Tower of Hanoi Problem in DSA C# - 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 below.

void TowerOfHanoi(int n, char from, char to, char aux) {
  if (n == 1) {
    Console.WriteLine($"Move disk 1 from {from} to {to}");
    return;
  }
  TowerOfHanoi(n - 1, from, aux, to);
  Console.WriteLine($"Move disk {n} from {from} to {to}");
  TowerOfHanoi(n - 1, aux, to, from);
}
    

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

Identify Repeating Operations

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

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

How Execution Grows With Input

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

Input Size (n)	Approx. Operations (moves)
1	1
2	3
3	7
4	15

Pattern observation: The moves double plus one each time, growing very fast as n increases.

Final Time Complexity

Time Complexity: O(2^n)

This means the time needed grows exponentially, doubling with each extra disk.

Common Mistake

[X] Wrong: "The time grows linearly because we just move each disk once."

[OK] Correct: Actually, smaller disks move many times during the process, causing the total moves to grow exponentially.

Interview Connect

Understanding this classic problem helps you think about recursion and exponential growth, skills useful in many coding challenges.

Self-Check

"What if we changed the problem to allow moving multiple disks at once? 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. 6

B. 7

C. 8

D. 9

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:

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

2. Which of the following is the correct C# method signature for a recursive Tower of Hanoi function?

easy

A. void TowerOfHanoi(int n, string fromRod, string toRod)

B. int TowerOfHanoi(int n, char fromRod, char toRod, char auxRod)

C. int TowerOfHanoi(int n, string fromRod, string toRod, string auxRod)

D. void TowerOfHanoi(int n, char fromRod, char toRod, char auxRod)

Solution

1. Step 1: Identify correct parameter types

   The Tower of Hanoi function typically uses integer for disk count and characters for rods.

2. Step 2: Check return type and parameters

   The function usually returns void and takes 3 rods as char parameters plus disk count as int.

3. Final Answer:

   void TowerOfHanoi(int n, char fromRod, char toRod, char auxRod) -> Option D

4. Quick Check:

   Correct signature uses void and char rods [OK]

Hint: Use void return and char for rods in signature [OK]

Common Mistakes:

• Using string instead of char for rods
• Returning int instead of void
• Missing auxiliary rod parameter

3. What will be the output sequence of moves for Tower of Hanoi with 2 disks from rod A to rod C using rod B as auxiliary?

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

Solution

1. Step 1: Understand the recursive move sequence for 2 disks

   Move smaller disk 1 to auxiliary rod B, then move disk 2 to target rod C, then move disk 1 from B to C.

2. Step 2: Verify the move order

   Sequence is: 1 from A to B, 2 from A to C, 1 from B to 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 order for 2 disks [OK]

Hint: For 2 disks, move smaller disk first to aux rod [OK]

Common Mistakes:

• Swapping disk numbers in moves
• Confusing auxiliary and target rods
• Reversing move order

4. Identify the error in this Tower of Hanoi recursive call snippet:

void Hanoi(int n, char from, char to, char aux) {
  if (n == 0) return;
  Hanoi(n - 1, from, aux, to);
  Console.WriteLine($"Move disk {n} from {from} to {to}");
  Hanoi(n - 1, aux, to, from);
}

medium

A. Base case should be if (n == 1) return;

B. Recursive calls have wrong rod order in parameters

C. No error, code is correct

D. Console.WriteLine should print disk n-1 instead of n

Solution

1. Step 1: Check base case correctness

   Base case n == 0 means no disks to move, which is correct to stop recursion.

2. Step 2: Verify recursive calls and print statement

   Recursive calls correctly move n-1 disks between rods, and print statement moves disk n from 'from' to 'to'.

3. Final Answer:

   No error, code is correct -> Option C

4. Quick Check:

   Base case and recursion are correctly implemented [OK]

Hint: Base case n==0 is correct to stop recursion [OK]

Common Mistakes:

• Changing base case to n==1 incorrectly
• Mixing up rod parameters in recursive calls
• Printing wrong disk number

5. You want to solve Tower of Hanoi for 4 disks but print only the moves where disk 1 is moved. How many such moves will be printed?

hard

A. 8

B. 15

C. 7

D. 4

Solution

1. Step 1: Calculate total moves for 4 disks

   Total moves = 2^4 - 1 = 15 moves.

2. Step 2: Count moves involving disk 1

   Disk 1 moves every alternate move, so number of moves = (total moves + 1) / 2 = (15 + 1)/2 = 8.

3. Final Answer:

   8 -> Option A

4. Quick Check:

   Disk 1 moves half the times plus one [OK]

Hint: Disk 1 moves on every alternate move, count half plus one [OK]

Common Mistakes:

• Assuming disk 1 moves equal to total moves
• Using total moves minus one
• Confusing disk 1 moves with total disks