Minimum Spanning Tree Prim's Algorithm in DSA C++ - Build from Scratch

Minimum Spanning Tree using Prim's Algorithm

📖 Scenario: You are working as a network engineer. You need to connect several computers with cables. Each cable has a cost. Your goal is to connect all computers with the least total cost.

🎯 Goal: Build a program that finds the minimum total cost to connect all computers using Prim's algorithm on a weighted graph.

📋 What You'll Learn

Create an adjacency matrix to represent the graph with exact weights

Create a variable to track the number of vertices

Implement Prim's algorithm to find the minimum spanning tree

Print the edges chosen and the total minimum cost

💡 Why This Matters

🌍 Real World

Network engineers use minimum spanning trees to design cost-efficient networks connecting computers or routers.

💼 Career

Understanding Prim's algorithm helps in roles involving network design, optimization, and graph-based problem solving.

1

Create the graph adjacency matrix

Create a 2D vector called graph with 5 vertices representing the following weighted edges:
graph = {{0, 2, 0, 6, 0}, {2, 0, 3, 8, 5}, {0, 3, 0, 0, 7}, {6, 8, 0, 0, 9}, {0, 5, 7, 9, 0}}

DSA C++

# Create the adjacency matrix graph with 5 vertices
# Your code here

Hint

Use vector> to create a 2D matrix with the exact values given.

2

Set the number of vertices

Create an integer variable called V and set it to the number of vertices in the graph, which is 5.

DSA C++

#include 
using namespace std;

vector> graph = {
    {0, 2, 0, 6, 0},
    {2, 0, 3, 8, 5},
    {0, 3, 0, 0, 7},
    {6, 8, 0, 0, 9},
    {0, 5, 7, 9, 0}
};

// Set the number of vertices V
# Your code here

Hint

Count the number of vertices in the graph and assign it to V.

3

Implement Prim's algorithm core logic

Write a function called primMST that takes graph and V as parameters and finds the minimum spanning tree using Prim's algorithm. Use arrays key, parent, and mstSet inside the function. The function should fill parent with the MST edges.

DSA C++

#include 
using namespace std;

vector> graph = {
    {0, 2, 0, 6, 0},
    {2, 0, 3, 8, 5},
    {0, 3, 0, 0, 7},
    {6, 8, 0, 0, 9},
    {0, 5, 7, 9, 0}
};

int V = 5;

// Write the primMST function below
# Your code here

Hint

Use arrays to track the minimum edge weights and visited vertices. Pick the smallest edge each time.

4

Print the MST edges and total cost

In main(), create a parent vector of size V. Call primMST(graph, V, parent). Then print each edge in the MST as parent[i] - i : weight and print the total minimum cost as Total minimum cost: X.

DSA C++

#include 
#include 
#include 
using namespace std;

vector> graph = {
    {0, 2, 0, 6, 0},
    {2, 0, 3, 8, 5},
    {0, 3, 0, 0, 7},
    {6, 8, 0, 0, 9},
    {0, 5, 7, 9, 0}
};

int V = 5;

int minKey(const vector& key, const vector& mstSet) {
    int min = INT_MAX, min_index = -1;
    for (int v = 0; v >& graph, int V, vector& parent) {
    vector key(V, INT_MAX);
    vector mstSet(V, false);
    key[0] = 0;
    parent[0] = -1;

    for (int count = 0; count  parent(V);
    primMST(graph, V, parent);

    // Print the MST edges and total cost
    // Your code here
}

Hint

Loop from 1 to V-1 and print edges from parent[i] to i with their weights. Sum the weights for total cost.

Practice

1. What is the main goal of Prim's Algorithm in graph theory?

easy

A. To find the shortest path between two nodes

B. To detect cycles in a graph

C. To find the minimum spanning tree of a connected weighted graph

D. To sort the vertices of a graph

Solution

1. Step 1: Understand Prim's Algorithm purpose

   Prim's Algorithm is designed to find a tree that connects all vertices with the minimum total edge weight.

2. Step 2: Differentiate from other graph algorithms

   It is not used for shortest path, cycle detection, or sorting vertices, which are different problems.

3. Final Answer:

   To find the minimum spanning tree of a connected weighted graph -> Option C

4. Quick Check:

   Prim's Algorithm = Minimum Spanning Tree [OK]

Hint: Prim's always builds a tree with minimum total edge weight [OK]

Common Mistakes:

• Confusing Prim's with shortest path algorithms
• Thinking it detects cycles
• Assuming it sorts nodes

2. Which of the following is the correct way to initialize the key array in Prim's Algorithm in C++?

easy

A. int key[V]; for(int i=0; i

B. int key[V] = {0};

C. int key[V]; for(int i=0; i

D. int key[V]; for(int i=0; i

Solution

1. Step 1: Understand key array role

   The key array stores minimum edge weights to include each vertex; initially, all are set to a very large value (infinity).

2. Step 2: Check initialization options

   Setting key[i] = INT_MAX correctly represents infinity; other options set to 0 or -1 are incorrect for this purpose.

