Table 1 Results on \((d_1, d_2)\)-sensitive bucketing functions of length-n sequence
From: Locality-sensitive bucketing functions for the edit distance
\((d_1, d_2)\)-sensitive | B | |B| | \(|f(\varvec{s})|\) | Ref. |
|---|---|---|---|---|
(1, 2) | \(\{1,\ldots , |B|\}\) | \(n|\Sigma |^{n-1}\)⁎⁎ | n⁎⁎ | Theorem 1 |
(1, 3) | \(\mathcal {S}_n\) | \(|\Sigma |^n\) | \(|N_n^1(\varvec{s})|=(|\Sigma |-1)n+1\) | Lemma 6 |
(1, 3) | \(B_n^i\) | \(|\Sigma |^{n-1}\)⁎ | \({\left\{ \begin{array}{ll}1&{}\text {if } \varvec{s}\in B\\ n&{}\text {if } \varvec{s}\not \in B\end{array}\right. }\)⁎ | |
(3, 5) | \(B_n^i\) | \(|\Sigma |^{n-1}\) | \(\le |N_n^2(\varvec{s})|\) | Theorem 2 |
\((r, 2r+1)\), \(r>1\) | \(B_n^i\) | \(|\Sigma |^{n-1}\) | \(\le |N_n^{r}(\varvec{s})|\) | |
\((2r-1, 2r+1)\), \(r\ge 3\) odd | \(\mathcal {S}_n\) | \(|\Sigma |^n\) | \(|N_n^r(\varvec{s})|\) | Lemma 6 |
\((2r, 2r+1)\), \(r\ge 2\) even | \(\mathcal {S}_n\) | \(|\Sigma |^n\) | \(|N_n^r(\varvec{s})|\) | Lemma 6 |