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A computer has a $256\text{-KByte}$, 4-way set associative, write back data cache with block size of $32\text{-Bytes}$. The processor sends $32\text{-bit}$ addresses to the cache controller. Each cache tag directory entry contains, in addition to address tag, $2$ valid bits, $1$ modified bit and $1$ replacement bit.

The number of bits in the tag field of an address is

1. $11$
2. $14$
3. $16$
4. $27$
edited | 3.4k views
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block size=32 bytes=2^5 bytes =2^8  bits means 8 bits needed uniquely identify a block.

no. of sets*lines per set*block size=cache size

2^x  * 4 * 2^8 =2^21(cache size=256 kb which is 2^21 bits)

2^x=2^11 (means 11 bits uniquely identify each set)

32 bit address is send to cache means total is 32 bit

32=tag+set+block offset

32=tag+11+8

tag=13

now additional bits required are 4 so total no. of bits required would be 17.

i know 17 is not there in options but tell me where the problem lies in this method.

Total cache size $= 256\ KB$
Cache block size $=32\text{ Bytes}$
So, number of cache entries $=\dfrac{ 256\ K}{32}=8\ K$

Number of sets in cache $=\dfrac{ 8\ K}{4}=2\ K$ as cache is $4\text{-way}$ associative.

So, $\log(2048) = 11\ bits$ are needed for accessing a set. Inside a set we need to identify the cache entry.

No. of memory block possible $=\dfrac{\text{Memory size}}{\text{Cache block size}}$

$=\dfrac{2^{32}}{32} = 2^{27}$.

So, no. of memory block that can go to a single cache set

$=\dfrac{2^{27}}{2^{11}}$

$=2^{16}.$

So, we need $16\text{ tag bits}$ along with each cache entry to identify which of the possible $2^{16}$ blocks is being mapped there.
edited
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So number of bits for tag field = 32- (11 +5) = 16.

5 bits cache offset and 11 bits to identify a set.
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here they are asking for tag bits which is 16.How the ans is coming 11????
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@Arjun
Can you please tell why you didn't consider additional 4 bits, these extra bits are also stored in tag memory.
We have solved so many questions when we considered tag bits + additional bits together
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Each cache tag directory entry contains, in addition to address tag, 2 valid bits, 1 modified bit and 1 replacement bit

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Means if we are given physical address and try to traslate it in virtual address, then we add or subtract extra bits
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@Vijay "Tag bits" mean address tag - that is the bits which are required for identifying a particular memory block in cache. Cache Tag directory includes the other bits as mentioned in question.
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@ Vijay Thakur

Because it says "number of bits in the tag field of an address". Those extra bits are not part of address, they are present in tag directory.

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@Arjun Suresh Sir can I say we add those additional bits when we translate virtual address to physical address or when calculating Tag directory.
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yes....
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@Arjun, why did you consider block size of main memory as equal to Cache block size ?

No. of memory block possible =Memory size / Cache block size
=232/32=227