Space requirements in the FKS scheme: A Loose Bound
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The important thing to realize is that the space is measured in terms
of "cells" of size log M, the space required to store an element of M.
Claim: The FKS scheme uses at most O(S) cells.
Proof. Here is the breakdown ---
Space for the first level table N: N = S cells.
Space for the next level tables: \sum (N_i choose 2) \leq 3N = 3S.
Number of Hash functions: at most N+1 = S+1
Each takes at most 2(log M) bits, and hence at most 2 cells.
Total space = 2(S+1)
Total number of cells = 6S + 2.
QED.