3. Final Answer:

   int key[V]; for(int i=0; i -> Option A

4. Quick Check:

   Initialize keys with INT_MAX for minimum edge tracking [OK]

Hint: Initialize keys with INT_MAX to represent infinity [OK]

Common Mistakes:

• Initializing keys with 0 instead of INT_MAX
• Using negative values incorrectly
• Not initializing keys at all

3. Consider the following graph with 4 vertices and edges with weights:

0 - 1 : 10
0 - 2 : 6
0 - 3 : 5
1 - 3 : 15
2 - 3 : 4

What is the total weight of the minimum spanning tree found by Prim's Algorithm starting from vertex 0?

medium

A. 21

B. 19

C. 20

D. 25

Solution

1. Step 1: Start from vertex 0 and pick minimum edges

   Edges from 0: (0-3)=5, (0-2)=6, (0-1)=10. Pick (0-3)=5 first.

2. Step 2: Add next minimum edge connecting new vertices

   From vertices {0,3}, edges are (3-2)=4, (3-1)=15, (0-2)=6, (0-1)=10. Pick (3-2)=4 next.

3. Step 3: Add last vertex with minimum edge

   From {0,3,2}, edges to 1 are (0-1)=10 and (3-1)=15. Pick (0-1)=10.

4. Step 4: Calculate total weight

   Total = 5 + 4 + 10 = 19.

5. Final Answer:

   19 -> Option B

6. Quick Check:

   Sum of chosen edges = 19 [OK]

Hint: Pick smallest edge connecting new vertex each step [OK]

Common Mistakes:

• Choosing edges with larger weights first
• Not updating the set of included vertices
• Adding edges that form cycles

4. The following C++ code snippet is intended to implement Prim's Algorithm but has a bug. What is the error?

const int V = 4;
int key[V] = {INT_MAX, INT_MAX, INT_MAX, INT_MAX};
bool mstSet[V] = {false, false, false, false};
key[0] = 0;
for (int count = 0; count < V - 1; count++) {
    int u = -1;
    for (int v = 0; v < V; v++) {
        if (!mstSet[v] && (u == -1 || key[v] < key[u]))
            u = v;
    }
    mstSet[u] = true;
    for (int v = 0; v < V; v++) {
        if (graph[u][v] && !mstSet[v] && graph[u][v] < key[v])
            key[v] = graph[u][v];
    }
}

medium

A. The condition 'graph[u][v]' should check for non-zero edges

B. The loop should run from 0 to V instead of V - 1

C. The key array is not initialized with INT_MAX

D. The variable 'u' is initialized incorrectly inside the loop

Solution

1. Step 1: Analyze the neighbor update condition

   The condition if(graph[u][v] && !mstSet[v] && graph[u][v] < key[v]) uses graph[u][v] as a boolean check (true if non-zero).

2. Step 2: Identify the bug

   This implicit check assumes positive weights and treats 0 as no edge. Standard implementations explicitly use graph[u][v] != 0 to clearly indicate checking for edge existence (non-zero weight).

3. Step 3: Rule out other options

   A: 'u' is correctly initialized to -1 inside each iteration. B: V-1 iterations build the tree correctly. C: key array is initialized with INT_MAX.

4. Final Answer:

   The condition 'graph[u][v]' should check for non-zero edges -> Option A

5. Quick Check:

   Edge check: graph[u][v] != 0 [OK]

Hint: Check edge existence condition carefully for zero weights [OK]

Common Mistakes:

• Misinitializing 'u' outside the loop
• Incorrect loop range
• Not initializing key array properly

5. Given a graph with vertices 0 to 5 and edges:

0-1: 3, 0-3: 1, 1-2: 1, 1-3: 3, 1-4: 10, 2-4: 5, 3-4: 6, 4-5: 2

If Prim's Algorithm starts from vertex 0, which edge will NOT be included in the minimum spanning tree?

hard

A. Edge 1-2 with weight 1

B. Edge 0-1 with weight 3

C. Edge 4-5 with weight 2

D. Edge 3-4 with weight 6

Solution

1. Step 1: Build MST step-by-step from vertex 0

   Start at 0: edges (0-3)=1, (0-1)=3. Pick (0-3)=1 first.

2. Step 2: Add next minimum edges connecting new vertices

   From {0,3}, edges: (3-1)=3, (3-4)=6, (0-1)=3. Pick (0-1)=3 or (3-1)=3 (same weight). Pick (0-1)=3.

3. Step 3: Continue adding minimum edges

   From {0,1,3}, edges: (1-2)=1, (1-4)=10, (3-4)=6. Pick (1-2)=1.

4. Step 4: Add edge to vertex 4

   From {0,1,2,3}, edges: (2-4)=5, (3-4)=6, (1-4)=10. Pick (2-4)=5.

5. Step 5: Add edge to vertex 5

   From {0,1,2,3,4}, edge: (4-5)=2.

6. Step 6: Identify excluded edge

   Edge (3-4)=6 is not chosen because (2-4)=5 is cheaper and connects 4 earlier.

7. Final Answer:

   Edge 3-4 with weight 6 -> Option D

8. Quick Check:

   Exclude heavier edge when cheaper path exists [OK]

Hint: Always pick smallest edge connecting new vertex, exclude heavier alternatives [OK]

Common Mistakes:

• Including heavier edges when cheaper ones exist
• Not updating candidate edges after each addition
• Assuming all edges connected to MST are included