Add edge cases for equilibrium index (#15372)

* Add edge cases for equilibrium index

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Refactor docstring in equilibrium_index_in_array.py

Updated docstring examples and formatting for clarity.

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

data_structures/arrays/equilibrium_index_in_array.py

@@ -1,54 +1,62 @@
 """
 Find the Equilibrium Index of an Array.
-Reference: https://www.geeksforgeeks.org/equilibrium-index-of-an-array/
 
-Python doctest can be run with the following command:
-python -m doctest -v equilibrium_index_in_array.py
-
-Given a sequence arr[] of size n, this function returns
-an equilibrium index (if any) or -1 if no equilibrium index exists.
-
-The equilibrium index of an array is an index such that the sum of
-elements at lower indexes is equal to the sum of elements at higher indexes.
+Reference:
+https://www.geeksforgeeks.org/equilibrium-index-of-an-array
 
+Python doctest can be run with:
 
+python -m doctest -v equilibrium_index_in_array.py
 
-Example Input:
-arr = [-7, 1, 5, 2, -4, 3, 0]
-Output: 3
+Given an array arr of size n, return an equilibrium index
+if one exists; otherwise return -1.
 
+An equilibrium index is an index where the sum of all
+elements to the left equals the sum of all elements
+to the right.
 """
 
 
 def equilibrium_index(arr: list[int]) -> int:
     """
-    Find the equilibrium index of an array.
-
+    Find the first equilibrium index of an array.
     Args:
-        arr (list[int]): The input array of integers.
-
+        arr: The input array of integers.
     Returns:
-        int: The equilibrium index or -1 if no equilibrium index exists.
-
+        The first equilibrium index, or -1 if none exists.
     Examples:
+        >>> equilibrium_index([])
+        -1
+        >>> equilibrium_index([5])
+        0
         >>> equilibrium_index([-7, 1, 5, 2, -4, 3, 0])
         3
+        >>> equilibrium_index([2, 4, 6, 8, 10, 3])
+        -1
         >>> equilibrium_index([1, 2, 3, 4, 5])
         -1
         >>> equilibrium_index([1, 1, 1, 1, 1])
         2
-        >>> equilibrium_index([2, 4, 6, 8, 10, 3])
-        -1
+        >>> equilibrium_index([0, 0, 0])
+        0
+        >>> equilibrium_index([-1, -1, -1])
+        1
+        >>> equilibrium_index([1, -1, 0])
+        2
+
+    Time Complexity:
+        O(n), where n is the length of the array.
+
+    Space Complexity:
+        O(1), using only constant extra space.
     """
     total_sum = sum(arr)
     left_sum = 0
-
     for i, value in enumerate(arr):
         total_sum -= value
         if left_sum == total_sum:
             return i
         left_sum += value
-
     return -1