Minimum Spanning Tree Kruskal's Algorithm in DSA Typescript - Execution Trace

Concept Flow - Minimum Spanning Tree Kruskal's Algorithm

Start with all edges sorted by weight

↓

Pick smallest edge not yet chosen

↓

Check if adding edge forms a cycle?

Yes→Discard edge

No↓

Add edge to MST

↓

Repeat until MST has (V-1) edges

↓

Done

Kruskal's algorithm sorts edges by weight and adds them one by one to the MST if they don't form a cycle, until MST spans all vertices.

Execution Sample

DSA Typescript

edges = [(A,B,1), (B,C,3), (A,C,2)]
sort edges by weight
for edge in edges:
  if no cycle with edge:
    add edge to MST

This code sorts edges by weight and adds them to MST if they don't create a cycle.

Execution Table

Step	Operation	Edge Considered	Cycle Check Result	MST Edges	Visual State
1	Sort edges by weight	-	-	[]	Edges sorted: (A,B,1), (A,C,2), (B,C,3)
2	Consider edge	(A,B,1)	No cycle	[(A,B,1)]	MST: A--1--B
3	Consider edge	(A,C,2)	No cycle	[(A,B,1), (A,C,2)]	MST: A--1--B and A--2--C
4	Consider edge	(B,C,3)	Cycle detected	[(A,B,1), (A,C,2)]	MST unchanged: A--1--B and A--2--C
5	Check MST size	Edges in MST = 2	MST complete	[(A,B,1), (A,C,2)]	Final MST: A--1--B and A--2--C

💡 MST has (V-1)=2 edges for 3 vertices, so algorithm stops.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	Final
edges	[(A,B,1), (B,C,3), (A,C,2)]	[(A,B,1), (A,C,2), (B,C,3)]	[(A,B,1), (A,C,2), (B,C,3)]	[(A,B,1), (A,C,2), (B,C,3)]	[(A,B,1), (A,C,2), (B,C,3)]	[(A,B,1), (A,C,2), (B,C,3)]
MST	[]	[]	[(A,B,1)]	[(A,B,1), (A,C,2)]	[(A,B,1), (A,C,2)]	[(A,B,1), (A,C,2)]
Union-Find sets	[A],[B],[C]	[A],[B],[C]	[A,B],[C]	[A,B,C]	[A,B,C]	[A,B,C]

Key Moments - 3 Insights

Why do we check for cycles before adding an edge?

Because adding an edge that forms a cycle breaks the tree property. See step 4 in execution_table where edge (B,C,3) is discarded due to cycle.

How do we know when to stop adding edges?

We stop when MST has (V-1) edges, where V is number of vertices. Step 5 shows MST has 2 edges for 3 vertices, so algorithm stops.

What data structure helps detect cycles efficiently?

Union-Find (Disjoint Set) tracks connected components. If two vertices are already connected, adding edge creates cycle. See variable_tracker for Union-Find sets.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what edges are in the MST?

A [(A,B,1), (B,C,3)]

B [(A,B,1)]

C [(A,B,1), (A,C,2)]

D []

💡 Hint

Check the MST Edges column at step 3 in execution_table.

At which step does the algorithm detect a cycle?

A Step 4

B Step 3

C Step 2

D Step 5

💡 Hint

Look at the Cycle Check Result column in execution_table.

If the edge (B,C,3) had weight 0.5, how would the MST change?

A Edge (A,B,1) would be discarded

B Edge (B,C,3) would be added first

C MST would have edges (B,C,3) and (A,C,2)

D No change in MST

💡 Hint

Edges are sorted by weight; check variable_tracker edges after Step 1.

Concept Snapshot

Kruskal's Algorithm for MST:
- Sort all edges by weight ascending
- Use Union-Find to detect cycles
- Add edges if no cycle formed
- Stop when MST has (V-1) edges
- Result is minimum spanning tree connecting all vertices

Full Transcript

Kruskal's algorithm builds a minimum spanning tree by sorting edges from smallest to largest weight. It picks edges one by one, adding them if they don't create a cycle. Cycle detection uses a Union-Find data structure to track connected components. The process stops when the MST contains one less edge than the number of vertices. This ensures the MST connects all vertices with minimum total edge weight.