




from __future__ import annotations


def 				_a       (				lowerCAmelCase						,	lowerCAmelCase   )->  int:
		SCREAMING_SNAKE_CASE_			,							SCREAMING_SNAKE_CASE_							=       0, len(lowerCAmelCase   ) - 1
		while start_ind <= end_ind:
				if array[start_ind] == search_item:
						return start_ind
				elif array[end_ind] == search_item:
						return end_ind
				else:
						start_ind += 1
						end_ind -= 1
    # returns -1 if search_item is not found in array
		return -1


if __name__ == "__main__":
						print(double_linear_search(list(range(1_0_0)), 4_0))





from __future__ import annotations


def 				_a       (				lowerCAmelCase   )->  list[int]:
		if not list_of_ints:
				return []

		if min(lowerCAmelCase   ) < 0:
				raise ValueError('All numbers must be positive'   )

		SCREAMING_SNAKE_CASE_							=       max(len(bin(lowerCAmelCase   )[2:]   ) for x in list_of_ints   )
		return _msd_radix_sort(lowerCAmelCase						,	lowerCAmelCase   )


def 				_a       (				lowerCAmelCase						,	lowerCAmelCase   )->  list[int]:
		if bit_position == 0 or len(lowerCAmelCase   ) in [0, 1]:
				return list_of_ints

		SCREAMING_SNAKE_CASE_							=       []
		SCREAMING_SNAKE_CASE_							=       []
		# Split numbers based on bit at bit_position from the right
		for number in list_of_ints:
				if (number >> (bit_position - 1)) & 1:
						# number has a one at bit bit_position
						ones.append(lowerCAmelCase   )
				else:
						# number has a zero at bit bit_position
						zeros.append(lowerCAmelCase   )

    # recursively split both lists further
		SCREAMING_SNAKE_CASE_							=       _msd_radix_sort(lowerCAmelCase						,	bit_position - 1   )
		SCREAMING_SNAKE_CASE_							=       _msd_radix_sort(lowerCAmelCase						,	bit_position - 1   )

		# recombine lists
		SCREAMING_SNAKE_CASE_							=       zeros
		res.extend(lowerCAmelCase   )

		return res


def 				_a       (				lowerCAmelCase   )->  Dict:
		SCREAMING_SNAKE_CASE_							=       len(lowerCAmelCase   )
		if not list_of_ints or length == 1:
				return

		if min(lowerCAmelCase   ) < 0:
				raise ValueError('All numbers must be positive'   )

		SCREAMING_SNAKE_CASE_							=       max(len(bin(lowerCAmelCase   )[2:]   ) for x in list_of_ints   )
		_msd_radix_sort_inplace(lowerCAmelCase						,	lowerCAmelCase						,	0						,	lowerCAmelCase   )


def 				_a       (				lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase   )->  Optional[int]:
		if bit_position == 0 or end_index - begin_index <= 1:
				return

		bit_position -= 1

		SCREAMING_SNAKE_CASE_							=       begin_index
		SCREAMING_SNAKE_CASE_							=       end_index - 1
		while i <= j:
				SCREAMING_SNAKE_CASE_							=       False
				if not (list_of_ints[i] >> bit_position) & 1:
						# found zero at the beginning
						i += 1
						SCREAMING_SNAKE_CASE_							=       True
				if (list_of_ints[j] >> bit_position) & 1:
						# found one at the end
						j -= 1
						SCREAMING_SNAKE_CASE_							=       True

				if changed:
						continue

				SCREAMING_SNAKE_CASE_			,							SCREAMING_SNAKE_CASE_							=       list_of_ints[j], list_of_ints[i]
				j -= 1
				if j != i:
						i += 1

		_msd_radix_sort_inplace(lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase   )
		_msd_radix_sort_inplace(lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase						,	lowerCAmelCase   )


if __name__ == "__main__":
						import doctest

						doctest.testmod()