Tower of Hanoi Problem in DSA Python - Time & Space Complexity

Time Complexity: Tower of Hanoi Problem

O(2^n)

Understanding Time Complexity

We want to understand how the time needed to solve the Tower of Hanoi grows as we add more disks.

How does the number of moves change when the number of disks increases?

Scenario Under Consideration

Analyze the time complexity of the recursive Tower of Hanoi solution.

def tower_of_hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk 1 from {source} to {target}")
        return
    tower_of_hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    tower_of_hanoi(n-1, auxiliary, target, source)

This code moves n disks from one peg to another using recursion.

Identify Repeating Operations

Look at the recursive calls that repeat the same process for smaller numbers of disks.

• Primary operation: Recursive calls to move n-1 disks.
• How many times: Each call makes two recursive calls plus one move.

How Execution Grows With Input

Each time we add a disk, the number of moves roughly doubles and then adds one more move.

Input Size (n)	Approx. Operations (Moves)
1	1
2	3
3	7
4	15
5	31

Pattern observation: The moves grow exponentially, doubling each time plus one extra move.

Final Time Complexity

Time Complexity: O(2^n)

This means the time needed grows very fast, doubling with each additional disk.

Common Mistake

[X] Wrong: "The time grows linearly because we just move disks one by one."

[OK] Correct: Each disk move depends on moving smaller stacks twice, so the total moves double each time, not just add one.

Interview Connect

Understanding this problem helps you think about recursion and exponential growth, which appear often in coding challenges.

Self-Check

"What if we changed the problem to move disks using four pegs instead of three? How would the time complexity change?"

Practice

1. What is the minimum number of moves required to solve the Tower of Hanoi problem with 3 disks?

easy

A. 5

B. 7

C. 9

D. 6

Solution

1. Step 1: Understand the formula for minimum moves

   The minimum moves required for n disks is 2^n - 1.

2. Step 2: Calculate for 3 disks

   For 3 disks, minimum moves = 2^3 - 1 = 8 - 1 = 7.

3. Final Answer:

   7 -> Option B

4. Quick Check:

   2^3 - 1 = 7 [OK]

Hint: Use formula 2^n - 1 for minimum moves [OK]

Common Mistakes:

• Using n instead of 2^n - 1
• Calculating 2*n - 1 instead of 2^n - 1
• Confusing number of disks with moves

2. Which of the following is the correct base case condition in a recursive Tower of Hanoi function in Python?

easy

A. if n == 0: return

B. if n < 0: return

C. if n > 1: call recursion

D. if n == 1: move disk

Solution

1. Step 1: Identify base case for recursion

   The base case stops recursion when only one disk is left, i.e., n == 1.

2. Step 2: Understand base case action

   When n == 1, the disk is moved directly from source to target.

3. Final Answer:

   if n == 1: move disk -> Option D

4. Quick Check:

   Base case is n == 1 [OK]

Hint: Base case is when only one disk remains (n == 1) [OK]

Common Mistakes:

• Using n == 0 as base case incorrectly
• Not defining a base case causing infinite recursion
• Using invalid conditions like n < 0

3. What will be the output of the following Python code snippet for Tower of Hanoi with 2 disks?

def hanoi(n, source, target, auxiliary):
    if n == 0:
        return
    hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    hanoi(n-1, auxiliary, target, source)

hanoi(2, 'A', 'C', 'B')

medium

A. Move disk 1 from A to B
Move disk 2 from A to C
Move disk 1 from B to C

B. Move disk 2 from A to C
Move disk 1 from A to B
Move disk 1 from B to C

C. Move disk 1 from A to C
Move disk 2 from A to B
Move disk 1 from C to B

D. Move disk 1 from B to C
Move disk 2 from A to C
Move disk 1 from A to B

Solution

1. Step 1: Trace recursive calls for n=2

   First, move disk 1 from A to B, then disk 2 from A to C, then disk 1 from B to C.

2. Step 2: Verify print statements order

   Prints match the moves: disk 1 A->B, disk 2 A->C, disk 1 B->C.

3. Final Answer:

   Move disk 1 from A to B Move disk 2 from A to C Move disk 1 from B to C -> Option A

4. Quick Check:

   Correct move sequence for 2 disks [OK]

Hint: Follow recursive calls step-by-step [OK]

Common Mistakes:

• Mixing source and auxiliary rods
• Printing moves in wrong order
• Confusing disk numbers

4. Identify the error in this Tower of Hanoi recursive function and select the correct fix:

def hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk {n} from {source} to {target}")
    else:
        hanoi(n-1, source, target, auxiliary)
        print(f"Move disk {n} from {source} to {target}")
        hanoi(n-1, auxiliary, target, source)

medium

A. Change first recursive call to hanoi(n-1, source, auxiliary, target)

B. Change base case to if n == 0

C. Swap print statements order

D. Remove the else block

Solution

1. Step 1: Analyze recursive calls

   The first recursive call incorrectly moves disks from source to target instead of auxiliary.

2. Step 2: Correct the first recursive call

   It should be hanoi(n-1, source, auxiliary, target) to move disks to auxiliary rod first.

3. Final Answer:

   Change first recursive call to hanoi(n-1, source, auxiliary, target) -> Option A

4. Quick Check:

   Correct recursive call order fixes the error [OK]

Hint: First recursive call moves disks to auxiliary rod [OK]

Common Mistakes:

• Calling recursion with wrong target rod
• Changing base case incorrectly
• Swapping print statements unnecessarily

5. You want to solve Tower of Hanoi for 4 disks but only have 3 rods. What is the minimum number of moves required, and why can't it be less than that?

hard

A. 14 moves, because the largest disk can be moved directly

B. 16 moves, because one extra move is needed for 4 disks

C. 15 moves, because 2^4 - 1 is the minimum for 3 rods

D. 7 moves, because only half the disks need moving

Solution

1. Step 1: Apply minimum moves formula for 3 rods

   Minimum moves for n disks with 3 rods is 2^n - 1, so for 4 disks: 2^4 - 1 = 15.

2. Step 2: Understand why moves can't be less

   With 3 rods, no shortcut exists; all disks must be moved following rules, so 15 is minimum.

3. Final Answer:

   15 moves, because 2^4 - 1 is the minimum for 3 rods -> Option C

4. Quick Check:

   Minimum moves = 2^n - 1 for 3 rods [OK]

Hint: Use 2^n - 1 formula for 3 rods minimum moves [OK]

Common Mistakes:

• Assuming fewer moves possible with 3 rods
• Adding extra moves unnecessarily
• Confusing number of rods with moves