[rawnlq]

1) Throw n balls into n bins, the bin for each ball chosen randomly

2) Throw n balls into n bins, two bin for each ball chosen randomly, always picking the bin with fewer balls in it

In both cases you will have n balls distributed over n bins in the end. But the number of balls in the largest bin will be different for the two processes above. In the first case the largest bin has more balls: O(log n / log log n) == O(log n). And the second case has just O(log log n) balls. So just adding an extra choice of bins made the expected largest bin exponentially smaller.

More rough intuition: if x of your bins are occupied, in the first case your next ball has x/n probability of queueing instead of finding an empty bin but in the second it's only (x/n)^2 chance to need to queue.

[adrianratnapala]

So that means the expectation value of the maximum scales as O(log n / log log n)?

[matahwoosh]

Generally, yes, but I think `O(log n / log log n) == O(log n)` is wrong.

log(n) / log(log(n)) = logx(n) (where x = log(n), wasn't sure how to describe logarithm base in a better way). So you get O(logx(n)). In general the logarithm base doesn't matter for Big-O when it's a constant, but I'm not sure you can apply the same thing to a base of log(n).