Calculating number of page faults for 2-d array

Question

I am trying to study for an exam..and I found this example but can\'t understand how they got the answer. Can anyone explain it please?

Question:

Answer 1

Suppose your systems allocated Two Frames for your process such that 200 * sizeof(int) of matrix can be keep in memory at a time. And allocation for matrix happens in Row Major Order.

In first loop A:

for (int j=0;j<100;j++)
   for (int i=0; i<100; i++)
     A[i][j] = 0;

loop access memory cells for matrix column-wise like:

A[0][0], A[2][0], A[3][0], ...A[0][2], A[0][3], A[0][4], ......
  ^        ^        ^   
      row changes               

At each iteration Row changes and allocation is in row major and each row takes one page. so code A will causes page fault for each alternative A[i][j] access so total number of page faults are = 100 * 100 / 2) = 5000.

Where as second code B:

for(int i=0; i<100; i++)
  for (int j=0; j<100; j++)
      A[i][j] = 0;

loop access memory cells for matrix row-wise on each iteration, like:

A[0][0], A[0][5], A[0][6],...,A[1][0], A[1][7], A[1][8],...,A[2][0], A[2][9],
     ^        ^        ^  
  column changes, row are same 

Access row wise (columns changes at read read row changes only after 100 read), One row loaded at time so page fault happens when row changes (for outer loop) and for each alternative row access a page fault happens so number of page faults = 100/2 = 50.

we can understand it another ways like:
In row major, Number of times row index changes we need new page to access because number of pages are small it page fault on each alternative index change in first A loop row index changes 100*100 times where as in B loop row index changes 100 times so page fault ration in both A/B = 100*100/100 = 100, and if number of page faults happens in A = 50,00 then in B number of page faults = 50,00/100 = 50.

Similarly you can calculate number of page-faults for Column-major order and because matrix has equal numbers of rows and cols result will be same.

The similar example is given in my book:
Download a pdf: operating system book Galvin Read chapter 9: Virtual Memory Section: 9.9.5 Program Structure.