Tower of Hanoi Problem in DSA Go - 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 following Tower of Hanoi recursive solution.

func TowerOfHanoi(n int, from, to, aux string) {
    if n == 1 {
        fmt.Println("Move disk 1 from", from, "to", to)
        return
    }
    TowerOfHanoi(n-1, from, aux, to)
    fmt.Println("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

Identify the loops, recursion, array traversals that repeat.

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

How Execution Grows With Input

Each time we add a disk, the number of moves roughly doubles plus one extra 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 we add a disk.

Final Time Complexity

Time Complexity: O(2^n)

This means the time needed doubles with each additional disk, growing very fast as n increases.

Common Mistake

[X] Wrong: "The time grows linearly with the number of disks because we just move each disk once."

[OK] Correct: Each disk move depends on moving smaller stacks multiple times, so moves double with each added disk, not just increase by one.

Interview Connect

Understanding this exponential growth helps you explain recursive problems clearly and shows you can analyze complex time patterns.

Self-Check

"What if we used an iterative approach instead of recursion? 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. 9

C. 8

D. 7

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 D

4. Quick Check:

   2^3 - 1 = 7 [OK]

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

Common Mistakes:

• Forgetting to subtract 1
• Using n instead of 2^n
• Confusing number of disks with moves

2. Which of the following Go function signatures correctly represents the Tower of Hanoi recursive function?

easy

A. func towerOfHanoi(n string, from int, to int, aux int)

B. func towerOfHanoi(n int, from string, to string, aux string)

C. func towerOfHanoi(n int, from int, to int)

D. func towerOfHanoi(n int, from string, to string)

Solution

1. Step 1: Identify parameters needed for Tower of Hanoi

   We need number of disks (int), and names of three rods (strings): from, to, auxiliary.

2. Step 2: Match function signature

   func towerOfHanoi(n int, from string, to string, aux string) matches with n as int and three string rods. Others miss parameters or have wrong types.

3. Final Answer:

   func towerOfHanoi(n int, from string, to string, aux string) -> Option B

4. Quick Check:

   Correct parameters and types = func towerOfHanoi(n int, from string, to string, aux string) [OK]

Hint: Tower of Hanoi needs 4 params: disks and 3 rods [OK]

Common Mistakes:

• Using wrong parameter types
• Missing auxiliary rod parameter
• Using fewer than 4 parameters

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

func towerOfHanoi(n int, from string, to string, aux string) {
    if n == 1 {
        fmt.Printf("Move disk 1 from %s to %s\n", from, to)
        return
    }
    towerOfHanoi(n-1, from, aux, to)
    fmt.Printf("Move disk %d from %s to %s\n", n, from, to)
    towerOfHanoi(n-1, aux, to, from)
}

func main() {
    towerOfHanoi(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 1 from A to C
Move disk 2 from A to B
Move disk 1 from C to B

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

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: Match output with options

   Move disk 1 from A to B Move disk 2 from A to C Move disk 1 from B to C matches the exact sequence of moves printed by the code.

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 = Move disk 1 from A to B Move disk 2 from A to C Move disk 1 from B to C [OK]

Hint: Trace recursion carefully for small n [OK]

Common Mistakes:

• Swapping auxiliary and target rods
• Printing moves in wrong order
• Confusing disk numbers

4. Identify the error in the following Tower of Hanoi Go function and select the correct fix:

func towerOfHanoi(n int, from string, to string, aux string) {
    if n == 0 {
        return
    }
    towerOfHanoi(n-1, from, to, aux)
    fmt.Printf("Move disk %d from %s to %s\n", n, from, to)
    towerOfHanoi(n-1, aux, to, from)
}

medium

A. Remove base case entirely

B. Change base case to if n == 1 and adjust recursive calls accordingly

C. Swap 'to' and 'aux' in first recursive call

D. Change print statement to use 'aux' instead of 'to'

Solution

1. Step 1: Analyze recursive calls and base case

   Base case with n==0 is okay, but first recursive call uses wrong rod order: it should move disks from 'from' to 'aux' using 'to' as auxiliary.

2. Step 2: Correct first recursive call

   Swap 'to' and 'aux' in first recursive call to towerOfHanoi(n-1, from, aux, to).

3. Final Answer:

   Swap 'to' and 'aux' in first recursive call -> Option C

4. Quick Check:

   Correct rod order in recursion = Swap 'to' and 'aux' in first recursive call [OK]

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

Common Mistakes:

• Confusing auxiliary and target rods
• Incorrect base case condition
• Wrong print statement parameters

5. You want to solve the Tower of Hanoi problem for 4 disks but print only the moves where disk 1 is moved. Which modification to the Go recursive function below will achieve this?

func towerOfHanoi(n int, from string, to string, aux string) {
    if n == 1 {
        fmt.Printf("Move disk 1 from %s to %s\n", from, to)
        return
    }
    towerOfHanoi(n-1, from, aux, to)
    fmt.Printf("Move disk %d from %s to %s\n", n, from, to)
    towerOfHanoi(n-1, aux, to, from)
}

hard

A. Add an if condition to print only when n == 1 inside the function

B. Change base case to n == 0 and print only in recursive calls

C. Remove the print statement in the base case

D. Print only when from == "A" and to == "C"

Solution

1. Step 1: Understand when disk 1 moves are printed

   Disk 1 moves are printed only in base case when n == 1.

2. Step 2: Modify print statements to print only for disk 1

   Add an if condition around print statements to print only when n == 1, skipping other disks.

3. Final Answer:

   Add an if condition to print only when n == 1 inside the function -> Option A

4. Quick Check:

   Print only disk 1 moves = Add an if condition to print only when n == 1 inside the function [OK]

Hint: Print only when disk number is 1 inside recursion [OK]

Common Mistakes:

• Removing base case print
• Printing based on rod names instead of disk number
• Changing base case incorrectly