Common Algorithms

Sequentially check each element until target is found

def linear_search(arr, target):
    """Search for target in array sequentially"""
    for i in range(len(arr)):
        if arr[i] == target:
            return i  # Return index where found
    return -1  # Not found

# Example usage
numbers = [64, 34, 25, 12, 22, 11, 90]
target = 22

result = linear_search(numbers, target)
if result != -1:
    print(f"Found {target} at index {result}")
else:
    print(f"{target} not found")

# Time complexity: O(n) - may need to check all elements
# Space complexity: O(1) - uses constant extra space

Efficiently search sorted data by repeatedly dividing search space in half

def binary_search(arr, target):
    """Search for target in sorted array using binary search"""
    left, right = 0, len(arr) - 1
    
    while left <= right:
        mid = (left + right) // 2
        
        if arr[mid] == target:
            return mid  # Found target
        elif arr[mid] < target:
            left = mid + 1  # Search right half
        else:
            right = mid - 1  # Search left half
    
    return -1  # Not found

# Example usage
sorted_numbers = [11, 12, 22, 25, 34, 64, 90]  # Must be sorted!
target = 25

result = binary_search(sorted_numbers, target)
if result != -1:
    print(f"Found {target} at index {result}")

# Time complexity: O(log n) - halves search space each step
# Space complexity: O(1) - uses constant extra space

Repeatedly step through list, compare adjacent elements and swap if needed

def bubble_sort(arr):
    """Sort array using bubble sort algorithm"""
    n = len(arr)
    
    # Traverse through all elements
    for i in range(n):
        swapped = False
        
        # Last i elements are already sorted
        for j in range(0, n - i - 1):
            # Swap if element is greater than next element
            if arr[j] > arr[j + 1]:
                arr[j], arr[j + 1] = arr[j + 1], arr[j]
                swapped = True
        
        # If no swapping happened, array is sorted
        if not swapped:
            break
    
    return arr

# Example usage
numbers = [64, 34, 25, 12, 22, 11, 90]
print("Original:", numbers)
sorted_numbers = bubble_sort(numbers.copy())
print("Sorted:", sorted_numbers)

# Time complexity: O(n²) worst/average case, O(n) best case
# Space complexity: O(1) - sorts in place

Build sorted array one element at a time by inserting elements in correct position

def insertion_sort(arr):
    """Sort array using insertion sort algorithm"""
    
    # Traverse from second element to end
    for i in range(1, len(arr)):
        key = arr[i]  # Current element to insert
        j = i - 1
        
        # Move elements greater than key one position ahead
        while j >= 0 and arr[j] > key:
            arr[j + 1] = arr[j]
            j -= 1
        
        # Insert key at correct position
        arr[j + 1] = key
    
    return arr

# Example usage
numbers = [64, 34, 25, 12, 22, 11, 90]
print("Original:", numbers)
sorted_numbers = insertion_sort(numbers.copy())
print("Sorted:", sorted_numbers)

# Time complexity: O(n²) worst case, O(n) best case
# Space complexity: O(1) - sorts in place

Divide array into halves, sort each half, then merge sorted halves

def merge_sort(arr):
    """Sort array using merge sort algorithm (divide and conquer)"""
    if len(arr) <= 1:
        return arr
    
    # Divide array into two halves
    mid = len(arr) // 2
    left_half = merge_sort(arr[:mid])
    right_half = merge_sort(arr[mid:])
    
    # Merge sorted halves
    return merge(left_half, right_half)

def merge(left, right):
    """Merge two sorted arrays into one sorted array"""
    result = []
    i = j = 0
    
    # Compare elements and merge in sorted order
    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
    
    # Add remaining elements
    result.extend(left[i:])
    result.extend(right[j:])
    
    return result

# Example usage
numbers = [64, 34, 25, 12, 22, 11, 90]
sorted_numbers = merge_sort(numbers)
print("Sorted:", sorted_numbers)

# Time complexity: O(n log n) all cases
# Space complexity: O(n) - needs additional memory