Add Count Inversions algorithm using Merge Sort

Author: usman-7455

sorts/count_inversion.py

@@ -0,0 +1,56 @@
+"""
+Count Inversions in an array using a modified Merge Sort algorithm.
+
+An inversion is a pair of indices (i, j) such that:
+    - i < j
+    - arr[i] > arr[j]
+This represents a disorder in the array relative to a sorted order.
+
+This implementation efficiently counts the number of such inversions
+in O(n log n) time by modifying the merge step of Merge Sort.
+
+Use Cases:
+----------
+- Measuring how far a sequence is from being sorted.
+- Evaluating disorder in datasets (e.g., in rankings or sequences).
+- Useful in computer vision, bioinformatics, and comparison of lists.
+- Core concept in Kendall tau distance and other similarity metrics.
+"""
+def count_inversions(arr):
+    def merge_sort(arr):
+        if len(arr) <= 1:
+            return arr, 0
+
+        mid = len(arr) // 2
+        left, inv_left = merge_sort(arr[:mid])
+        right, inv_right = merge_sort(arr[mid:])
+        merged, inv_split = merge(left, right)
+
+        total_inversions = inv_left + inv_right + inv_split
+        return merged, total_inversions
+
+    def merge(left, right):
+        i = j = inv_count = 0
+        merged = []
+
+        while i < len(left) and j < len(right):
+            if left[i] <= right[j]:
+                merged.append(left[i])
+                i += 1
+            else:
+                merged.append(right[j])
+                j += 1
+                inv_count += len(left) - i  # All remaining in left > right[j]
+
+        merged += left[i:]
+        merged += right[j:]
+        return merged, inv_count
+
+    _, total_inversions = merge_sort(arr)
+    return total_inversions
+
+
+# Example usage
+if __name__ == "__main__":
+    sample = [2, 4, 1, 3, 5]
+    print(f"Inversion count: {count_inversions(sample)}")