Comment by gnulinux
2 years ago
> So we've seen that this works, but why would anyone think to do this in the first place? The Power of Two Random Choices: A Survey of Techniques and Results by Mitzenmacher, Richa, and Sitaraman has a great explanation. The mathematical intuition is that if we (randomly) throw n balls into n bins, the maximum number of balls in any bin is O(log n / log log n) with high probability, which is pretty much just O(log n). But if (instead of choosing randomly) we choose the least loaded of k random bins, the maximum is O(log log n / log k) with high probability, i.e., even with two random choices, it's basically O(log log n) and each additional choice only reduces the load by a constant factor.
This sounds very powerful! And intuitively doesn't make any sense to me. So, say I have n choices, I do not know which choice is better. Does this result say that it's more efficient to randomly choose k and find the best among them (even for k=2), instead of randomly choosing a single choice? Could someone point me in the right direction?
> This sounds very powerful! And intuitively doesn't make any sense to me. So, say I have n choices, I do not know which choice is better. Does this result say that it's more efficient to randomly choose k and find the best among them (even for k=2), instead of randomly choosing a single choice? Could someone point me in the right direction?
Yes, let's consider the k=2 vs selecting a random item from a set of N items:
Selecting one item uniformly randomly from a set of N is identical to selecting two distinct items and then picking one of those two uniformly randomly.
So if you select items A and B, then pick the best item, you end up with an item of MAX(A,B) utility instead of MEAN(A,B) utility. MAX(A,B) >= MEAN(A,B) should hopefully be obvious.
Random numbers sound possibly expensive for 'small' caches. Though I /offhand/ I guess if there's a hardware RNG source it doesn't need to be cryptographically secure or trusted as long as it's fast for this task.
Similarly if updates are bursty an idle cull could use a bitvector to mark free slots within a fixed array for rather low overhead and a burst of speedy inserts when load picks up. Something close is probably already necessary overhead for when the cache isn't yet populated.
You've explained that wonderfully, thank you! I'm not the original commenter, but was also confused by my intuition here. That makes a lot more sense now :)
It didnt make any sense to me.
I think an intuitive perspective is that if you pick one item randomly, you cant use the LRU signal. So, you pick two items. Now you get to also use LRU. Notice that it doesnt add any value to pick three random items and then apply LRU.
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