Minimum spanning tree (Prim's) in Data Structures Theory - Time & Space Complexity

Time Complexity: Minimum spanning tree (Prim's)

O(n^2)

Understanding Time Complexity

We want to understand how the time needed to find a minimum spanning tree using Prim's algorithm changes as the graph grows.

Specifically, how does the work increase when the number of nodes and edges increases?

Scenario Under Consideration

Analyze the time complexity of the following Prim's algorithm code snippet.

function primMST(graph) {
  let mstSet = new Set();
  let key = Array(graph.length).fill(Infinity);
  key[0] = 0;
  for (let count = 0; count < graph.length; count++) {
    let u = minKey(key, mstSet);
    mstSet.add(u);
    for (let v = 0; v < graph.length; v++) {
      if (graph[u][v] && !mstSet.has(v) && graph[u][v] < key[v]) {
        key[v] = graph[u][v];
      }
    }
  }
  return key;
}
    

This code finds the minimum spanning tree by repeatedly adding the closest vertex not yet in the tree.

Identify Repeating Operations

Look at the loops and repeated steps:

• Primary operation: The outer loop runs once for each vertex to add it to the MST.
• How many times: For each vertex, the inner loop checks all vertices to update keys.

How Execution Grows With Input

As the number of vertices (n) grows, the algorithm checks all vertices for each addition to the MST.

Input Size (n)	Approx. Operations
10	About 100 checks
100	About 10,000 checks
1000	About 1,000,000 checks

Pattern observation: The work grows roughly with the square of the number of vertices.

Final Time Complexity

Time Complexity: O(n^2)

This means if you double the number of vertices, the time needed roughly quadruples.

Common Mistake

[X] Wrong: "Prim's algorithm always runs in linear time because it just adds one vertex at a time."

[OK] Correct: Each addition requires checking all vertices to find the minimum edge, so the total work grows faster than just the number of vertices.

Interview Connect

Understanding how Prim's algorithm scales helps you explain your choices and shows you grasp how graph size affects performance.

Self-Check

"What if we used a priority queue (min-heap) to select the next vertex? How would the time complexity change?"

Practice

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

easy

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

B. To find the shortest path between two vertices

C. To detect cycles in a graph

D. To sort vertices based on their degree

Solution

1. Step 1: Understand Prim's algorithm purpose

   Prim's algorithm builds a tree that connects all vertices with the minimum sum of edge weights.

2. Step 2: Compare with other graph problems

   Unlike shortest path or cycle detection, Prim's focuses on spanning all vertices with minimal total edge cost.

3. Final Answer:

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

4. Quick Check:

   Prim's algorithm = Minimum spanning tree [OK]

Hint: Prim's always finds the cheapest tree connecting all points [OK]

Common Mistakes:

• Confusing minimum spanning tree with shortest path
• Thinking it detects cycles
• Assuming it sorts vertices

2. Which data structure is commonly used in Prim's algorithm to select the next edge with the smallest weight?

easy

A. Priority queue (min-heap)

B. Stack

C. Queue

D. Hash table

Solution

1. Step 1: Identify how Prim's picks edges

   Prim's algorithm always picks the edge with the smallest weight connecting the tree to a new vertex.

2. Step 2: Choose the right data structure

   A priority queue (min-heap) efficiently retrieves the smallest edge weight at each step.

3. Final Answer:

   Priority queue (min-heap) -> Option A

4. Quick Check:

   Smallest edge selection = Priority queue [OK]

Hint: Use a min-heap to always pick the cheapest edge next [OK]

Common Mistakes:

• Using stack or queue which don't sort by weight
• Confusing hash table with priority queue
• Ignoring edge weights in selection

3. Consider the following weighted graph edges:
A-B: 2, A-C: 3, B-C: 1, B-D: 4, C-D: 5
Using Prim's algorithm starting from vertex A, what is the total weight of the minimum spanning tree?

medium

A. 5

B. 8

C. 7

D. 6

Solution

1. Step 1: Start from vertex A and pick smallest edge

   Edges from A: A-B(2), A-C(3). Pick A-B with weight 2.

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

   From B and A, edges are B-C(1), B-D(4), A-C(3). Pick B-C with weight 1.

3. Step 3: Add next smallest edge to connect remaining vertex D

   Edges from B, C, A to D: B-D(4), C-D(5). Pick B-D with weight 4.

4. Step 4: Calculate total weight

   Total = 2 + 1 + 4 = 7.

5. Final Answer:

   7 -> Option C

6. Quick Check:

   Sum edges 2+1+4 = 7 [OK]

Hint: Add edges with smallest weight connecting new vertices [OK]

Common Mistakes:

• Choosing edges without checking if vertex is already included
• Adding edges that form cycles
• Forgetting to sum all chosen edges

4. In the following code snippet for Prim's algorithm, what is the error?

graph = { 'A': {'B': 2, 'C': 3}, 'B': {'A': 2, 'C': 1, 'D': 4}, 'C': {'A': 3, 'B': 1, 'D': 5}, 'D': {'B': 4, 'C': 5} }
visited = set()
keys = {v: float('inf') for v in graph}
keys['A'] = 0
while len(visited) < len(graph):
    u = min((k for k in keys if k not in visited), key=lambda x: keys[x])
    visited.add(u)
    for v, w in graph[u].items():
        if v not in visited and w < keys[v]:
            keys[v] = w

medium

A. The keys dictionary is not updated correctly

B. The algorithm does not track parent nodes to build the MST

C. The min function call is incorrect

D. The visited set is never updated

Solution

1. Step 1: Analyze the code's MST construction

   The code updates keys but does not store which vertex connects to which, so MST edges are unknown.

2. Step 2: Identify missing MST parent tracking

   Prim's algorithm requires tracking parent nodes to reconstruct the MST edges, which is missing here.

3. Final Answer:

   The algorithm does not track parent nodes to build the MST -> Option B

4. Quick Check:

   Missing parent tracking = MST incomplete [OK]

Hint: Track parents to know MST edges, not just weights [OK]

Common Mistakes:

• Assuming keys alone build MST
• Ignoring parent node tracking
• Thinking visited set is not updated

5. You have a graph with vertices representing cities and edges representing roads with costs. If some roads have zero cost, how does Prim's algorithm handle these edges when building the minimum spanning tree?

hard

A. It always includes zero-cost edges first, possibly creating multiple disconnected trees

B. It causes an error because zero-cost edges are invalid

C. It ignores zero-cost edges because they don't reduce total cost

D. It treats zero-cost edges as the cheapest and includes them first to connect vertices

Solution

1. Step 1: Understand edge weight role in Prim's

   Prim's algorithm always picks the edge with the smallest weight connecting new vertices.

2. Step 2: Consider zero-cost edges

   Zero-cost edges are the cheapest possible, so they will be chosen first to connect vertices without increasing total cost.

3. Step 3: Confirm no errors or ignoring

   Zero-cost edges are valid and included; they do not cause errors or get ignored.

4. Final Answer:

   It treats zero-cost edges as the cheapest and includes them first to connect vertices -> Option D

5. Quick Check:

   Zero-cost edges = chosen first [OK]

Hint: Zero-cost edges are cheapest, so Prim picks them first [OK]

Common Mistakes:

• Thinking zero-cost edges cause errors
• Assuming zero-cost edges are ignored
• Believing zero-cost edges create disconnected trees