updated

data_structures/arrays/median_two_array.py

@@ -1,18 +1,18 @@
-"""
-https://www.enjoyalgorithms.com/blog/median-of-two-sorted-arrays
-"""
-
-
 def find_median_sorted_arrays(nums1: list[int], nums2: list[int]) -> float:
     """
-    Find the median of two arrays.
+    Find the median of two sorted arrays.
+
+    This implementation uses binary search to achieve an optimal solution in O(log(min(m, n))) time.
 
     Args:
-        nums1: The first array.
-        nums2: The second array.
+        nums1: The first sorted array.
+        nums2: The second sorted array.
 
     Returns:
-    The median of the two arrays.
+        The median of the two sorted arrays.
+
+    Raises:
+        ValueError: If both input arrays are empty.
 
     Examples:
         >>> find_median_sorted_arrays([1, 3], [2])
@@ -25,7 +25,7 @@ def find_median_sorted_arrays(nums1: list[int], nums2: list[int]) -> float:
         0.0
 
         >>> find_median_sorted_arrays([], [])
-        Traceback (most recent call last):
+        Traceback (most recent last):
             ...
         ValueError: Both input arrays are empty.
 
@@ -41,19 +41,104 @@ def find_median_sorted_arrays(nums1: list[int], nums2: list[int]) -> float:
     if not nums1 and not nums2:
         raise ValueError("Both input arrays are empty.")
 
-    # Merge the arrays into a single sorted array.
-    merged = sorted(nums1 + nums2)
-    total = len(merged)
+    # Insert 1: Ensure the smaller array is nums1
+    if len(nums1) > len(nums2):
+        nums1, nums2 = nums2, nums1
 
-    if total % 2 == 1:  # If the total number of elements is odd
-        return float(merged[total // 2])  # then return the middle element
+    m, n = len(nums1), len(nums2)
+    low, high = 0, m
 
-    # If the total number of elements is even, calculate
-    # the average of the two middle elements as the median.
-    middle1 = merged[total // 2 - 1]
-    middle2 = merged[total // 2]
-    return (float(middle1) + float(middle2)) / 2.0
+    while low <= high:
+        partition1 = (low + high) // 2
+        partition2 = (m + n + 1) // 2 - partition1
 
+        # Handle edges of the arrays with infinities
+        max_left1 = float('-inf') if partition1 == 0 else nums1[partition1 - 1]
+        min_right1 = float('inf') if partition1 == m else nums1[partition1]
+
+        max_left2 = float('-inf') if partition2 == 0 else nums2[partition2 - 1]
+        min_right2 = float('inf') if partition2 == n else nums2[partition2]
+
+        # Insert 2: Debugging: Log partition indices and elements (useful for large arrays)
+        # print(f"partition1: {partition1}, partition2: {partition2}")
+        # print(f"max_left1: {max_left1}, min_right1: {min_right1}")
+        # print(f"max_left2: {max_left2}, min_right2: {min_right2}")
+
+        # Check if we have found the correct partition
+        if max_left1 <= min_right2 and max_left2 <= min_right1:
+            if (m + n) % 2 == 1:
+                return max(max_left1, max_left2)
+            else:
+                return (max(max_left1, max_left2) + min(min_right1, min_right2)) / 2.0
+        elif max_left1 > min_right2:
+            high = partition1 - 1
+        else:
+            low = partition1 + 1
+
+    # Insert 3: Remove redundant exception, already handled at the beginning
+    # raise ValueError("Input arrays are not sorted.")  # This line is no longer necessary.
+
+def merge_sorted_arrays(nums1: list[int], nums2: list[int]) -> list[int]:
+    """
+    Merge two sorted arrays into a single sorted array in O(m + n) time.
+
+    Args:
+        nums1: The first sorted array.
+        nums2: The second sorted array.
+
+    Returns:
+        A new sorted array containing all elements of nums1 and nums2.
+
+    Examples:
+        >>> merge_sorted_arrays([1, 3], [2])
+        [1, 2, 3]
+
+        >>> merge_sorted_arrays([1, 2], [3, 4])
+        [1, 2, 3, 4]
+
+        >>> merge_sorted_arrays([0, 0], [0, 0])
+        [0, 0, 0, 0]
+
+        >>> merge_sorted_arrays([], [1])
+        [1]
+
+        >>> merge_sorted_arrays([], [])
+        []
+    """
+    # Insert 4: Edge case: If one array is empty, just return the other array
+    if not nums1: 
+        return nums2
+    if not nums2:
+        return nums1
+
+    i, j = 0, 0
+    merged = []
+
+    while i < len(nums1) and j < len(nums2):
+        if nums1[i] < nums2[j]:
+            merged.append(nums1[i])
+            i += 1
+        else:
+            merged.append(nums2[j])
+            j += 1
+
+    # Insert 5: Append remaining elements from either nums1 or nums2
+    merged.extend(nums1[i:])
+    merged.extend(nums2[j:])
+    return merged
+
+
+def is_sorted(nums: list[int]) -> bool:
+    """
+    Helper function to check if the array is sorted.
+
+    Args:
+        nums: The array to check.
+
+    Returns:
+        True if the array is sorted, False otherwise.
+    """
+    return all(nums[i] <= nums[i + 1] for i in range(len(nums) - 1))
 
 if __name__ == "__main__":
     import doctest