BEGIN

 FOR ALL v ∈ V

  d[v][v] ← 0

 FOR ALL (u,v) ∈ E

  d[u][v] ← w(u,v)

 for k from 1 to |V|

  for i from 1 to |V|

   for j from 1 to |V|

    if d[i][j] > d[i][k] + d[k][j]

     d[i][j] ← d[i][k] + d[k][j]

    end if

END

#include <stdio.h>
#include <limits.h>

#define V 4 // Number of vertices

void floydWarshall(int graph[V][V]) {
    int dist[V][V];

    // Initialize distance matrix
    for (int i = 0; i < V; i++) {
        for (int j = 0; j < V; j++) {
            dist[i][j] = graph[i][j];
        }
    }

    // Main algorithm
    for (int k = 0; k < V; k++) {
        for (int i = 0; i < V; i++) {
            for (int j = 0; j < V; j++) {
                if (dist[i][k] != INT_MAX && dist[k][j] != INT_MAX &&
                    dist[i][j] > dist[i][k] + dist[k][j]) {
                    dist[i][j] = dist[i][k] + dist[k][j];
                }
            }
        }
    }

    // Print the shortest distances
    printf("Shortest distances between vertices:\n");
    for (int i = 0; i < V; i++) {
        for (int j = 0; j < V; j++) {
            if (dist[i][j] == INT_MAX) {
                printf("INF ");
            } else {
                printf("%d ", dist[i][j]);
            }
        }
        printf("\n");
    }
}

int main() {
    int graph[V][V] = {
        {0, 5, INT_MAX, 10},
        {INT_MAX, 0, 3, INT_MAX},
        {INT_MAX, INT_MAX, 0, 1},
        {INT_MAX, INT_MAX, INT_MAX, 0}
    };

    floydWarshall(graph);

    return 0;
}

BEGIN

  d(v[1]) ← 0

  FOR j = 2,..,n DO

    d(v[j]) ← ∞

  FOR i = 1,..,(|V|-1) DO

    FOR ALL (u,v) in E DO

      d(v) ← min(d(v), d(u)+l(u,v))

  FOR ALL (u,v) in E DO

    IF d(v) > d(u) + l(u,v) DO

      Message: "Negative Circle"

END

#include <stdio.h>
#include <limits.h>

#define MAX_VERTICES 100
#define MAX_EDGES 100

typedef struct {
    int u, v, weight;
} Edge;

int numVertices, numEdges;
int distance[MAX_VERTICES];
Edge edges[MAX_EDGES];

void bellmanFord(int startVertex) {
    distance[startVertex] = 0;

    for (int j = 1; j <= numVertices; j++) {
        if (j != startVertex) {
            distance[j] = INT_MAX;
        }
    }

    for (int i = 1; i <= numVertices - 1; i++) {
        for (int k = 0; k < numEdges; k++) {
            int u = edges[k].u;
            int v = edges[k].v;
            int weight = edges[k].weight;

            if (distance[u] != INT_MAX && distance[u] + weight < distance[v]) {
                distance[v] = distance[u] + weight;
            }
        }
    }

    for (int k = 0; k < numEdges; k++) {
        int u = edges[k].u;
        int v = edges[k].v;
        int weight = edges[k].weight;

        if (distance[u] != INT_MAX && distance[u] + weight < distance[v]) {
            printf("Negative Circle\n");
            return;
        }
    }
}

int main() {
    scanf("%d %d", &numVertices, &numEdges);

    for (int i = 0; i < numEdges; i++) {
        scanf("%d %d %d", &edges[i].u, &edges[i].v, &edges[i].weight);
    }

    int startVertex;
    scanf("%d", &startVertex);

    bellmanFord(startVertex);

    for (int i = 1; i <= numVertices; i++) {
        printf("Vertex %d: Distance = %d\n", i, distance[i]);
    }

    return 0;
}

Dijkstra(G,s):
	 for each vertex u ∈ G.V - {s}
	 u.d = ∞
	 u.pred = null
	 u.done = false
	 s.d = 0
	Q = G.V
	 while Q ≠ ∅
	 u = Q.extractMin()
	 u.done = true
	 for each v ∈ G.Adj[u]
	 Relax(u,v)
Relax(G,u,v):
	if u.d + G.edge(u,v).w < v.d
	 v.d = u.d + G.edge(u,v).w
	v.pred = u

