) with all values as except dist[s].

• Calculate the shortest distances iteratively. Repeat |V|- 1 times for each node except s

  • Repeat for each edge connecting vertices u and v

    • If dist[v] > (dist[u] + weight of edge u-v), Then

      • Update dist[v] = dist[u] + weight of edge u-v

• The array dist[] contains the shortest path from s to every other node.

Dijkstras Algorithm

Input A graph representing the network; and a source node, s

Output A shortest path tree, spt[], with s as the root node.

Initializations

• An array of distances dist[] of size |V| (number of nodes), where dist[s] = 0 and dist[u] = (infinite), where u represents a node in the graph except s.

• An array, Q, containing all nodes in the graph. When the algorithm runs into completion, Q will become empty.

• An empty set, S, to which the visited nodes will be added. When the algorithm runs into completion, S will contain all the nodes in the graph.

• Repeat while Q is not empty

  • Remove from Q, the node, u having the smallest dist[u] and which is not in S. In the first run, dist[s] is removed.

  • Add u to S, marking u as visited.

  • For each node v which is adjacent to u, update dist[v] as

    • If (dist[u] + weight of edge u-v) < dist[v], Then

      • Update dist[v] = dist[u] + weight of edge u-v

• The array dist[] contains the shortest path from s to every other node.

Floyd Warshall Algorithm

Input A cost adjacency matrix, adj[][], representing the paths between the nodes in the network.

Output A shortest path cost matrix, cost[][], showing the shortest paths in terms of cost between each pair of nodes in the graph.

• Populate cost[][] as follows:

  • If adj[][] is empty Then cost[][] = (infinite)

  • Else cost[][] = adj[][]

• N = |V|, where V represents the set of nodes in the network.

• Repeat for k = 1 to N

  • Repeat for i = 1 to N

    • Repeat for j = 1 to N

      • If cost[i][k] + cost[k][j] < cost[i][j], Then

        • Update cost[i][j] := cost[i][k] + cost[k][j]

• The matrix cost[][] contains the shortest cost from each node, i , to every other node, j.