random cover of a range

[Jason House]

Andrei Alexandrescu Wrote:

> bearophile wrote:
> > Andrei Alexandrescu:
> >> Say at some point there are k available (not taken) slots out of
> >> "n". There is a k/n chance that a random selection finds an
> >> unoccupied slot. The average number of random trials needed to find
> >> an unoccupied slot is proportional to 1/(k/n) = n/k. So the total
> >> number of random trials to span the entire array is quadratic.
> >> Multiplying that by 0.9 leaves it quadratic.
> > 
> > It's like in hashing: if you want to fill 99% of the available space
> > in a hash, then you take ages to find empty slots. But if you fill it
> > only at 75-90%, then on average you need only one or two tries to
> > find an empty slot. So your time is linear, with a small
> > multiplicative constant. When the slots start to get mostly full, you
> > change algorithm, copying the empty slots elsewhere.
> 
> Well I don't buy it. If you make a point, you need to be more precise 
> than such hand-waving. It's not like in hashing. It's like in the 
> algorithm we discuss. If you make a clear point that your performance is 
> better than O(n*n) by stopping at 90% then make it. I didn't go through 
> much formalism, but my napkin says you're firmly in quadratic territory.
> 
> Andrei

Retrying when 90% full gives you a geometric series for the number of tries: 1+0.1+0.1^2+0.1^3+...
Ignoring the math trick to get 1/(1-p), it's easy to see it's 1.111111... You're firmly in linear territory.