# Program to find valid matrix given row and column sums in Python

PythonServer Side ProgrammingProgramming

Suppose we have two arrays rowSum and colSum with non-negative values where rowSum[i] has the sum of the elements in the ith row and colSum[j] has the sum of the elements in the jth column of a 2D matrix. We have to find any matrix with non-negative values of size (rowSum size x colSum size) that satisfies the given rowSum and colSum values.

So, if the input is like rowSum = [13,14,12] colSum = [9,13,17], then the output will be

 9 4 0 0 9 5 0 0 12

To solve this, we will follow these steps −

• matrix := create empty matrix
• visited := a new set
• Define a function minimum() . This will take r,c
• min_total := infinity
• type := blank string
• for i in range 0 to size of r - 1, do
• if r[i] < min_total, then
• index := i
• type := 'row'
• min_total := r[i]
• for i in range 0 to size of c - 1, do
• if c[i] < min_total, then
• min_total := c[i]
• type := 'col'
• index := i
• if type is same as 'row', then
• r[index] := infinity
• for i in range 0 to size of c - 1, do
• if c[i] is not same as infinity and c[i] >= min_total, then
• c[i] := c[i] - min_total
• matrix[index, i] := min_total
• come out from loop
• if type is same as 'col', then
• c[index] := infinity
• for i in range 0 to size of r - 1, do
• if r[i] is not same as infinity and r[i] >= min_total, then
• r[i] := r[i] - min_total
• matrix[i, index] := min_total
• come out from loop
• insert pair (index,type) into visited
• From the main method do the following −
• while size of visited is not same as size of r +len(c) , do
• minimum(r, c)
• return matrix

## Example

Let us see the following implementation to get better understanding −

def solve(r, c):
matrix = [*len(c) for _ in range(len(r))]
visited = set()

def minimum(r,c):
min_total = float('inf')

type = ''
for i in range(len(r)):
if(r[i] < min_total):
index = i
type = 'row'
min_total = r[i]

for i in range(len(c)):
if(c[i] < min_total):
min_total = c[i]
type = 'col'
index = i

if(type == 'row'):
r[index] = float('inf')

for i in range(len(c)):
if(c[i] != float('inf') and c[i] >= min_total):
c[i] -= min_total
matrix[index][i] = min_total
break

if(type == 'col'):
c[index] = float('inf')
for i in range(len(r)):
if(r[i] != float('inf') and r[i] >= min_total):
r[i] -= min_total
matrix[i][index] = min_total
break

while len(visited) != len(r)+len(c):
minimum(r,c)

return matrix

rowSum = [13,14,12]
colSum = [9,13,17]
print(solve(rowSum, colSum))

## Input

[13,14,12], [9,13,17]

## Output

[[9, 4, 0], [0, 9, 5], [0, 0, 12]]