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Minimum Spanning Trees (MST) are a fundamental concept in graph theory. They represent the subset of edges that connects all the vertices in a graph with the minimum possible total edge weight, while ensuring there are no cycles. There are two main algorithms for finding MSTs: Kruskal's Algorithm and Prim's Algorithm. In this explanation, we'll focus on Prim's Algorithm.
A graph is typically represented as G = (V, E), where V is the set of vertices and E is the set of edges. An MST is a tree that spans all the vertices in the graph and has the smallest sum of edge weights. If the graph is connected, an MST exists, and it ensures that the total cost of connecting all nodes is minimized.
To visualize this, consider a graph where each edge has a weight. The MST would be the selection of edges that connects every node without forming any loops and with the least overall cost. The bold lines in the image illustrate the path taken by the MST.
Prim’s algorithm works by maintaining two sets of vertices: one that is already included in the MST (set A), and another that is not yet part of the MST (set B). Initially, set A contains only one vertex (often chosen arbitrarily, such as vertex 1), and set B contains the rest. At each step, the algorithm finds the edge with the smallest weight that connects a vertex in set A to a vertex in set B, adds that edge to the MST, and moves the connected vertex from set B to set A. This process continues until all vertices are in set A.
The basic idea of the algorithm can be optimized using a priority queue or an array to keep track of the minimum edge weights connecting the current MST to the remaining vertices. By maintaining an array `pCost[i]`, which stores the minimum weight to connect vertex `i` to the MST, the algorithm can efficiently find the next vertex to add.
This optimization reduces the time complexity from O(n³) to O(n²), making it more efficient for larger graphs. The algorithm iteratively selects the vertex with the smallest pCost value, updates the pCost array based on the newly added vertex, and continues until all vertices are part of the MST.
Here is a sample implementation of Prim's algorithm in C++:
```cpp
#include
#include
#include
using namespace std;
const int MAX = 1024;
const int INF = 2147483647; // Set maximum weight
int N, M;
vector> pMap[MAX]; // Adjacency list
void Prim();
int main() {
cin >> N >> M;
for (int i = 1; i <= M; i++) {
int u, v, w;
cin >> u >> v >> w;
pMap[u].push_back(make_pair(v, w));
pMap[v].push_back(make_pair(u, w));
}
Prim();
return 0;
}
void Prim() {
int nCost = 0;
vector pMST; // Stores nodes in MST
int pCost[MAX]; // Minimum cost to connect to set A
pMST.push_back(1); // Start with node 1
pCost[1] = 0;
// Initialize pCost for other nodes
for (int i = 2; i <= N; i++) {
pCost[i] = INF;
}
// Initialize pCost for neighbors of node 1
for (int i = 0; i < pMap[1].size(); i++) {
pCost[pMap[1][i].first] = pMap[1][i].second;
}
for (int i = 1; i <= N - 1; i++) {
int nVertex = 0, nWeight = INF;
// Find the vertex with the minimum cost
for (int j = 1; j <= N; j++) {
if (nWeight > pCost[j] && pCost[j] != 0) {
nVertex = j;
nWeight = pCost[j];
}
}
pCost[nVertex] = 0;
pMST.push_back(nVertex);
nCost += nWeight;
// Update pCost for neighbors of the new vertex
for (int j = 0; j < pMap[nVertex].size(); j++) {
if (pCost[pMap[nVertex][j].first] != 0 &&
pCost[pMap[nVertex][j].first] > pMap[nVertex][j].second) {
pCost[pMap[nVertex][j].first] = pMap[nVertex][j].second;
}
}
}
cout << "MST Cost is " << nCost << endl;
cout << "The vertices in MST are ";
for (int i = 0; i < pMST.size(); i++) {
cout << pMST[i] << " ";
}
cout << endl;
}
```
This code reads input for a graph, builds an adjacency list, and then applies Prim's algorithm to compute the MST. It outputs both the total cost of the MST and the list of vertices included in it. This approach is efficient and widely used in various applications, including network design and clustering problems.
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