prompter: how to write hungarian algorithm?
assistant: The Hungarian algorithm is a method for solving assignment problems. It is named after its inventor, George Dantzig, who published it in 1951. The algorithm is based on the idea of finding a minimum-cost matching between two sets of items.

The Hungarian algorithm can be used to solve a variety of problems, including:

The assignment problem: Given a set of workers and a set of jobs, find a way to assign each worker to a job so that the total cost is minimized.
The transportation problem: Given a set of suppliers and a set of customers, find a way to transport goods from the suppliers to the customers so that the total cost is minimized.
The minimum-cost flow problem: Given a network of nodes and edges, find a way to flow a certain amount of fluid from one node to another so that the total cost is minimized.
The Hungarian algorithm is a polynomial-time algorithm, meaning that it takes time that is proportional to the cube of the number of items in the problem. It is also a greedy algorithm, meaning that it always makes the choice that looks best at the moment, without considering the long-term consequences.

The Hungarian algorithm has been shown to be optimal in many cases, but it is not always the best choice for solving assignment problems. In some cases, other algorithms, such as the network simplex algorithm, can find a better solution.

Here is a sample implementation in python.
```python

import numpy as np

def min_zero_row(zero_mat, mark_zero):
	
	'''
	The function can be splitted into two steps:
	#1 The function is used to find the row which containing the fewest 0.
	#2 Select the zero number on the row, and then marked the element corresponding row and column as False
	'''

	#Find the row
	min_row = [99999, -1]

	for row_num in range(zero_mat.shape[0]): 
		if np.sum(zero_mat[row_num] == True) > 0 and min_row[0] > np.sum(zero_mat[row_num] == True):
			min_row = [np.sum(zero_mat[row_num] == True), row_num]

	# Marked the specific row and column as False
	zero_index = np.where(zero_mat[min_row[1]] == True)[0][0]
	mark_zero.append((min_row[1], zero_index))
	zero_mat[min_row[1], :] = False
	zero_mat[:, zero_index] = False

def mark_matrix(mat):

	'''
	Finding the returning possible solutions for LAP problem.
	'''

	#Transform the matrix to boolean matrix(0 = True, others = False)
	cur_mat = mat
	zero_bool_mat = (cur_mat == 0)
	zero_bool_mat_copy = zero_bool_mat.copy()

	#Recording possible answer positions by marked_zero
	marked_zero = []
	while (True in zero_bool_mat_copy):
		min_zero_row(zero_bool_mat_copy, marked_zero)
	
	#Recording the row and column positions seperately.
	marked_zero_row = []
	marked_zero_col = []
	for i in range(len(marked_zero)):
		marked_zero_row.append(marked_zero[i][0])
		marked_zero_col.append(marked_zero[i][1])

	#Step 2-2-1
	non_marked_row = list(set(range(cur_mat.shape[0])) - set(marked_zero_row))
	
	marked_cols = []
	check_switch = True
	while check_switch:
		check_switch = False
		for i in range(len(non_marked_row)):
			row_array = zero_bool_mat[non_marked_row[i], :]
			for j in range(row_array.shape[0]):
				#Step 2-2-2
				if row_array[j] == True and j not in marked_cols:
					#Step 2-2-3
					marked_cols.append(j)
					check_switch = True

		for row_num, col_num in marked_zero:
			#Step 2-2-4
			if row_num not in non_marked_row and col_num in marked_cols:
				#Step 2-2-5
				non_marked_row.append(row_num)
				check_switch = True
	#Step 2-2-6
	marked_rows = list(set(range(mat.shape[0])) - set(non_marked_row))

	return(marked_zero, marked_rows, marked_cols)

def adjust_matrix(mat, cover_rows, cover_cols):
	cur_mat = mat
	non_zero_element = []

	#Step 4-1
	for row in range(len(cur_mat)):
		if row not in cover_rows:
			for i in range(len(cur_mat[row])):
				if i not in cover_cols:
					non_zero_element.append(cur_mat[row][i])
	min_num = min(non_zero_element)

	#Step 4-2
	for row in range(len(cur_mat)):
		if row not in cover_rows:
			for i in range(len(cur_mat[row])):
				if i not in cover_cols:
					cur_mat[row, i] = cur_mat[row, i] - min_num
	#Step 4-3
	for row in range(len(cover_rows)):  
		for col in range(len(cover_cols)):
			cur_mat[cover_rows[row], cover_cols[col]] = cur_mat[cover_rows[row], cover_cols[col]] + min_num
	return cur_mat

def hungarian_algorithm(mat): 
	dim = mat.shape[0]
	cur_mat = mat

	#Step 1 - Every column and every row subtract its internal minimum
	for row_num in range(mat.shape[0]): 
		cur_mat[row_num] = cur_mat[row_num] - np.min(cur_mat[row_num])
	
	for col_num in range(mat.shape[1]): 
		cur_mat[:,col_num] = cur_mat[:,col_num] - np.min(cur_mat[:,col_num])
	zero_count = 0
	while zero_count < dim:
		#Step 2 & 3
		ans_pos, marked_rows, marked_cols = mark_matrix(cur_mat)
		zero_count = len(marked_rows) + len(marked_cols)

		if zero_count < dim:
			cur_mat = adjust_matrix(cur_mat, marked_rows, marked_cols)

	return ans_pos

def ans_calculation(mat, pos):
	total = 0
	ans_mat = np.zeros((mat.shape[0], mat.shape[1]))
	for i in range(len(pos)):
		total += mat[pos[i][0], pos[i][1]]
		ans_mat[pos[i][0], pos[i][1]] = mat[pos[i][0], pos[i][1]]
	return total, ans_mat

def main():

	'''Hungarian Algorithm: 
	Finding the minimum value in linear assignment problem.
	Therefore, we can find the minimum value set in net matrix 
	by using Hungarian Algorithm. In other words, the maximum value
	and elements set in cost matrix are available.'''

	#The matrix who you want to find the minimum sum
	cost_matrix = np.array([[7, 6, 2, 9, 2],
				[6, 2, 1, 3, 9],
				[5, 6, 8, 9, 5],
				[6, 8, 5, 8, 6],
				[9, 5, 6, 4, 7]])
	ans_pos = hungarian_algorithm(cost_matrix.copy())#Get the element position.
	ans, ans_mat = ans_calculation(cost_matrix, ans_pos)#Get the minimum or maximum value and corresponding matrix.

	#Show the result
	print(f"Linear Assignment problem result: {ans:.0f}\n{ans_mat}")

	#If you want to find the maximum value, using the code as follows: 
	#Using maximum value in the cost_matrix and cost_matrix to get net_matrix
	profit_matrix = np.array([[7, 6, 2, 9, 2],
				[6, 2, 1, 3, 9],
				[5, 6, 8, 9, 5],
				[6, 8, 5, 8, 6],
				[9, 5, 6, 4, 7]])
	max_value = np.max(profit_matrix)
	cost_matrix = max_value - profit_matrix
	ans_pos = hungarian_algorithm(cost_matrix.copy())#Get the element position.
	ans, ans_mat = ans_calculation(profit_matrix, ans_pos)#Get the minimum or maximum value and corresponding matrix.
	#Show the result
	print(f"Linear Assignment problem result: {ans:.0f}\n{ans_mat}")

if __name__ == '__main__':
	main()
```