Prim's Algorithm Algorithm
Prim's Algorithm is a widely-used greedy algorithm in computer science that helps in finding the minimum spanning tree (MST) for a connected, undirected, weighted graph. The main objective of this algorithm is to determine the most efficient way to connect all the vertices in a graph, such that the total weight of the edges is minimized, and no cycles are formed. Developed by mathematician Vojtěch Jarník in 1930 and later rediscovered by computer scientist Robert C. Prim in 1957, the algorithm has found numerous applications in real-world scenarios, including network design, transportation planning, and cluster analysis.
The algorithm starts by selecting an arbitrary vertex from the graph and adds the lightest edge connected to the selected vertex into the MST. In each subsequent step, the algorithm selects the smallest-weighted edge that links a vertex in the MST to a vertex outside the MST, and adds the new edge and vertex to the MST. This process is repeated until all the vertices in the graph are included in the MST. Throughout the execution of Prim's Algorithm, it maintains a priority queue of vertices not yet part of the MST, ordered by the weight of the edges connecting them to the MST. This ensures that the algorithm always considers the smallest-weighted edges first, leading to an overall minimum spanning tree.
/*
Petar 'PetarV' Velickovic
Algorithm: Prim's Algorithm
*/
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <iostream>
#include <vector>
#include <list>
#include <string>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <complex>
#define MAX_N 101
using namespace std;
typedef long long lld;
int n;
struct Node
{
vector<int> adj;
vector<int> weight;
};
Node graf[MAX_N];
bool mark[MAX_N];
struct edge
{
int u, v;
double w;
bool operator <(const edge &a) const
{
return (w>a.w);
}
};
priority_queue<edge> pq_prim;
//Primov algoritam za racunanje minimalnog stabla razapinjanja datog tezinskog grafa
//Slozenost: O((V+E)log V)
inline int Prim()
{
int xt = 0;
int ret = 0;
int amt = 0;
while (amt < n-1)
{
mark[xt]=true;
for (int i=0;i<graf[xt].adj.size();i++)
{
if (!mark[graf[xt].adj[i]])
{
edge E;
E.u = xt;
E.v = graf[xt].adj[i];
E.w = graf[xt].weight[i];
pq_prim.push(E);
}
}
while (!pq_prim.empty())
{
edge X = pq_prim.top();
pq_prim.pop();
if (!mark[X.v])
{
ret += X.w;
xt = X.v;
amt++;
break;
}
}
}
return ret;
}
int main()
{
n = 7;
graf[0].adj.push_back(1);
graf[0].weight.push_back(7);
graf[1].adj.push_back(0);
graf[1].weight.push_back(7);
graf[0].adj.push_back(3);
graf[0].weight.push_back(5);
graf[3].adj.push_back(0);
graf[3].weight.push_back(5);
graf[1].adj.push_back(2);
graf[1].weight.push_back(8);
graf[2].adj.push_back(1);
graf[2].weight.push_back(8);
graf[1].adj.push_back(3);
graf[1].weight.push_back(9);
graf[3].adj.push_back(1);
graf[3].weight.push_back(9);
graf[1].adj.push_back(4);
graf[1].weight.push_back(7);
graf[4].adj.push_back(1);
graf[4].weight.push_back(7);
graf[2].adj.push_back(4);
graf[2].weight.push_back(5);
graf[4].adj.push_back(2);
graf[4].weight.push_back(5);
graf[3].adj.push_back(4);
graf[3].weight.push_back(15);
graf[4].adj.push_back(3);
graf[4].weight.push_back(15);
graf[3].adj.push_back(5);
graf[3].weight.push_back(6);
graf[5].adj.push_back(3);
graf[5].weight.push_back(6);
graf[4].adj.push_back(5);
graf[4].weight.push_back(8);
graf[5].adj.push_back(4);
graf[5].weight.push_back(8);
graf[4].adj.push_back(6);
graf[4].weight.push_back(9);
graf[6].adj.push_back(4);
graf[6].weight.push_back(9);
graf[5].adj.push_back(6);
graf[5].weight.push_back(11);
graf[6].adj.push_back(5);
graf[6].weight.push_back(11);
printf("%d\n",Prim());
return 0;
}
