Minimum Spanning Tree Prim's Algorithm in DSA Rust - 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 for the Minimum Spanning Tree.

📋 What You'll Learn

Create a graph using an adjacency list with exact edges and weights

Create a vector to track visited nodes

Implement Prim's algorithm core logic using a priority queue

Print the total minimum cost after running the algorithm

💡 Why This Matters

🌍 Real World

Network design, connecting computers or cities with minimum cable or road cost.

💼 Career

Useful for software engineers working on network optimization, infrastructure planning, and graph algorithms.

1

Create the graph as an adjacency list

Create a variable called graph as a vector of vectors of tuples. The graph should have 5 nodes (0 to 4). Add these edges with weights exactly:
0 - 1 (2), 0 - 3 (6), 1 - 2 (3), 1 - 3 (8), 1 - 4 (5), 2 - 4 (7), 3 - 4 (9). Each edge should be added in both directions (undirected graph).

DSA Rust

# Your code here to create the graph adjacency list

Hint

Use a Vec> where each inner vector holds tuples of (neighbor_node, weight).

2

Create a visited vector to track nodes

Create a mutable vector called visited of type Vec with 5 elements, all set to false. This will track which nodes have been visited.

DSA Rust

let graph: Vec> = vec![
    vec![(1, 2), (3, 6)],
    vec![(0, 2), (2, 3), (3, 8), (4, 5)],
    vec![(1, 3), (4, 7)],
    vec![(0, 6), (1, 8), (4, 9)],
    vec![(1, 5), (2, 7), (3, 9)]
];

// Your code here to create the visited vector

Hint

Use vec![false; 5] to create a vector with 5 false values.

3

Implement Prim's algorithm core logic

Use std::collections::BinaryHeap as a min-heap by pushing tuples with negative weights. Create a mutable variable total_cost set to 0. Start from node 0 by pushing (0, 0) into the heap. While the heap is not empty, pop the smallest edge. If the node is not visited, mark it visited and add the edge weight to total_cost. Then push all edges from this node to the heap if their destination is not visited.

DSA Rust

let graph: Vec> = vec![
    vec![(1, 2), (3, 6)],
    vec![(0, 2), (2, 3), (3, 8), (4, 5)],
    vec![(1, 3), (4, 7)],
    vec![(0, 6), (1, 8), (4, 9)],
    vec![(1, 5), (2, 7), (3, 9)]
];

let mut visited: Vec = vec![false; 5];

// Your code here to implement Prim's algorithm core logic

Hint

Use a max-heap with negative weights to simulate a min-heap. Remember to mark nodes visited before adding their edges.

4

Print the total minimum cost

Write a println! statement to print the total_cost variable.

DSA Rust

use std::collections::BinaryHeap;

let graph: Vec> = vec![
    vec![(1, 2), (3, 6)],
    vec![(0, 2), (2, 3), (3, 8), (4, 5)],
    vec![(1, 3), (4, 7)],
    vec![(0, 6), (1, 8), (4, 9)],
    vec![(1, 5), (2, 7), (3, 9)]
];

let mut visited: Vec = vec![false; 5];

let mut heap = BinaryHeap::new();
let mut total_cost = 0;

heap.push((0, 0));

while let Some((neg_weight, node)) = heap.pop() {
    if visited[node] {
        continue;
    }
    visited[node] = true;
    total_cost += -neg_weight;

    for &(neighbor, weight) in &graph[node] {
        if !visited[neighbor] {
            heap.push((-weight as i32, neighbor));
        }
    }
}

// Your code here to print total_cost

Hint

Use println!("{}", total_cost); to display the final minimum cost.

Practice

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

easy

A. To detect cycles in a graph

B. To find the minimum spanning tree connecting all vertices with minimum total edge weight

C. To find the shortest path between two vertices

D. To sort vertices in topological order

Solution

1. Step 1: Understand Prim's Algorithm purpose

   Prim's Algorithm builds a tree that connects all vertices with the smallest possible sum of edge weights.

2. Step 2: Compare with other graph algorithms

   Shortest path algorithms find paths, cycle detection finds loops, and topological sort orders vertices, which are different goals.

