$.$Access pattern$$ is defined as the order in which the elements within a data

structure are referenced. The access pattern can have a major impact on

performance when using a memory system that employs caches.

Consider a system that has a direct$-$mapped D$-$cache $($data cache$)$ containing $4096$

lines. Each line is $256$ bytes in size. The D$-$cache is initially empty. The matrix X with

$256$ rows and $256$ columns is to be processed. The matrix resides in memory

starting at address $0$x$40010000.$ Each matrix element is $32$ bits. Registers are used

for the array indices and no code optimizations are performed.

a$)(3)$ If the matrix is stored in row major order, what is the miss ratio for the data

cache based on the matrix accesses made by the following code?

for $($j$=0$; j$<256$; j$=$j$+2$;$)\{$

for $\{$i$=0$; i$<256$; i$=$i$+2\}$

X$[$i$,$j$]=$ X$[$i$,$j$]+$ X$[$i$+1,$j$]$;

$\}$

$\}$

b$)(3)$ If the matrix is stored instead in column major order, what is the hit ratio for

the cache based on the matrix accesses made by the following code?

for $($i$=0$; i$<256$; i$=$i$+2$;$)\{$

for $\{$j$=0$; j$<256$; j$=$j$+2\}$

X$[$i$,$j$]=$ X$[$i$,$j$]+$ X$[$i$+1,$j$]$;

$\}$

$\}$