#include <stdio.h>
#include <stdlib.h>
#include <limits.h>

#define MAX_VERTICES 1000

struct Edge {
    int v;
    int weight;
    struct Edge *next;
};

struct Vertex {
    int d;
    int pred;
    int done;
    struct Edge *adjList;
};

struct Vertex vertices[MAX_VERTICES];
int numVertices;

void relax(int u, int v, int weight) {
    if (vertices[u].d + weight < vertices[v].d) {
        vertices[v].d = vertices[u].d + weight;
        vertices[v].pred = u;
    }
}

int extractMin(int Q[]) {
    int min = INT_MAX;
    int minIndex = -1;

    for (int i = 0; i < numVertices; i++) {
        if (!vertices[i].done && vertices[i].d < min) {
            min = vertices[i].d;
            minIndex = i;
        }
    }

    return minIndex;
}

void dijkstra(int source) {
    for (int u = 0; u < numVertices; u++) {
        vertices[u].d = INT_MAX;
        vertices[u].pred = -1;
        vertices[u].done = 0;
    }

    vertices[source].d = 0;

    int Q[MAX_VERTICES];
    for (int i = 0; i < numVertices; i++) {
        Q[i] = i;
    }

    while (1) {
        int u = extractMin(Q);
        if (u == -1) break;

        vertices[u].done = 1;
        
        struct Edge *edge = vertices[u].adjList;
        while (edge != NULL) {
            relax(u, edge->v, edge->weight);
            edge = edge->next;
        }
    }
}

int main() {
    int numEdges;
    printf("Enter the number of vertices and edges: ");
    scanf("%d %d", &numVertices, &numEdges);

    for (int i = 0; i < numVertices; i++) {
        vertices[i].adjList = NULL;
    }

    printf("Enter edge information (u v weight):\n");
    for (int i = 0; i < numEdges; i++) {
        int u, v, weight;
        scanf("%d %d %d", &u, &v, &weight);

        struct Edge *newEdge = malloc(sizeof(struct Edge));
        newEdge->v = v;
        newEdge->weight = weight;
        newEdge->next = vertices[u].adjList;
        vertices[u].adjList = newEdge;
    }

    int source;
    printf("Enter the source vertex: ");
    scanf("%d", &source);

    dijkstra(source);

    printf("Shortest paths from source vertex %d:\n", source);
    for (int i = 0; i < numVertices; i++) {
        if (i != source) {
            printf("Vertex %d: Distance = %d, Pred = %d\n", i, vertices[i].d, vertices[i].pred);
        }
    }

    return 0;
}

MST-Kruskal(G)
	    for each v ∈ G.V
 Make-Set(v)
	   for each (u,v) by weight
if Find-Set(u) ≠ Find-Set(v)
Union(u,v)
(u,v).inMST = True

#include <stdio.h>
#include <stdlib.h>

#define MAX_VERTICES 1000

struct Edge {
    int u, v, weight;
};

struct Subset {
    int parent;
    int rank;
};

struct Edge edges[MAX_VERTICES];
struct Subset subsets[MAX_VERTICES];

int findSet(int i) {
    if (subsets[i].parent != i)
        subsets[i].parent = findSet(subsets[i].parent);
    return subsets[i].parent;
}

void unionSets(int x, int y) {
    int xRoot = findSet(x);
    int yRoot = findSet(y);

    if (subsets[xRoot].rank < subsets[yRoot].rank)
        subsets[xRoot].parent = yRoot;
    else if (subsets[xRoot].rank > subsets[yRoot].rank)
        subsets[yRoot].parent = xRoot;
    else {
        subsets[yRoot].parent = xRoot;
        subsets[xRoot].rank++;
    }
}

