Difference between revisions of "Open Problems:58"
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| − | 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. | + | 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 $h(S_1)$. | |
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| − | '''Question''' | ||
Latest revision as of 05:51, 18 April 2013
| Suggested by | Rasmus Pagh |
|---|---|
| Source | Dortmund 2012 |
| Short link | https://sublinear.info/58 |
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 $h(S_1)$.
