This flow shows how a weighted graph is built by adding nodes and edges with weights, then using those weights for calculations like shortest paths.
Execution Sample
Data Structures Theory
Graph G = {}
Add nodes: A, B, C
Add edges with weights:
A-B: 4
B-C: 3
A-C: 7
This example builds a weighted graph with three nodes and edges that have weights representing cost or distance.
Analysis Table
Step
Operation
Graph Nodes
Edges with Weights
Visual State
1
Initialize empty graph
{}
{}
{}
2
Add node A
{A}
{}
{A}
3
Add node B
{A, B}
{}
{A} {B}
4
Add node C
{A, B, C}
{}
{A} {B} {C}
5
Add edge A-B with weight 4
{A, B, C}
{(A-B,4)}
A--4--B {C}
6
Add edge B-C with weight 3
{A, B, C}
{(A-B,4), (B-C,3)}
A--4--B--3--C
7
Add edge A-C with weight 7
{A, B, C}
{(A-B,4), (B-C,3), (A-C,7)}
A--4--B--3--C
\_______7__/
8
Graph ready for algorithms
{A, B, C}
{(A-B,4), (B-C,3), (A-C,7)}
Weighted graph complete
💡 All nodes and weighted edges added; graph is ready for use.
State Tracker
Variable
Start
After Step 2
After Step 3
After Step 4
After Step 5
After Step 6
After Step 7
Final
Nodes
{}
{A}
{A, B}
{A, B, C}
{A, B, C}
{A, B, C}
{A, B, C}
{A, B, C}
Edges
{}
{}
{}
{}
{(A-B,4)}
{(A-B,4), (B-C,3)}
{(A-B,4), (B-C,3), (A-C,7)}
{(A-B,4), (B-C,3), (A-C,7)}
Key Insights - 3 Insights
Why do edges have weights and what do they represent?
Edges have weights to represent cost, distance, or any measure between nodes. This is shown in execution_table rows 5-7 where weights like 4, 3, and 7 are assigned to edges.
Can a weighted graph have edges without weights?
In a weighted graph, every edge must have a weight. If edges have no weights, it's just a normal (unweighted) graph. Our example assigns weights at steps 5-7, confirming this.
How is the graph visually represented with weights?
The visual state in execution_table shows edges labeled with their weights, like 'A--4--B', helping to understand the cost between nodes.
Visual Quiz - 3 Questions
Test your understanding
Look at the execution_table at step 6. What edges and weights does the graph have?
A{(A-B,4), (A-C,7)}
B{(A-B,4), (B-C,3)}
C{(B-C,3), (A-C,7)}
D{(A-B,3), (B-C,4)}
💡 Hint
Check the 'Edges with Weights' column at step 6 in the execution_table.
At which step are all nodes added to the graph?
AStep 4
BStep 5
CStep 2
DStep 7
💡 Hint
Look at the 'Graph Nodes' column to see when nodes A, B, and C all appear.
If the edge A-C weight changed from 7 to 2 at step 7, how would the visual state change?
AEdge A-B weight changes to 2
BNo change in visual state
CA--4--B--3--C with A-C edge labeled 2 instead of 7
DNode C is removed
💡 Hint
Refer to the visual state in execution_table step 7 and imagine changing the weight label.
Concept Snapshot
Weighted Graphs:
- Graph with nodes connected by edges that have weights.
- Weights represent cost, distance, or value between nodes.
- Used in algorithms like shortest path.
- Visualize edges labeled with weights.
- Essential for real-world problems like maps or networks.
Full Transcript
A weighted graph is a set of nodes connected by edges that have numbers called weights. These weights show cost or distance between nodes. We start by creating an empty graph, add nodes, then add edges with weights. The graph can then be used for calculations like finding the shortest path. The execution table shows step-by-step how nodes and weighted edges are added. The variable tracker shows how nodes and edges grow over time. Key moments explain why weights are important and how the graph looks visually. The quiz tests understanding of graph structure and weights at different steps.