Minimum Spanning Tree Prim's Algorithm in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a Minimum Spanning Tree (MST)?

A Minimum Spanning Tree is a subset of edges in a connected, weighted graph that connects all vertices together without any cycles and with the minimum possible total edge weight.

beginner

What is the main idea behind Prim's Algorithm?

Prim's Algorithm builds the MST by starting from any vertex and repeatedly adding the smallest edge that connects a vertex in the MST to a vertex outside it until all vertices are included.

beginner

In Prim's Algorithm, how do we choose the next edge to add?

We choose the edge with the smallest weight that connects a vertex already in the MST to a vertex not yet included.

intermediate

What data structure is commonly used to efficiently select the next edge in Prim's Algorithm?

A priority queue (often implemented as a min-heap) is used to quickly find and extract the edge with the smallest weight.

intermediate

Why does Prim's Algorithm always produce a Minimum Spanning Tree?

Because it always picks the smallest edge connecting the growing MST to a new vertex, ensuring no cycles and minimal total weight by the greedy choice property.

What does Prim's Algorithm start with?

A Any single vertex

B All edges sorted by weight

C All vertices at once

D The edge with the largest weight

Prim's Algorithm starts from any single vertex and grows the MST by adding edges.

Which data structure helps Prim's Algorithm pick the next smallest edge efficiently?

A Stack

B Queue

C Priority Queue

D Linked List

A priority queue (min-heap) is used to efficiently select the smallest edge.

Prim's Algorithm is an example of which type of algorithm?

A Dynamic Programming

B Greedy

C Divide and Conquer

D Backtracking

Prim's Algorithm uses a greedy approach by always choosing the smallest edge.

What condition must the graph satisfy for Prim's Algorithm to work correctly?

A Graph must be connected

B Graph must have negative weights

C Graph must be directed

D Graph must be disconnected

Prim's Algorithm requires the graph to be connected to reach all vertices.

What does Prim's Algorithm add to the MST at each step?

A A vertex with the largest degree

B A random edge

C All edges connected to the current vertex

D An edge with the smallest weight connecting MST to a new vertex

At each step, Prim's Algorithm adds the smallest edge connecting the MST to a new vertex.

Explain how Prim's Algorithm builds a Minimum Spanning Tree step-by-step.

Think about growing the tree one edge at a time by choosing the smallest connecting edge.

Describe why Prim's Algorithm guarantees the minimum total weight for the spanning tree.

Focus on how choosing the smallest edge at each step leads to overall minimum weight.

Practice

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

easy

A. To sort the vertices of a graph

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

C. To detect cycles in a graph

D. To find the shortest path between two nodes

Solution

1. Step 1: Understand Prim's algorithm purpose

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

2. Step 2: Compare with other graph problems

   Shortest path and cycle detection are different problems; Prim's focuses on minimum spanning tree.

3. Final Answer:

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

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 vertices

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] = {INT_MAX};

D. int key[V] = NULL;

Solution

1. Step 1: Understand key array purpose

   The key array stores minimum edge weights to include vertices; initially set to a very large value (infinity).

2. Step 2: Check initialization methods

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

3. Final Answer:

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

4. Quick Check:

   Initialize keys to INT_MAX using loop [OK]

Hint: Use a loop to set all keys to INT_MAX, not zero [OK]

Common Mistakes:

• Initializing key array with zeros
• Using incorrect array initialization syntax
• Assigning NULL to an int array

3. Consider the following adjacency matrix for a graph with 4 vertices:

int graph[4][4] = {
  {0, 2, 0, 6},
  {2, 0, 3, 8},
  {0, 3, 0, 0},
  {6, 8, 0, 0}
};

If Prim's algorithm starts from vertex 0, what is the total weight of the minimum spanning tree?

medium

A. 14

B. 9

C. 11

D. 10

Solution

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

   Edges chosen: 0-1 (2), 1-2 (3), 0-3 (6)

2. Step 2: Sum the weights of chosen edges

   Total weight = 2 + 3 + 6 = 11

3. Step 3: Re-evaluate chosen edges for MST

   Actually, the MST edges are 0-1 (2), 1-2 (3), and 1-3 (8) or 0-3 (6) but 0-3 (6) is better than 1-3 (8). So edges are 0-1 (2), 1-2 (3), 0-3 (6) totaling 11.

4. Step 4: Check if a smaller total weight is possible

   Check edges: 0-1 (2), 1-2 (3), 1-3 (8) sum to 13; 0-1 (2), 1-2 (3), 3-4 (not present); so 11 is correct.

5. Final Answer:

   11 -> Option C

6. Quick Check:

   Sum edges 2+3+6 = 11 [OK]

Hint: Add smallest edges connecting new vertices stepwise [OK]

Common Mistakes:

• Including edges that form cycles
• Adding edges with zero weight incorrectly
• Starting from wrong vertex

4. The following code snippet is part of Prim's algorithm implementation. Identify the error:

for (int count = 0; count < V - 1; count++) {
  int u = minKey(key, mstSet);
  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 loop should run from 1 to V instead of 0 to V-1

B. mstSet[u] should be set to false instead of true

C. The key array is not updated correctly inside the inner loop

D. The condition 'graph[u][v]' should be 'graph[u][v] != 0'

Solution

1. Step 1: Analyze the condition 'graph[u][v]'

   In C, non-zero values are true, zero is false. Using 'graph[u][v]' works but is less clear; better to explicitly check 'graph[u][v] != 0'.

2. Step 2: Check other parts for correctness

   Loop range and key update logic are correct; mstSet[u] should be true to mark visited.

3. Final Answer:

   The condition 'graph[u][v]' should be 'graph[u][v] != 0' -> Option D

4. Quick Check:

   Explicit non-zero check improves clarity [OK]

Hint: Use explicit '!= 0' for clarity in conditions [OK]

Common Mistakes:

• Misunderstanding loop ranges
• Incorrectly toggling mstSet flags
• Not updating key array properly

5. Given a graph with vertices and edges as below:

Vertices: 5
Edges:
0 - 1 (2)
0 - 3 (6)
1 - 2 (3)
1 - 3 (8)
1 - 4 (5)
2 - 4 (7)
3 - 4 (9)

If Prim's algorithm starts from vertex 0, which edges will be included in the minimum spanning tree?

hard

A. Edges: 0-1, 1-2, 1-4, 0-3

B. Edges: 0-1, 1-3, 3-4, 2-4

C. Edges: 0-3, 3-4, 1-4, 1-2

D. Edges: 0-1, 1-2, 2-4, 3-4

Solution

1. Step 1: Start from vertex 0 and pick smallest edges stepwise

   Choose 0-1 (2), then from {0,1} choose 1-2 (3), then 1-4 (5), finally 0-3 (6).

2. Step 2: Confirm no cycles and all vertices included

   All vertices 0,1,2,3,4 are connected with minimum total weight edges without cycles.

3. Final Answer:

   Edges: 0-1, 1-2, 1-4, 0-3 -> Option A

4. Quick Check:

   Pick smallest edges connecting new vertices [OK]

Hint: Always pick smallest edge connecting visited to unvisited vertices [OK]

Common Mistakes:

• Choosing edges with larger weights first
• Including edges that create cycles
• Missing to connect all vertices