Kernal — Data Structures & Algorithms

Recursion & Divide and Conquer

Recursion is a function that calls itself on smaller subproblems until reaching a base case.

Anatomy of Recursion

``python


def recurse(n):
if n <= 0:          # Base case - stops recursion
return
recurse(n - 1)      # Recursive call on smaller input

Towers of Hanoi

Move n disks from source → destination using an auxiliary peg.

Rules: Only move one disk at a time. Never place a larger disk on a smaller one. Solution: Move n-1 disks to auxiliary, move largest to destination, move n-1 from auxiliary to destination. Recurrence: T(n) = 2T(n-1) + 1 → O(2ⁿ)

Merge Sort (Divide and Conquer)

Divide: Split array in half
Conquer: Recursively sort each half
Combine: Merge sorted halves

Recurrence: T(n) = 2T(n/2) + O(n) → O(n log n)

Quick Sort

Partition: Pick pivot, place it in correct position
Conquer: Recursively sort left and right partitions

Average: O(n log n) | Worst: O(n²) - when pivot is always min or max Space: O(log n) - in-place sort

Code Example

python

def tower_of_hanoi(n, source='A', target='C', auxiliary='B'):
    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)

print("Towers of Hanoi (3 disks):")
tower_of_hanoi(3)

def merge_sort(arr):
    if len(arr) <= 1:
        return arr
    mid = len(arr) // 2
    left = merge_sort(arr[:mid])
    right = merge_sort(arr[mid:])
    return merge(left, right)

def merge(left, right):
    result = []
    i = j = 0
    while i < len(left) and j < len(right):
        if left[i] <= right[j]:
            result.append(left[i]); i += 1
        else:
            result.append(right[j]); j += 1
    result.extend(left[i:])
    result.extend(right[j:])
    return result

def quick_sort(arr):
    if len(arr) <= 1:
        return arr
    pivot = arr[len(arr) // 2]
    left = [x for x in arr if x < pivot]
    middle = [x for x in arr if x == pivot]
    right = [x for x in arr if x > pivot]
    return quick_sort(left) + middle + quick_sort(right)

data = [38, 27, 43, 3, 9, 82, 10]
print("Merge:", merge_sort(data.copy()))
print("Quick:", quick_sort(data.copy()))

Recursion & Divide and Conquer

Recursion & Divide and Conquer

Anatomy of Recursion

Towers of Hanoi

Merge Sort (Divide and Conquer)

Quick Sort

Code Example

Practice Problems