"""simple docstring"""







def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	global f  # a global dp table for knapsack
	if f[i][j] < 0:
		if j < wt[i - 1]:
			_lowercase :    List[Any]							=							mf_knapsack(i - 1     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE			)
		else:
			_lowercase :    Union[str, Any]							=							max(
			    mf_knapsack(i - 1     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE			)     ,mf_knapsack(i - 1     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,j - wt[i - 1]			) + val[i - 1]     ,)
		_lowercase :    Optional[int]							=							val
	return f[i][j]


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Optional[Any]							=							[[0] * (w + 1) for _ in range(n + 1			)]

	for i in range(1     ,n + 1			):
		for w_ in range(1     ,w + 1			):
			if wt[i - 1] <= w_:
				_lowercase :    int							=							max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]     ,dp[i - 1][w_]			)
			else:
				_lowercase :    int							=							dp[i - 1][w_]

	return dp[n][w_], dp


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	if not (isinstance(_SCREAMING_SNAKE_CASE     ,(list, tuple)			) and isinstance(_SCREAMING_SNAKE_CASE     ,(list, tuple)			)):
		raise ValueError(
		    'Both the weights and values vectors must be either lists or tuples'			)

	_lowercase :    int							=							len(_SCREAMING_SNAKE_CASE			)
	if num_items != len(_SCREAMING_SNAKE_CASE			):
		_lowercase :    int							=							(
		    'The number of weights must be the same as the number of values.\n'
		    F'''But got {num_items} weights and {len(_SCREAMING_SNAKE_CASE			)} values'''
		)
		raise ValueError(_SCREAMING_SNAKE_CASE			)
	for i in range(_SCREAMING_SNAKE_CASE			):
		if not isinstance(wt[i]     ,_SCREAMING_SNAKE_CASE			):
			_lowercase :    Tuple							=							(
			    'All weights must be integers but got weight of '
			    F'''type {type(wt[i]			)} at index {i}'''
			)
			raise TypeError(_SCREAMING_SNAKE_CASE			)

	_lowercase				,      _lowercase :    Optional[int]							=							knapsack(_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE			)
	_lowercase :    Dict							=							set()
	_construct_solution(_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE			)

	return optimal_val, example_optional_set


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	if i > 0 and j > 0:
		if dp[i - 1][j] == dp[i][j]:
			_construct_solution(_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,i - 1     ,_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE			)
		else:
			optimal_set.add(_SCREAMING_SNAKE_CASE			)
			_construct_solution(_SCREAMING_SNAKE_CASE     ,_SCREAMING_SNAKE_CASE     ,i - 1     ,j - wt[i - 1]     ,_SCREAMING_SNAKE_CASE			)


if __name__ == "__main__":
						SCREAMING_SNAKE_CASE									=						[3, 2, 4, 4]
						SCREAMING_SNAKE_CASE									=						[4, 3, 2, 3]
						SCREAMING_SNAKE_CASE									=						4
						SCREAMING_SNAKE_CASE									=						6
						SCREAMING_SNAKE_CASE									=						[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
						SCREAMING_SNAKE_CASE									=						knapsack(w, wt, val, n)
						print(optimal_solution)
						print(mf_knapsack(n, wt, val, w))  # switched the n and w

						# testing the dynamic programming problem with example
						# the optimal subset for the above example are items 3 and 4
						SCREAMING_SNAKE_CASE									=						knapsack_with_example_solution(w, wt, val)
						assert optimal_solution == 8
						assert optimal_subset == {3, 4}
						print('optimal_value = ', optimal_solution)
						print('An optimal subset corresponding to the optimal value', optimal_subset)




"""simple docstring"""







def      __lowerCAmelCase(		__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    str							=							[0 for i in range(len(__UpperCAmelCase			)			)]

	# initialize interval's left pointer and right pointer
	_lowercase				,      _lowercase :    str							=							0, 0

	for i in range(1     ,len(__UpperCAmelCase			)			):
		# case when current index is inside the interval
		if i <= right_pointer:
			_lowercase :    Union[str, Any]							=							min(right_pointer - i + 1     ,z_result[i - left_pointer]			)
			_lowercase :    Any							=							min_edge

		while go_next(__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):
			z_result[i] += 1

		# if new index's result gives us more right interval,
		# we've to update left_pointer and right_pointer
		if i + z_result[i] - 1 > right_pointer:
			_lowercase				,      _lowercase :    Optional[int]							=							i, i + z_result[i] - 1

	return z_result


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	return i + z_result[i] < len(__UpperCAmelCase			) and s[z_result[i]] == s[i + z_result[i]]


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Union[str, Any]							=							0
	# concatenate 'pattern' and 'input_str' and call z_function
	# with concatenated string
	_lowercase :    List[str]							=							z_function(pattern + input_str			)

	for val in z_result:
		# if value is greater then length of the pattern string
		# that means this index is starting position of substring
		# which is equal to pattern string
		if val >= len(__UpperCAmelCase			):
			answer += 1

	return answer


if __name__ == "__main__":
						import doctest

						doctest.testmod()



