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Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal
Authors
Elazar Goldenberg
Tomasz Kociumaka
Robert Krauthgamer
Barna Saha
Publication date
24 November 2021
Publisher
View
on
arXiv
Abstract
We study the problem of approximating edit distance in sublinear time. This is formalized as a promise problem
(
k
,
k
c
)
(k,k^c)
(
k
,
k
c
)
-Gap Edit Distance, where the input is a pair of strings
X
,
Y
X,Y
X
,
Y
and parameters
k
,
c
>
1
k,c>1
k
,
c
>
1
, and the goal is to return YES if
E
D
(
X
,
Y
)
β€
k
ED(X,Y)\leq k
E
D
(
X
,
Y
)
β€
k
and NO if
E
D
(
X
,
Y
)
>
k
c
ED(X,Y)> k^c
E
D
(
X
,
Y
)
>
k
c
. Recent years have witnessed significant interest in designing sublinear-time algorithms for Gap Edit Distance. We resolve the non-adaptive query complexity of Gap Edit Distance, improving over several previous results. Specifically, we design a non-adaptive algorithm with query complexity
O
~
(
n
k
c
β
0.5
)
\tilde{O}(\frac{n}{k^{c-0.5}})
O
~
(
k
c
β
0.5
n
β
)
, and further prove that this bound is optimal up to polylogarithmic factors. Our algorithm also achieves optimal time complexity
O
~
(
n
k
c
β
0.5
)
\tilde{O}(\frac{n}{k^{c-0.5}})
O
~
(
k
c
β
0.5
n
β
)
whenever
c
β₯
1.5
c\geq 1.5
c
β₯
1.5
. For
1
<
c
<
1.5
1<c<1.5
1
<
c
<
1.5
, the running time of our algorithm is
O
~
(
n
k
2
c
β
1
)
\tilde{O}(\frac{n}{k^{2c-1}})
O
~
(
k
2
c
β
1
n
β
)
. For the restricted case of
k
c
=
Ξ©
(
n
)
k^c=\Omega(n)
k
c
=
Ξ©
(
n
)
, this matches a known result [Batu, Erg\"un, Kilian, Magen, Raskhodnikova, Rubinfeld, and Sami, STOC 2003], and in all other (nontrivial) cases, our running time is strictly better than all previous algorithms, including the adaptive ones
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Last time updated on 10/02/2022