Open Problems:58

2013-04-18T05:51:41Z

Pagh: Added example of why linearity is useful.

{{Header
|source=dortmund12
|who=Rasmus Pagh
}}
Given $S\subseteq\{1,..n\}$, we would like to construct a fingerprint so that later, given fingerprints of two sets, we can check the equality of the two sets. There are (at least) two possible solutions to the problem:
# $h(S)=\left(\sum_{i\in S} x^i\right)\bmod p$ for random $x\in \Z_p$. Update time would be roughly $\log p=\Omega(\log n)$. One would like to obtain a better update time.
# $h(S)=\left(\prod_{i\in S} (x-i)\right) \bmod p$, and random $x$. Insertion can be done in constant time. But the fingerprint is not linear.

'''Question:''' Can we construct a fingerprint that achieves constant update time and is linear, while using $O(\log n)$ random bits? Ideally updates would include insertions and deletions. Linearity would imply, for example, that if $S_1 \subseteq S_2$ we can compute $h(S_2\backslash S_1)$ in constant time, as the difference of $h(S_2)$ and $h(S_1)$.

Open Problems in Sublinear Algorithms - User contributions [en]

Open Problems:58