Set row and column to 1's

Given a matrix of size n x m filled with 0's and 1's

e.g.:

1 1 0 1 0

0 0 0 0 0

0 1 0 0 0

1 0 1 1 0

if the matrix has 1 at (i,j), fill the column j and row i with 1's

i.e., we get:

1 1 1 1 1

1 1 1 1 0

1 1 1 1 1

1 1 1 1 1

Required complexity: O(n*m) time and O(1) space

NOTE: you are not allowed to store anything except '0' or '1' in the matrix entries

Here is a solution in python pseudocode that uses 2 extra bools of storage. I think it is more clear than I could do in English.

def scanRow(i):
    return 0 if row i is all zeroes, else 1

def scanColumn(j):
    return 0 if col j is all zeroes, else 1

# we're going to use the first row and column
# of the matrix to store row and column scan values,
# but we need aux storage to deal with the overlap
firstRow = scanRow(0)
firstCol = scanCol(0)

# scan each column and store result in 1st row - O(mn) work
for col in range(1, n):
    matrix[0, col] = scanColumn(col)

# now row 0 tells us whether each column is all zeroes or not
# it's also the correct output unless row 0 contained a 1 originally

# do the same for rows into column 0 - O(mn) work
for row in range(1, m):
    matrix[row, 0] = scanRow(row)

matrix[0,0] = firstRow or firstCol

# now deal with the rest of the values - O(mn) work
for row in range(1, m):
    for col in range(1, n):
        matrix[row, col] = matrix[0, col] or matrix[row, 0]


# 3 O(mn) passes!

# go back and fix row 0 and column 0
if firstRow:
    # set row 0 to all ones

if firstCol:
    # set col 0 to all ones