# Expected maximum number of collisions for universal hash function

If we hash a set $$S$$ of $$n$$ keys into a table of size $$n$$ with a universal hash function $$h$$, what is the expected maximum number of keys that collide?

We break down this computation into a sequence of easier steps, as follows. Let $$A_j$$ be the event that at least one slot in the hash table has ≥ j keys. We compute the largest $$j$$ for which $$Prob[A_j] ≤ 1/2$$; that $$j$$ is our answer. Calculating $$A_j$$ directly is not straightforward, so we proceed as follows.

(a) Let $$A^1_j$$ be the event that the table slot 1 gets ≥ $$j$$ keys under $$h$$. Supposing you know $$Prob[A^1_j]$$, give an upper bound on $$Prob[A_j]$$.

(b) Let $$B$$ be the event that a fixed subset $$C ⊂ S$$ of size $$|C| = j$$ hashes into slot 1. That is, each key of $$C$$ maps to slot 1 under $$h$$. Calculate the probability $$Prob[B]$$.

(c) Use $$Prob[B]$$ to get an upper bound on the probability $$Prob[A^1_j]$$.

(d) Compute the largest value of j for which $$Prob[A^1_j] ≤ \frac{1}{2n}$$. Explain how in combination with (a), this $$j$$ is the expected maximum number of collisions.

I am not sure how to solve this problem using the steps shown. I have managed to find some resources to solve the slot-size bound for hash chaining but I am unable to follow the given steps logically.

• It should be that you are computing the largest $j$ such that $Prob [A_j] \ge \frac 12$ because the probability is a decreasing function of $j$ Jan 25 '19 at 1:42

For $$a$$ the point is that $$Prob[A_j] \le nProb[A_j^1]$$ because each slot has the same chance to have at least $$j$$ entries. It will actually be less than this because the right side counts cases where two slots each have $$j$$ entries twice while the left side counts them once. We were asked for an upper bound, so this is not a problem.
For $$b$$ the idea is to assume that each key maps to a slot randomly, so the chance a given key maps to slot $$1$$ is $$\frac 1n$$
For $$c$$ you just use $$b$$ and multiply by the number of subsets of size $$j$$