from sys import maxsize


def array_equalization(vector: list[int], step_size: int) -> int:
 """
 This algorithm equalizes all elements of the input vector
 to a common value, by making the minimal number of
 "updates" under the constraint of a step size (step_size).

 >>> array_equalization([1, 1, 6, 2, 4, 6, 5, 1, 7, 2, 2, 1, 7, 2, 2], 4)
 4
 >>> array_equalization([22, 81, 88, 71, 22, 81, 632, 81, 81, 22, 92], 2)
 5
 >>> array_equalization([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 5)
 0
 >>> array_equalization([22, 22, 22, 33, 33, 33], 2)
 2
 >>> array_equalization([1, 2, 3], 0)
 Traceback (most recent call last):
 ValueError: Step size must be positive and non-zero.
 >>> array_equalization([1, 2, 3], -1)
 Traceback (most recent call last):
 ValueError: Step size must be positive and non-zero.
 >>> array_equalization([1, 2, 3], 0.5)
 Traceback (most recent call last):
 ValueError: Step size must be an integer.
 >>> array_equalization([1, 2, 3], maxsize)
 1
 """
 if step_size <= 0:
 raise ValueError("Step size must be positive and non-zero.")
 if not isinstance(step_size, int):
 raise ValueError("Step size must be an integer.")

 unique_elements = set(vector)
 min_updates = maxsize

 for element in unique_elements:
 elem_index = 0
 updates = 0
 while elem_index < len(vector):
 if vector[elem_index] != element:
 updates += 1
 elem_index += step_size
 else:
 elem_index += 1
 min_updates = min(min_updates, updates)

 return min_updates


if __name__ == "__main__":
 from doctest import testmod

 testmod()