Question

What is the pseudocode for Belman-ford dijkstra and floyd warshall routing algorithms for shortest path

Answer 1

SOLUTION:-

SOLUTION:- Pseudocode for Belman-ford algorithm for shortest path i Bellman-Ford Algorithm is an algorithm for single source shortest path where edges can be negative (but if there is a cycle with negative weight, then this problem will be NP). Pseudocode Bellman-Ford(int v) dul- inf for each vertex j for step-1 to n for all edges like e j = e.first // first end k e.second// second end we.weight if d[k] > d[j] w if stepn then return "Negative cycle found"

Pseudocode for Diikstra algorithm for shortest path: Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a graph. Pseudocodei dijkstra(v): d[j]-inf for each vertex j snew empty set while s.size) < n for each j in V-s //The set of vertices is V then ydjl, u-j insert u into s // Relaxing process for each j in adj[u] d]-min(di], d[u]+ w(u,j))

Floyd-Warshall Algorithm is an algorithm based on dynamic programming technique to compute the shortest path between all pair of nodes in a graph Pseudocode: Floyd-Warshal( d[v][u]-inf for each pair (v,u) d[v] [v] = 0 for each vertex v for m 1 to n for i = 1 to n for j 1 to n d[i min(d[ij], d[i][m]d[m]i])

Bellman-Ford: The main concept is to relax all the edges exactly n-1 times. If in the n-th step, we relax an edge, then we have a negative cycle (this is if and only if). The steps are given below: 1- The outer loop will traverse through o to N. 2- Loop over all edges, check if the next node distance > current node distance edge weight, in this case update the next node distance to "current node distance + edge weight". This algorithm depends on the relaxation principle where the shortest distance for all vertices is gradually replaced by more accurate values until eventually reaching the optimum solution. In the starting, all vertices have a distance of "Infinity", but only the distance of the source vertex o, then update all the connected vertices with the new distances (source vertex distance + edge weights), then apply the same concept for the new vertices with new distances & so on. Dijkstra: Step 1: Set the distance to the source to o and the distance to the remaining vertices to infinity. Step 2: Set the current vertex to the source. Step 3: Flag the current vertex as visited. Step 4: For all vertices adjacent to the current vertex, set the distance from the source to the adjacent vertex equal to the minimum of its present distance and the sum of the weight of the edge from the current vertex to the adjacent vertex and the distance from the source to the current vertex. Step 5: From the set of unvisited vertices, arbitrarily set one as the new current vertex, provided that there exists an edge to it such that it is the minimum of all edges from a vertex in the set of visited vertices to a vertex in the set of unvisited vertices. To reiterate: The new current vertex must be unvisited and have a minimum weight edges from a visited vertex to it. This can be done trivially by looping through all visited vertices and all adjacent unvisited vertices to those visited vertices, keeping the vertex with the minimum weight edge connecting it. Step 6: Repeat steps 3-5 until all vertices are flagged as visited.

Floyd-Warshal: STEP1: Initialize the shortest paths between any 2 vertices with Infinity. STEP2: Find all pair shortest paths that use O intermediate vertices, then find the shortest paths that use 1 intermediate vertex and so on, until using all N vertices as intermediate nodes. STEP3: Minimize the shortest paths between any pairs in the previous operation STEP4: For any 2 vertices i and j, one should actually minimize the distances between this pair using the first K nodes, so the shortest path will be: minimum( D[i][k]D[k]jl, D[i]])
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