int compareEdges(const void *a, const void *b) {
    return ((struct Edge *)a)->weight - ((struct Edge *)b)->weight;
}

void kruskalMST(struct Edge graph[], int V, int E) {
    qsort(graph, E, sizeof(struct Edge), compareEdges);

    for (int v = 0; v < V; v++) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }

    int mstEdges = 0;
    int mstWeight = 0;

    for (int e = 0; e < E; e++) {
        int u = graph[e].u;
        int v = graph[e].v;

        int uRoot = findSet(u);
        int vRoot = findSet(v);

        if (uRoot != vRoot) {
            unionSets(uRoot, vRoot);
            mstEdges++;
            mstWeight += graph[e].weight;
            printf("(%d, %d) in MST\n", u, v);
        }

        if (mstEdges == V - 1)
            break;
    }

    printf("MST Weight: %d\n", mstWeight);
}

int main() {
    int V, E;
    printf("Enter number of vertices and edges: ");
    scanf("%d %d", &V, &E);

    for (int i = 0; i < E; i++) {
        printf("Enter edge (u v weight): ");
        scanf("%d %d %d", &edges[i].u, &edges[i].v, &edges[i].weight);
    }

    kruskalMST(edges, V, E);

    return 0;
}

MST-Prim(G,r)
for each u ∈ G.V
       u.key = ∞
	   u.pred = nil
	   r.key = 0
	   Q = G.V
while Q ≠ ∅
	 u = ExtractMin(Q)
	 for each v ∈ G.Adj[u]
	 if v ∈ Q and w(u,v) < v.key
	 v.pred = u
	 v.key = w(u,v)

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <limits.h>

// Define the maximum number of vertices in the graph
#define MAX_VERTICES 100

// Structure to represent a vertex
struct Vertex {
    int key;
    int pred;
};

// Function to extract the vertex with the minimum key from the queue
int extractMin(struct Vertex Q[], int size, bool visited[]) {
    int minIndex = -1;
    int minValue = INT_MAX;
    
    for (int i = 0; i < size; i++) {
        if (!visited[i] && Q[i].key < minValue) {
            minValue = Q[i].key;
            minIndex = i;
        }
    }
    
    return minIndex;
}

// Main Prim's algorithm function
void MST_Prim(int G[MAX_VERTICES][MAX_VERTICES], int numVertices, int r) {
    struct Vertex vertices[MAX_VERTICES];
    bool visited[MAX_VERTICES] = {false};
    
    // Initialize vertices
    for (int u = 0; u < numVertices; u++) {
        vertices[u].key = INT_MAX;
        vertices[u].pred = -1;
    }
    
    vertices[r].key = 0;
    
    for (int i = 0; i < numVertices; i++) {
        int u = extractMin(vertices, numVertices, visited);
        visited[u] = true;
        
        for (int v = 0; v < numVertices; v++) {
            if (G[u][v] && !visited[v] && G[u][v] < vertices[v].key) {
                vertices[v].pred = u;
                vertices[v].key = G[u][v];
            }
        }
    }
    
    // Print the minimum spanning tree
    printf("Edge   Weight\n");
    for (int i = 0; i < numVertices; i++) {
        if (vertices[i].pred != -1) {
            printf("%d - %d   %d\n", vertices[i].pred, i, vertices[i].key);
        }
    }
}

int main() {
    int numVertices, r;
    int G[MAX_VERTICES][MAX_VERTICES];
    
    printf("Enter the number of vertices: ");
    scanf("%d", &numVertices);
    
    printf("Enter the adjacency matrix:\n");
    for (int i = 0; i < numVertices; i++) {
        for (int j = 0; j < numVertices; j++) {
            scanf("%d", &G[i][j]);
        }
    }
    
    printf("Enter the root vertex: ");
    scanf("%d", &r);
    
    MST_Prim(G, numVertices, r);
    
    return 0;
}