Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11225 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909225281650688 |
|---|---|
| author | Koucký, Michal Saks, Michael |
| author_facet | Koucký, Michal Saks, Michael |
| contents | Edit distance is an important measure of string similarity. It counts the number of insertions, deletions and substitutions one has to make to a string $x$ to get a string $y$. In this paper we design an almost linear-size sketching scheme for computing edit distance up to a given threshold $k$. The scheme consists of two algorithms, a sketching algorithm and a recovery algorithm. The sketching algorithm depends on the parameter $k$ and takes as input a string $x$ and a public random string $ρ$ and computes a sketch $sk_ρ(x;k)$, which is a digested version of $x$. The recovery algorithm is given two sketches $sk_ρ(x;k)$ and $sk_ρ(y;k)$ as well as the public random string $ρ$ used to create the two sketches, and (with high probability) if the edit distance $ED(x,y)$ between $x$ and $y$ is at most $k$, will output $ED(x,y)$ together with an optimal sequence of edit operations that transforms $x$ to $y$, and if $ED(x,y) > k$ will output LARGE. The size of the sketch output by the sketching algorithm on input $x$ is $k{2^{O(\sqrt{\log(n)\log\log(n)})}}$ (where $n$ is an upper bound on length of $x$). The sketching and recovery algorithms both run in time polynomial in $n$. The dependence of sketch size on $k$ is information theoretically optimal and improves over the quadratic dependence on $k$ in schemes of Kociumaka, Porat and Starikovskaya (FOCS'2021), and Bhattacharya and Koucký (STOC'2023). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost Linear Size Edit Distance Sketch Koucký, Michal Saks, Michael Data Structures and Algorithms Edit distance is an important measure of string similarity. It counts the number of insertions, deletions and substitutions one has to make to a string $x$ to get a string $y$. In this paper we design an almost linear-size sketching scheme for computing edit distance up to a given threshold $k$. The scheme consists of two algorithms, a sketching algorithm and a recovery algorithm. The sketching algorithm depends on the parameter $k$ and takes as input a string $x$ and a public random string $ρ$ and computes a sketch $sk_ρ(x;k)$, which is a digested version of $x$. The recovery algorithm is given two sketches $sk_ρ(x;k)$ and $sk_ρ(y;k)$ as well as the public random string $ρ$ used to create the two sketches, and (with high probability) if the edit distance $ED(x,y)$ between $x$ and $y$ is at most $k$, will output $ED(x,y)$ together with an optimal sequence of edit operations that transforms $x$ to $y$, and if $ED(x,y) > k$ will output LARGE. The size of the sketch output by the sketching algorithm on input $x$ is $k{2^{O(\sqrt{\log(n)\log\log(n)})}}$ (where $n$ is an upper bound on length of $x$). The sketching and recovery algorithms both run in time polynomial in $n$. The dependence of sketch size on $k$ is information theoretically optimal and improves over the quadratic dependence on $k$ in schemes of Kociumaka, Porat and Starikovskaya (FOCS'2021), and Bhattacharya and Koucký (STOC'2023). |
| title | Almost Linear Size Edit Distance Sketch |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2406.11225 |