Shortest Path in DAG Using Topological Order in DSA C - Execution Trace

Concept Flow - Shortest Path in DAG Using Topological Order

Build graph with edges and weights

↓

Find topological order of nodes

↓

Initialize distances: start node=0, others=∞

↓

For each node in topological order

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Relax edges from current node

↓

Update shortest distances

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End: shortest paths found

We first build the graph, then find a topological order. We initialize distances and relax edges in that order to find shortest paths.

Execution Sample

DSA C

1. Create graph edges with weights
2. Get topological order
3. Set distance[start] = 0, others = infinity
4. For each node in order, relax edges
5. Print shortest distances

This code finds shortest paths from a start node in a DAG by processing nodes in topological order.

Execution Table

Step	Operation	Current Node	Distances Array	Relaxed Edge	Distance Update	Visual State
1	Initialize distances	N/A	[0, ∞, ∞, ∞, ∞]	N/A	Start node distance set to 0	0 -> ∞ -> ∞ -> ∞ -> ∞
2	Process node 0	0	[0, ∞, ∞, ∞, ∞]	0->1 (weight 3)	Distance[1] = min(∞, 0+3)=3	0 -> 3 -> ∞ -> ∞ -> ∞
3	Process node 0	0	[0, 3, ∞, ∞, ∞]	0->2 (weight 6)	Distance[2] = min(∞, 0+6)=6	0 -> 3 -> 6 -> ∞ -> ∞
4	Process node 1	1	[0, 3, 6, ∞, ∞]	1->2 (weight 4)	Distance[2] = min(6, 3+4)=6 (no change)	0 -> 3 -> 6 -> ∞ -> ∞
5	Process node 1	1	[0, 3, 6, ∞, ∞]	1->3 (weight 4)	Distance[3] = min(∞, 3+4)=7	0 -> 3 -> 6 -> 7 -> ∞
6	Process node 2	2	[0, 3, 6, 7, ∞]	2->4 (weight 2)	Distance[4] = min(∞, 6+2)=8	0 -> 3 -> 6 -> 7 -> 8
7	Process node 3	3	[0, 3, 6, 7, 8]	3->4 (weight 1)	Distance[4] = min(8, 7+1)=8 (no change)	0 -> 3 -> 6 -> 7 -> 8
8	Process node 4	4	[0, 3, 6, 7, 8]	No outgoing edges	No update	0 -> 3 -> 6 -> 7 -> 8
9	End	N/A	[0, 3, 6, 7, 8]	N/A	All nodes processed	0 -> 3 -> 6 -> 7 -> 8

💡 All nodes processed in topological order; shortest distances finalized.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 5	After Step 6	After Step 8	Final
Distances	[0, ∞, ∞, ∞, ∞]	[0, 3, ∞, ∞, ∞]	[0, 3, 6, ∞, ∞]	[0, 3, 6, 7, ∞]	[0, 3, 6, 7, 8]	[0, 3, 6, 7, 8]	[0, 3, 6, 7, 8]
Current Node	N/A	0	0	1	2	4	N/A

Key Moments - 3 Insights

Why do we initialize distances with infinity except the start node?

Because we don't know the shortest path to other nodes yet. Setting them to infinity means they are unreachable until updated. See execution_table step 1.

Why do we process nodes in topological order?

Topological order ensures all edges from earlier nodes to later nodes are relaxed in correct sequence, so shortest paths are found without revisiting nodes. See execution_table steps 2-8.

Why does relaxing an edge sometimes not update the distance?

Because the new calculated distance is not smaller than the current distance. For example, step 4 shows no update since 6 ≤ 7.

Visual Quiz - 3 Questions

Look at the execution_table at step 5, what is the distance to node 3?

A 4

B ∞

C 7

D 3

💡 Hint

Check the Distances Array column at step 5 in execution_table.

At which step does the distance to node 4 first get updated?

A Step 7

B Step 6

C Step 4

D Step 3

💡 Hint

Look for the first Distance Update involving node 4 in execution_table.

If the edge 1->3 had weight 2 instead of 4, what would be the distance to node 3 after step 5?

A 5

B 7

C 3

D 6

💡 Hint

Calculate distance[3] = min(∞, distance[1] + new weight) using values from execution_table.

Concept Snapshot

Shortest Path in DAG Using Topological Order:
- Build graph with weighted edges
- Find topological order of nodes
- Initialize distances: start=0, others=∞
- Relax edges in topological order
- Update distances if shorter path found
- Result: shortest distances from start node

Full Transcript

This visualization shows how to find shortest paths in a Directed Acyclic Graph (DAG) using topological order. We start by building the graph and finding a topological order of nodes. We initialize distances with zero for the start node and infinity for others. Then, we process each node in topological order, relaxing edges to update distances if a shorter path is found. The execution table tracks each step, showing current node, distances array, edges relaxed, and updates made. Key moments clarify why distances start at infinity, why topological order is important, and why some edge relaxations do not update distances. The visual quiz tests understanding of distances at specific steps and effects of edge weight changes. The concept snapshot summarizes the method in a few lines.