$[30]$ Consider a $2048$KB$,4-$way set associative write allocate, write back cache using $64-$byte lines

and LRU replacement $($implemented using counters$).$ Addresses are $36-$bits. Express your answers

in bytes or KB $($kilobytes$).$

a$)[10]$ How much storage required to implement the tag array for this cache? Remember "tag

array" refers to not just the tags but all other bookkeeping bits necessary to implement the

cache.

b$)[10]$ To reduce thrashing you consider making the cache $16-$way set associative while keeping

the overall size of the data portion of the cache the same. How much more $($or less$)$ space is

now needed? Show your work. Aside from the increased storage, what other resources that

would require a larger die are necessary?

c$)[10]$ As an alternative to increasing the associativity, you consider adding a $32$ entry, fully

associative victim cache using true LRU replacement $($again implemented with counters$).$ How

much more space is required by the victim cache $($including both data and tag arrays$)$ than the

original design? What other resources are necessary?