Consider that the main memory of a computer is 256 words, with each word of 8 bits in size. The cache memory of this computer is 16 blocks, having a block size of 32 bits. Find the cache addresses for the main memory addresses 5Ah and 8Bh given the following cache mapping schemes: (i)Associative cache mapping (ii)Direct cache mapping (iii)Two-way set associative cache mapping

To find the cache addresses for the given main memory addresses under different cache mapping schemes, let’s consider each mapping scheme:

(i) Associative Cache Mapping:

In associative cache mapping, any block of main memory can be placed in any block of cache memory. To find the cache address for a given main memory address, we perform a search in the cache for the block containing the desired address.

For the main memory address 5Ah:

• Main memory address: 5Ah
• Cache address: Search all blocks in the cache for the block containing the main memory address 5Ah.

For the main memory address 8Bh:

• Main memory address: 8Bh
• Cache address: Search all blocks in the cache for the block containing the main memory address 8Bh.

(ii) Direct Cache Mapping:

In direct cache mapping, each block of main memory is mapped to a specific block in the cache. The mapping is done using the least significant bits of the main memory address.

Given that the cache size is 16 blocks, we need 4 bits to address each block in the cache.

For the main memory address 5Ah:

• Main memory address: 01011010
• Block number in cache: 0101 (from bits 3 to 6)
• Cache address: Block number 5

For the main memory address 8Bh:

• Main memory address: 10001011
• Block number in cache: 1000 (from bits 3 to 6)
• Cache address: Block number 8

(iii) Two-way Set Associative Cache Mapping:

In two-way set associative cache mapping, the cache is divided into sets, and each set contains two blocks. Each block in main memory is mapped to one of the sets in the cache. Within each set, the block placement can be associative.

Given that the cache size is 16 blocks and there are 8 sets (16 blocks / 2 blocks per set), we need 3 bits to address each set in the cache.

For the main memory address 5Ah:

• Main memory address: 01011010
• Set number in cache: 010 (from bits 3 to 5)
• Cache address: Search both blocks in set number 2 for the block containing the main memory address 5Ah.

For the main memory address 8Bh:

• Main memory address: 10001011
• Set number in cache: 000 (from bits 3 to 5)
• Cache address: Search both blocks in set number 0 for the block containing the main memory address 8Bh.