Imagine that I've got a big $N \times N$ boolean matrix (all entries are $0$ or $1$). At the beginning, the matrix is full of zeros.
Then, I execute the following algorithm
Matrix M empty of type N x N
while (there is a 0 in M) {
Position pos;
do {
pos = random position in M
} while (pos is 1);
// Here pos is 0
mark pos as 1;
}
I want the cost of the previous algorithm (suppose that "there is a $0$ in $M$" has no cost, and that "random position in M" has constant, i.e. $\Theta(1)$, cost).
My intuition says that it is $\Theta(\cdot)$ of $$ 1 + \textrm{Geom}\left(\frac{N^2 -1}{N^2}\right) + \textrm{Geom}\left(\frac{N^2 -2}{N^2}\right) + \cdots + \textrm{Geom}\left(\frac{1}{N^2}\right) $$
