Matrix Transpose

March 20, 2015

Transposing a matrix turns rows into columns and columns into rows; for instance the transpose of the matrix below left is the matrix below right:

11 12 13 14 11 21 31
21 22 23 24 12 22 32
31 32 33 34 13 23 33
14 24 34

That’s easy enough to do when everything fits in memory, but when the matrix is stored in a file and is too big to fit in memory, things get rather harder.

Your task is to write a program that transposes a matrix to large to fit in memory. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

5 Responses to “Matrix Transpose”

Rutger said

March 20, 2015 at 10:09 AM

In Python, assuming that we cán fit one row in memory..

file_like_object = []
file_like_object.append([11,12,13,14,11,21,31])
file_like_object.append([21,22,23,24,12,22,32])
file_like_object.append([31,32,33,34,13,23,33])
file_like_object.append([14,24,34])


max_row_length = 1
i = 0
while i < max_row_length:
	for row in file_like_object:
		if len(row) > max_row_length:
			max_row_length = len(row)
		if i < len(row):
			print row[i],
	print
	i += 1

matthew said
March 20, 2015 at 1:24 PM
If it’s a square matrix with the row and column lengths 2^n, we can transpose in just n linear scans through the array, zipping together the front and back halves (like a riffle shuffle):

zipl (a:s) (b:t) = a:b:zipl s t
zipl _ _ = []

trans 0 m s = s
trans n m s = trans (n-1) m (zipl (take m s) (drop m s))

test n = trans n (2^(2*n-1)) [0..2^(2*n)]
main = print (test 2)
matthew said
March 20, 2015 at 4:45 PM
Actually, this will work for any matrix where the number of rows is a power of two. Tidying up the previous solution (now trans m n s expects m to be half the total array size, n is the number of rows, s is the array):

zipl (a:s) (b:t) = a:b:zipl s t
zipl _ _ = []

trans m 1 s = s
trans m n s = trans m (n `div` 2) (zipl (take m s) (drop m s))

main =
print [0..23] >>
print (trans 12 8 [0..23]) >>
print (trans 12 4 [0..23])

matthew said

March 20, 2015 at 4:50 PM

Just trying to get some nicer formatting:

zipl (a:s) (b:t) = a:b:zipl s t
zipl _ _ = []

trans m 1 s = s
trans m n s = trans m (n `div` 2) (zipl (take m s) (drop m s))

main =
print [0..23] >>
print (trans 12 8 [0..23]) >>
print (trans 12 4 [0..23])

matthew said

March 21, 2015 at 9:00 AM

And where the number of columns is 2^n, we can invert the process – output all even elements, then all odd elements, repeat n times. In Haskell:

pairs (a:b:s) = (a,b):pairs s
pairs _ = []

trans 0 s = s
trans n s = trans (n-1) (uncurry (++) $ unzip $ pairs s)
 
main =
     print [0..23] >>
     print (trans 1 [0..23]) >>
     print (trans 2 [0..23]) >>
     print (trans 3 [0..23])

Or maybe this corresponds better to how we might do it with files:

evens (a:s) = a : odds s
evens _ = []
odds (_:s) = evens s
odds _ = []

trans 0 s = s
trans n s = trans (n-1) (evens s ++ odds s)

main =
     print [0..23] >>
     print (trans 1 [0..23]) >>
     print (trans 2 [0..23]) >>
     print (trans 3 [0..23])

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