Minimum Spanning Tree Prim's Algorithm in DSA Rust - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to initialize the priority queue for Prim's algorithm.

DSA Rust

let mut pq = std::collections::BinaryHeap::new();
pq.push([1]);

Drag options to blanks, or click blank then click option'

A(usize::MAX, usize::MAX)

B(0, 0)

C(usize::MAX, 0)

D(0, usize::MAX)

Attempts:

3 left

💡 Hint

Common Mistakes

Using maximum usize values instead of zero for the starting cost.

Swapping the order of cost and vertex in the tuple.

We start Prim's algorithm by pushing the starting vertex with cost 0 into the priority queue.

2fill in blank

medium

Complete the code to mark a vertex as visited in Prim's algorithm.

DSA Rust

let mut visited = vec![false; n];
visited[[1]] = true;

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An

B1

C0

Dusize::MAX

Attempts:

3 left

💡 Hint

Common Mistakes

Using 1 instead of 0 as the starting vertex index.

Using n or usize::MAX which are out of bounds.

The starting vertex is usually vertex 0, so we mark visited[0] as true after adding it to the MST.

3fill in blank

hard

Fix the error in the edge relaxation step of Prim's algorithm.

DSA Rust

for &(cost, to) in &graph[[1]] {
    if !visited[to] && cost < dist[to] {
        dist[to] = cost;
        pq.push(([2], to));
    }
}

Drag options to blanks, or click blank then click option'

Acost

B-cost

Cdist[to]

D-dist[to]

Attempts:

3 left

💡 Hint

Common Mistakes

Pushing cost without negation causes wrong priority order.

Using dist[to] instead of cost in the priority queue.

We iterate over edges from the current vertex. We push (-cost, to) because Rust's BinaryHeap is a max-heap, so we negate cost to simulate a min-heap.

4fill in blank

hard

Fill both blanks to correctly pop the next vertex and check if it is visited.

DSA Rust

while let Some(([1], [2])) = pq.pop() {
    let cost = -cost;
    if visited[to] {
        continue;
    }

Drag options to blanks, or click blank then click option'

Acost

Bto

Cvertex

Dweight

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong variable names causing confusion.

Not negating cost after popping.

We pop (cost, to) from the priority queue. We negate cost to get the original positive cost. Then we check if 'to' vertex is visited.

5fill in blank

hard

Fill all three blanks to update the MST cost and mark the vertex visited.

DSA Rust

mst_cost += [1];
visited[[2]] = true;
current_vertex = [3];

Drag options to blanks, or click blank then click option'

Acost

Bto

Ccurrent_vertex

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Adding wrong variable to mst_cost.

Marking wrong vertex as visited.

Not updating current_vertex correctly.

Add the edge cost to mst_cost, mark the 'to' vertex visited, and update current_vertex to 'to' for the next iteration.

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