How to map the indexes of a matrix to a 1-dimensional array (C++)?

Question

I have an 8x8 matrix, like this:

char matrix[8][8];

Also, I have an array of 64 elements, like this:

char array[64];

Then I have drawn the matrix as a table, and filled the cells with numbers, each number being incremented from left to right, top to bottom.

If I have, say, indexes 3 (column) and 4 (row) into the matrix, I know that it corresponds to the element at position 35 in the array, as it can be seen in the table that I've drawn. I believe there is some sort of formula to translate the 2 indexes of the matrix into a single index of the array, but I can't figure out what it is.

Any ideas?

Answer 1

The way most languages store multi-dimensional arrays is by doing a conversion like the following:

If matrix has size, n (rows) by m (columns), and we're using "row-major ordering" (where we count along the rows first) then:

matrix[ i ][ j ] = array[ i*m + j ].

Here i goes from 0 to (n-1) and j from 0 to (m-1).

So it's just like a number system of base 'm'. Note that the size of the last dimension (here the number of rows) doesn't matter.

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For a conceptual understanding, think of a (3x5) matrix with 'i' as the row number, and 'j' as the column number. If you start numbering from i,j = (0,0) --> 0. For 'row-major' ordering (like this), the layout looks like:

           |-------- 5 ---------|
  Row      ______________________   _ _
   0      |0    1    2    3    4 |   |
   1      |5    6    7    8    9 |   3
   2      |10   11   12   13   14|  _|_
          |______________________|
Column     0    1    2    3    4 

As you move along the row (i.e. increase the column number), you just start counting up, so the Array indices are 0,1,2.... When you get to the second row, you already have 5 entries, so you start with indices 1*5 + 0,1,2.... On the third row, you have 2*5 entries already, thus the indices are 2*5 + 0,1,2....

For higher dimension, this idea generalizes, i.e. for a 3D matrix L by N by M:

matrix[ i ][ j ][ k ] = array[ i*(N*M) + j*M + k ]

and so on.

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For a really good explanation, see: http://www.cplusplus.com/doc/tutorial/arrays/; or for some more technical aspects: http://en.wikipedia.org/wiki/Row-major_order