Suppose we have total $n$ lists, each of them contain $c_i$ elements, which are from 1 to $c_i$. I want to store them in the following way:
We have a large two-dimension $N*C$ array $A$. For each list $l_i$ of $n$ lists, firstly, randomly choose index $r_i$ of $N$. Then, for list $l_i$, storing its $c_i$ elements in the first $c_i$ slots of $A[r_i]$. But the bad thing is, later on, when another list $l_j$ is assigned to the same slot $r_i$, collision happens and we counter the collision is $min(c_i, c_j)$.
We denote total elements of $n$ lists is $q=\sum_{i=1}^{n} c_i$ and all $c_i < C$. I want to calculate the collision probability, and want to see whether the collision is negligible to $N*C$ if $N=exp(\lambda) $, $n=poly(\lambda)$, $q=poly(\lambda)$ and $C=poly(\lambda)$.