3. Final Answer:

   To find the minimum spanning tree connecting all vertices with minimum total edge weight -> Option B

4. Quick Check:

   Prim's Algorithm = Minimum Spanning Tree [OK]

Hint: Prim's builds minimum spanning tree by adding cheapest edges [OK]

Common Mistakes:

• Confusing Prim's with shortest path algorithms
• Thinking it detects cycles
• Mixing it with topological sorting

2. Which Rust data structure is best suited to efficiently pick the next minimum edge in Prim's Algorithm?

easy

A. Vec (Vector)

B. LinkedList

C. HashMap

D. BinaryHeap (Min-Heap)

Solution

1. Step 1: Identify operation needed in Prim's Algorithm

   Prim's needs to repeatedly select the edge with the smallest weight efficiently.

2. Step 2: Choose data structure for minimum extraction

   A BinaryHeap (min-heap) allows quick access to the smallest element, unlike Vec or LinkedList which require scanning.

3. Final Answer:

   BinaryHeap (Min-Heap) -> Option D

4. Quick Check:

   Min-Heap = Efficient minimum edge selection [OK]

Hint: Use min-heap to get smallest edge fast in Prim's [OK]

Common Mistakes:

• Using Vec which is slow for minimum extraction
• Choosing HashMap which is for key-value, not min extraction
• Using LinkedList which is inefficient for this

3. What is the output of the following Rust code snippet implementing Prim's Algorithm on a graph with edges [(0,1,2), (1,2,3), (0,2,1)] starting from vertex 0?

let edges = vec![(0,1,2), (1,2,3), (0,2,1)];
// Prim's MST total weight calculation
// Output the total weight of MST

medium

A. 3

B. 5

C. 4

D. 6

Solution

1. Step 1: List edges and weights

   Edges: (0-1)=2, (1-2)=3, (0-2)=1. Start from vertex 0.

2. Step 2: Build MST using Prim's Algorithm

   Start at 0, pick smallest edge (0-2)=1, then pick next smallest connecting edge (0-1)=2. Total weight = 1 + 2 = 3.

3. Step 3: Check if all vertices connected

   Vertices 0,1,2 connected with edges (0-2) and (0-1). Edge (1-2) with weight 3 is not needed.

4. Final Answer:

   3 -> Option A

5. Quick Check:

   MST weight = 1 + 2 = 3 [OK]

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

Common Mistakes:

• Adding all edges instead of MST edges
• Choosing heavier edges first
• Not starting from given start vertex

4. Identify the bug in this Rust snippet implementing Prim's Algorithm:

let mut visited = vec![false; n];
let mut heap = BinaryHeap::new();
heap.push(Reverse((0, 0)));
while let Some(Reverse((cost, u))) = heap.pop() {
    if visited[u] { continue; }
    visited[u] = true;
    for &(v, w) in &graph[u] {
        if !visited[v] {
            heap.push(Reverse((w, v)));
        }
    }
}

medium

A. The heap should be a max-heap, not min-heap

B. The heap should store (cost, vertex) as (u, cost) instead

C. The visited check is missing before pushing neighbors

D. The graph adjacency list should store (vertex, weight) not (weight, vertex)

Solution

1. Step 1: Analyze visited check placement

   Currently, neighbors are pushed without checking if visited, causing duplicates in heap.

2. Step 2: Fix by checking visited before pushing neighbors

   Adding a visited check before pushing neighbors avoids unnecessary heap entries and improves efficiency.

3. Final Answer:

   The visited check is missing before pushing neighbors -> Option C

4. Quick Check:

   Check visited before pushing neighbors to heap [OK]

Hint: Check visited before pushing neighbors to avoid duplicates [OK]

Common Mistakes:

• Ignoring visited check before pushing neighbors
• Confusing min-heap with max-heap
• Swapping vertex and weight in tuples

5. Given a weighted undirected graph with vertices 0 to 4 and edges:

(0,1,10), (0,2,6), (0,3,5), (1,3,15), (2,3,4), (3,4,2)

Using Prim's Algorithm starting from vertex 0, what is the total weight of the minimum spanning tree?

hard

A. 21

B. 17

C. 27

D. 22

Solution

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

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

2. Step 2: Add edges connecting new vertices with minimum weight

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

3. Step 3: Continue adding minimum edges connecting unvisited vertices

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

4. Step 4: Add last vertex 1 with minimum edge

   Visited {0,2,3,4}, edges: (0-1)=10, (1-3)=15. Pick (0-1)=10.

5. Step 5: Calculate total weight

   Total = 5 + 2 + 4 + 10 = 21.

6. Final Answer:

   21 -> Option A

7. Quick Check:

   Sum of MST edges = 21 [OK]

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

Common Mistakes:

• Choosing heavier edges before lighter ones
• Not updating visited set correctly
• Adding edges that create cycles