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116 lines
3.1 KiB
C++
116 lines
3.1 KiB
C++
//brief-- Johnson's algorithm*
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//*Details-
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//*finding shortest paths between
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//*every pair of vertices in a given
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//*weighted directed Graph and weights may be negative.
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//---Complexity----
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//O(V^2logV + V*E)
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//---Application-----
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//1.A new vertex is added to the graph, and it is connected by edges of zero weight to all other vertices in the graph.
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//2.All edges go through a reweighting process that eliminates negative weight edges.
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//3.The added vertex from step 1 is removed and Dijkstra's algorithm is run on every node in the graph.
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//------Algorithm Idea-----
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//1. Let the given graph be G.
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//Add a new vertex s to the graph,
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//add edges from new vertex to all vertices of G.
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//Let the modified graph be G’.
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//2.Run Bellman-Ford algorithm on G’ with s as source.
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//Let the distances calculated by Bellman-Ford be h[0], h[1], .. h[V-1].
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//If we find a negative weight cycle, then return.
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//Note that the negative weight cycle cannot be created by new vertex s as there is no edge to s.
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//All edges are from s.
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//3.Reweight the edges of original graph.
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//For each edge (u, v), assign the new weight as “original weight + h[u] – h[v]”.
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//4.Remove the added vertex s and run Dijkstra’s algorithm for every vertex.
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//-----Pseudo Code------
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// 1.
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// create G` where G`.V = G.V + {s},
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// G`.E = G.E + ((s, u) for u in G.V), and
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// weight(s, u) = 0 for u in G.V
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// 2.
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//if Bellman-Ford(s) == False
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//return "The input graph has a negative weight cycle"
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// else:
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// for vertex v in G`.V:
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// h(v) = distance(s, v) computed by Bellman-Ford
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// for edge (u, v) in G`.E:
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// weight`(u, v) = weight(u, v) + h(u) - h(v)
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// D = new matrix of distances initialized to infinity
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// for vertex u in G.V:
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// run Dijkstra(G, weight`, u) to compute distance`(u, v) for all v in G.V
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// for each vertex v in G.V:
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// D_(u, v) = distance`(u, v) + h(v) - h(u)
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// return D
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#include<iostream>
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#define INF 9999
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using namespace std;
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int min(int a, int b);
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int cost[10][10], adj[10][10]; // declaring two 2-D array
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inline int min(int a, int b){
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/**Returns the minimum value*/
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return (a<b)?a:b;
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}
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main() {
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int vert, edge, i, j, k, c; // declaring variables
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cout << "Enter no of vertices: ";
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cin >> vert;
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cout << "Enter no of edges: ";
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cin >> edge;
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cout << "Enter the EDGE Costs:\n";
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for (k = 1; k <= edge; k++) {
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/**take the input and store it into adj and cost matrix*/
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cin >> i >> j >> c;
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adj[i][j] = cost[i][j] = c;
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}
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for (i = 1; i <= vert; i++)
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for (j = 1; j <= vert; j++) {
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if (adj[i][j] == 0 && i != j)
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adj[i][j] = INF;
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/**if there is no edge, put infinity*/
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}
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for (k = 1; k <= vert; k++)
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for (i = 1; i <= vert; i++)
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for (j = 1; j <= vert; j++)
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adj[i][j] = min(adj[i][j], adj[i][k] + adj[k][j]);
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/**find minimum path from i to j through k*/
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cout << "Resultant adj matrix\n";
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for (i = 1; i <= vert; i++) {
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for (j = 1; j <= vert; j++) {
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if (adj[i][j] != INF)
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cout << adj[i][j] << " ";
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}
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cout << "\n";
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}
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return 0;
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}
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