## digitalmars.D - Re: random cover of a range

- Jason House <jaaon.james.house gmail.com> Feb 13 2009

Andrei Alexandrescu Wrote:Jason House wrote:Andrei Alexandrescu Wrote:

No. Your wording sounds like you're doing stuff that's way off, but the resulting math is correct. My calculation would be based on the average length of a sequence of 1's (k/(n-k)). That means the work is 1+k/(n-k) = n/(n-k).

Well my wording means this: in an array of length n with k "holes" randomly distributed, the probability one slot is a a no-hole is (n-k)/n. What we want is to find the first no-hole starting from a random position in the array. How many steps do we do on average? That is the same as the average number of steps of rolling a fair dice with (n-k) faces until we obtain a particular face. And the average number of steps is IIRC 1/p = n/(n-k).Given that O(n*log(n)) is the theoretical best you can do, having a result that is < O(n*log(n)) is highly suspect. The sum 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ... is in fact O(log(n)).

Ever so pedantic. :o) I meant "<=" because I wasn't aware of the best theoretical bound. Do you have a pointer? Thanks.

I was thinking of the theoretical bound on sorting being n*log(n) but that does not apply in this case. The bound on sum(1/x) is pretty simple. The discrete sampling of 1/n can be made into either an over or under approximation of the integral depending on how you shift the starting points. That means sum(1/x) = ln(n)+C, where is 0 < C < 1I have the feeling there is something clever to do after half of the array was covered. After that, the probability of a random element being uncovered falls below 0.5. I also have the feeling that interesting things can be done if the length of the range has certain values, such as the period of certain generators with certain parameters. I don't have the time to look into that now. Anyone versed in e.g. linear congruential generators with a given period? One way or another, I'll add RandomCover to std.random. Thanks Leonardo, Denis, Steve, and Jason. Andrei

Feb 13 2009