Add new array algorithms : Maximum Circular Subarray and Sliding Window Maximum

data_structures/arrays/max_subarray_circular.py

@@ -0,0 +1,63 @@
+
+def max_subarray_circular(nums: list[int]) -> int:
+    """
+    Find the maximum possible sum of a subarray in a circular array.
+
+    Args:
+        nums: List of integers representing the circular array
+
+    Returns:
+        Maximum sum possible considering the array as circular
+
+    Examples:
+        >>> max_subarray_circular([1, -2, 3, -2])
+        3
+        >>> max_subarray_circular([5, -3, 5])
+        10
+        >>> max_subarray_circular([-3, -2, -3])
+        -2
+    """
+    if not nums:
+        return 0
+
+    # Case 1: Get the maximum sum using Kadane's algorithm
+    def kadane(arr):
+        curr_sum = max_sum = arr[0]
+        for num in arr[1:]:
+            curr_sum = max(num, curr_sum + num)
+            max_sum = max(max_sum, curr_sum)
+        return max_sum
+
+    # Case 1: The maximum sum without wrapping
+    max_straight = kadane(nums)
+
+    # If all numbers are negative, return the maximum straight sum
+    if max_straight < 0:
+        return max_straight
+
+    # Case 2: The maximum sum with wrapping
+    # This can be found by subtracting the minimum sum subarray from total sum
+    total = sum(nums)
+    # Invert the array and find the maximum sum to get the minimum sum
+    max_inverted = kadane([-x for x in nums])
+    max_wrapped = total + max_inverted  # Adding because we inverted the numbers
+
+    return max(max_straight, max_wrapped)
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()
+
+"""
+This module implements an algorithm to find the maximum sum of a subarray in a circular array.
+A circular array means the array can wrap around itself.
+
+For example:
+[1, -2, 3, -2] -> Maximum sum is 3
+[5, -3, 5] -> Maximum sum is 10 because [5, 5] is possible with wrapping
+[-3, -2, -3] -> Maximum sum is -2
+
+Time Complexity: O(n)
+Space Complexity: O(1)
+"""

data_structures/arrays/sliding_window_maximum.py

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+"""
+This module implements an algorithm to find the maximum element in all possible
+windows of size k in an array using a deque for O(n) time complexity.
+
+For example:
+[1, 3, -1, -3, 5, 3, 6, 7], k=3 -> [3, 3, 5, 5, 6, 7]
+Window position        Max
+---------------       -----
+[1  3  -1] -3  5  3  6  7   ->  3
+ 1 [3  -1  -3] 5  3  6  7   ->  3
+ 1  3 [-1  -3  5] 3  6  7   ->  5
+ 1  3  -1 [-3  5  3] 6  7   ->  5
+ 1  3  -1  -3 [5  3  6] 7   ->  6
+ 1  3  -1  -3  5 [3  6  7]  ->  7
+
+Time Complexity: O(n)
+Space Complexity: O(k)
+"""
+
+from collections import deque
+from typing import List
+
+
+def max_sliding_window(nums: List[int], k: int) -> List[int]:
+    """
+    Find maximum elements in all possible windows of size k.
+
+    Args:
+        nums: List of integers
+        k: Size of the sliding window
+
+    Returns:
+        List of maximum elements from each window
+
+    Examples:
+        >>> max_sliding_window([1, 3, -1, -3, 5, 3, 6, 7], 3)
+        [3, 3, 5, 5, 6, 7]
+        >>> max_sliding_window([1], 1)
+        [1]
+        >>> max_sliding_window([1, -1], 1)
+        [1, -1]
+        >>> max_sliding_window([1, 2, 3, 4], 4)
+        [4]
+    """
+    if not nums or k <= 0:
+        return []
+    if k == 1:
+        return nums
+
+    result = []
+    # Deque will store indices of potential maximum values
+    window = deque()
+
+
+    for i in range(k):
+
+        while window and nums[i] >= nums[window[-1]]:
+            window.pop()
+        window.append(i)
+
+
+    for i in range(k, len(nums)):
+        # First element in deque is the largest in previous window
+        result.append(nums[window[0]])
+
+        # Remove elements outside current window
+        while window and window[0] <= i - k:
+            window.popleft()
+
+        # Remove smaller elements from back
+        while window and nums[i] >= nums[window[-1]]:
+            window.pop()
+
+        window.append(i)
+
+    # Add maximum element of last window
+    result.append(nums[window[0]])
+
+    return result
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()