Let's simply start with a simple algorithm pointed to by Wikipedia and that seems to have been widely used :

```pseudo-LRU

two-way set associative - one bit

indicates which line of the two has been reference more recently

four-way set associative - three bits

each bit represents one branch point in a binary decision tree; let 1
represent that the left side has been referenced more recently than the
right side, and 0 vice-versa

are all 4 lines valid?
/       \
yes        no, use an invalid line
|
|
|
bit_0 == 0?            state | replace      ref to | next state
/       \             ------+--------      -------+-----------
y         n             00x  |  line_0      line_0 |    11_
/           \            01x  |  line_1      line_1 |    10_
bit_1 == 0?    bit_2 == 0?      1x0  |  line_2      line_2 |    0_1
/    \          /    \        1x1  |  line_3      line_3 |    0_0
y      n        y      n
/        \      /        \        ('x' means       ('_' means unchanged)
line_0  line_1  line_2  line_3      don't care)

(see Figure 3-7, p. 3-18, in Intel Embedded Pentium Processor Family Dev.
Manual, 1998, http://www.intel.com/design/intarch/manuals/273204.htm)

note that there is a 6-bit encoding for true LRU for four-way set associative

bit 0: bank[1] more recently used than bank[0]
bit 1: bank[2] more recently used than bank[0]
bit 2: bank[2] more recently used than bank[1]
bit 3: bank[3] more recently used than bank[0]
bit 4: bank[3] more recently used than bank[1]
bit 5: bank[3] more recently used than bank[2]

this results in 24 valid bit patterns within the 64 possible bit patterns
(4! possible valid traces for bank references)

e.g., a trace of 0 1 2 3, where 0 is LRU and 3 is MRU, is encoded as 111111

you can implement a state machine with a 256x6 ROM (6-bit state encoding
appended with a 2-bit bank reference input will yield a new 6-bit state),
and you can implement an LRU bank indicator with a 64x2 ROM```

Of course the 1998 link on the Intel website has long been broken but this gives us a first approximation :

• 2-sets uses 1 bit. This can't be more simple or easy and the logic is truly minimal. Go for it everytime you can :-)
• 4-sets is more complex. There are only 3 bits if pseudo-LRU is good enough for you, but true LRU now has to be distinguished and grows as N!, so you'll need 6 bits and a 256-bits ROM.

How can one build larger systems ?

Wikipedia lists many strategies but it is desirable to get "most" of the true-LRU benefits without the size, time and costs.