DFS traversal and applications in Data Structures Theory - Step-by-Step Execution

Concept Flow - DFS traversal and applications

Start at a chosen node

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Mark node as visited

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Explore each unvisited neighbor

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For each neighbor:

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If unvisited

→Recursive DFS call

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Backtrack when no unvisited neighbors

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Repeat until all reachable nodes visited

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DFS complete

DFS starts at a node, marks it visited, explores neighbors recursively, and backtracks when stuck, until all reachable nodes are visited.

Execution Sample

Data Structures Theory

def dfs(graph, node, visited):
    visited.add(node)
    for neighbor in graph[node]:
        if neighbor not in visited:
            dfs(graph, neighbor, visited)

This code visits nodes in a graph using DFS, marking nodes visited and recursively exploring neighbors.

Analysis Table

Step	Operation	Current Node	Visited Set	Stack (Call Trace)	Action
1	Start DFS	A	{}	[]	Visit A, add to visited and stack
2	Visit neighbors of A	A	{A}	[A]	Check neighbors B, C
3	Visit neighbor B	B	{A}	[A, B]	B not visited, recurse into B
4	Visit neighbors of B	B	{A, B}	[A, B]	Check neighbors D
5	Visit neighbor D	D	{A, B}	[A, B, D]	D not visited, recurse into D
6	Visit neighbors of D	D	{A, B, D}	[A, B, D]	No unvisited neighbors, backtrack
7	Back to B	B	{A, B, D}	[A, B]	All neighbors visited, backtrack
8	Back to A	A	{A, B, D}	[A]	Visit next neighbor C
9	Visit neighbor C	C	{A, B, D}	[A, C]	C not visited, recurse into C
10	Visit neighbors of C	C	{A, B, D, C}	[A, C]	Check neighbors E, F
11	Visit neighbor E	E	{A, B, D, C}	[A, C, E]	E not visited, recurse into E
12	Visit neighbors of E	E	{A, B, D, C, E}	[A, C, E]	No unvisited neighbors, backtrack
13	Back to C	C	{A, B, D, C, E}	[A, C]	Visit next neighbor F
14	Visit neighbor F	F	{A, B, D, C, E}	[A, C, F]	F not visited, recurse into F
15	Visit neighbors of F	F	{A, B, D, C, E, F}	[A, C, F]	No unvisited neighbors, backtrack
16	Back to C	C	{A, B, D, C, E, F}	[A, C]	All neighbors visited, backtrack
17	Back to A	A	{A, B, D, C, E, F}	[A]	All neighbors visited, DFS complete
18	End	-	{A, B, D, C, E, F}	[]	All reachable nodes visited, stop

💡 All nodes reachable from A are visited; recursion unwinds completely.

State Tracker

Variable	Start	After Step 1	After Step 5	After Step 10	After Step 15	Final
visited	{}	{A}	{A, B, D}	{A, B, D, C}	{A, B, D, C, E, F}	{A, B, D, C, E, F}
stack	[]	[A]	[A, B, D]	[A, C]	[A, C, F]	[]
current_node	-	A	D	C	F	-

Key Insights - 3 Insights

Why does DFS backtrack after visiting node D at step 6?

At step 6, node D has no unvisited neighbors left, so DFS finishes exploring D and returns to the previous call at B (see execution_table row 6).

Why is the visited set important in DFS?

The visited set prevents revisiting nodes and infinite loops. For example, at step 3, B is added to visited so it won't be revisited later (execution_table row 3).

How does DFS explore all neighbors of a node before backtracking?

DFS recursively visits each unvisited neighbor before backtracking, as seen at node C visiting E and F in steps 10-16 (execution_table rows 10-16).

Visual Quiz - 3 Questions

Look at the execution_table at step 5. What nodes are in the visited set?

A {A, B}

B {A, B, D}

C {A}

D {}

💡 Hint

Check the 'Visited Set' column at step 5 in the execution_table.

At which step does DFS first backtrack from node D?

A Step 6

B Step 7

C Step 5

D Step 8

💡 Hint

Look for the step where the action says 'No unvisited neighbors, backtrack' for node D.

If node E had an unvisited neighbor G, how would the visited set change after step 11?

A {A, B, D, C, E}

B {A, B, D, C, E, F, G}

C {A, B, D, C, E, G}

D {A, B, D, C, E, F}

💡 Hint

Consider that DFS visits neighbors recursively, so G would be added after E at step 11.

Concept Snapshot

DFS (Depth-First Search) explores graph nodes by going deep along each branch before backtracking.
Start at a node, mark visited, recursively visit unvisited neighbors.
Uses a stack implicitly via recursion.
Prevents cycles by tracking visited nodes.
Useful for pathfinding, cycle detection, and topological sorting.

Full Transcript

Depth-First Search (DFS) starts at a chosen node, marks it visited, then explores each unvisited neighbor recursively. When a node has no unvisited neighbors, DFS backtracks to the previous node to explore other neighbors. This process continues until all reachable nodes are visited. The visited set prevents revisiting nodes and infinite loops. The recursion stack tracks the path being explored. DFS is useful for tasks like finding paths, detecting cycles, and ordering nodes in graphs.