Add function to find duplicate number using Floyd's algorithm

Implement a function to detect a duplicate in a list using Floyd's Tortoise and Hare algorithm. Includes unittests to validate functionality with various test cases.

Author: nicholasadamou

Linked Lists/find_the_duplicate_number.py

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+# https://neetcode.io/problems/find-duplicate-integer
+
+import unittest
+
+
+def find_duplicate(nums: list[int]) -> int:
+    # Use Floyd's Tortoise and Hare algorithm to detect cycles.
+    # This algorithm is used here to find the duplicate number in the list.
+    # The problem guarantees that there is exactly one duplicate number.
+
+    # Initialize both pointers to the start of the list.
+    slow = nums[0]
+    fast = nums[0]
+
+    # Phase 1: Finding the intersection point of the two pointers.
+    # Move 'slow' by one step and 'fast' by two steps until they meet.
+    # Since there is a duplicate, a cycle must exist, and they will meet inside the cycle.
+    while True:
+        slow = nums[slow]  # Move slow pointer by one step.
+        fast = nums[nums[fast]]  # Move fast pointer by two steps.
+
+        if slow == fast:
+            # Pointers have met, indicating a cycle.
+            break
+
+    # Phase 2: Finding the entrance to the cycle,
+    # which is the duplicate number.
+    slow = nums[0]  # Move slow pointer back to the start of the list.
+
+    # Move both pointers at the same speed until they meet again.
+    # The meeting point is the start of the cycle, i.e., the duplicate number.
+    while slow != fast:
+        slow = nums[slow]  # Move slow pointer by one step.
+        fast = nums[fast]  # Move fast pointer by one step.
+
+    return slow  # The duplicate number.
+
+
+class Test(unittest.TestCase):
+    def test(self):
+        self.assertEqual(find_duplicate([1, 3, 4, 2, 2]), 2)
+        self.assertEqual(find_duplicate([3, 1, 3, 4, 2]), 3)
+        self.assertEqual(find_duplicate([1, 1]), 1)
+        self.assertEqual(find_duplicate([1, 1, 2]), 1)