Minimum spanning tree (Prim's) in Data Structures Theory - Step-by-Step Execution

Concept Flow - Minimum spanning tree (Prim's)

Start with any node

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Add node to MST set

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Find all edges from MST to non-MST nodes

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Pick edge with smallest weight

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Add connected node to MST

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Repeat until all nodes included

Prim's algorithm builds the minimum spanning tree by starting from one node and repeatedly adding the smallest edge that connects the tree to a new node.

Execution Sample

Data Structures Theory

Graph edges:
A-B:2, A-C:3, B-C:1, B-D:4, C-D:5
Start at A
Add smallest edge connecting MST to new node
Repeat until all nodes included

This example shows Prim's algorithm starting at node A and adding edges step-by-step to build the minimum spanning tree.

Analysis Table

Step	Operation	MST Nodes	Edges Considered	Edge Chosen	MST Edges	Visual State
1	Start at node A	[A]	Edges from A: A-B(2), A-C(3)	None	None	MST: A
2	Choose smallest edge from MST to non-MST	[A]	A-B(2), A-C(3)	A-B(2)	[A-B]	MST: A - B
3	Add node B to MST	[A, B]	Edges from A,B: A-C(3), B-C(1), B-D(4)	B-C(1)	[A-B, B-C]	MST: A - B - C
4	Add node C to MST	[A, B, C]	Edges from A,B,C: B-D(4), C-D(5)	B-D(4)	[A-B, B-C, B-D]	MST: A - B - C | D
5	Add node D to MST	[A, B, C, D]	All nodes included	None	[A-B, B-C, B-D]	MST complete: A-B, B-C, B-D
6	Stop	[A, B, C, D]	No edges left	None	[A-B, B-C, B-D]	All nodes connected, MST done

💡 All nodes are included in MST, no edges left to add

State Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
MST Nodes	[]	[A]	[A, B]	[A, B, C]	[A, B, C, D]	[A, B, C, D]
Edges Considered	Edges from A	Edges from A,B	Edges from A,B,C	All nodes included	None	None
MST Edges	[]	[A-B]	[A-B, B-C]	[A-B, B-C, B-D]	[A-B, B-C, B-D]	[A-B, B-C, B-D]

Key Insights - 3 Insights

Why do we only consider edges that connect MST nodes to non-MST nodes?

Because Prim's algorithm grows the MST by adding new nodes connected by the smallest edge from the existing MST. Considering edges only between MST and non-MST nodes prevents cycles and ensures all nodes get connected.

What happens if we pick an edge that connects two nodes already in MST?

Picking such an edge would create a cycle, which is not allowed in a spanning tree. The execution_table shows edges chosen always connect MST to a new node.

Why does the algorithm stop when all nodes are included in MST?

Because the MST must include all nodes with the minimum total edge weight. Once all nodes are in MST, no more edges need to be added, as shown in the last step of the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3. Which edge is chosen to add node C to MST?

A A-C with weight 3

B B-C with weight 1

C B-D with weight 4

D C-D with weight 5

💡 Hint

Check the 'Edge Chosen' column at Step 3 in the execution_table.

At which step does the MST include all nodes?

A Step 5

B Step 3

C Step 4

D Step 2

💡 Hint

Look at the 'MST Nodes' column to see when all nodes [A, B, C, D] are included.

If the edge B-C had weight 6 instead of 1, which edge would be chosen at Step 3?

A B-D with weight 4

B B-C with weight 6

C A-C with weight 3

D C-D with weight 5

💡 Hint

Compare edge weights in 'Edges Considered' at Step 3 and pick the smallest.

Concept Snapshot

Prim's Algorithm for MST:
- Start from any node
- Add smallest edge connecting MST to new node
- Repeat until all nodes included
- Avoid cycles by only connecting MST to non-MST nodes
- Result is a tree connecting all nodes with minimum total edge weight

Full Transcript

Prim's algorithm builds a minimum spanning tree by starting at any node and repeatedly adding the smallest edge that connects the current tree to a new node. At each step, it considers edges from nodes already in the MST to nodes not yet included. It picks the smallest such edge and adds the connected node to the MST. This process repeats until all nodes are included. The execution table shows step-by-step how nodes and edges are added, ensuring no cycles form and the total edge weight is minimized. Key points include only considering edges connecting MST to non-MST nodes and stopping when all nodes are included. The visual quiz tests understanding of edge choices and MST growth.