The 32nd Workshop on Combinatorial Mathematics and Computation Theory
An Efficient Algorithm for Computing Non-overlapping Inversion
and Transposition Distance
Toan Thang Ta, Cheng-Yao Lin and Chin Lung Lu
Department of Computer Science
National Tsing Hua University, Hsinchu 30013, Taiwan
{toanthanghy, begoingto0830}@gmail.com, cllu@cs.nthu.edu.tw
given pattern with non-overlapping reversals,
where n is the length of the text and m is the
length of the pattern. In this problem, two
equal-length strings are said to have a match with
non-overlapping reversals if one string can be
transformed into the other using any finite
sequence of non-overlapping reversals. It should
be noted that the number of the used
non-overlapping reversals in the algorithm
proposed by Cantone et al. [2] is not required to be
less than or equal to a fixed non-negative integer.
In [2], Cantone et al. also presented another
algorithm whose average-case time complexity is
ܱሺ݊ሻ. Cantone et al. [3] studied the same problem
by considering both non-overlapping reversals and
transpositions, where they called transpositions as
translocations and the lengths of two exchanged
adjacent fragments are constrained to be
equivalent. They finally designed an algorithm to
solve this problem in ܱሺ݊݉ଶ ሻ time and ܱሺ݉ଶ ሻ
space. For the above problem, Grabowski et at. [6]
gave another algorithm whose worst-case time and
space are ܱሺ݊݉ଶ ሻ and ܱሺ݉ሻ , respectively.
Moreover, they proved that their algorithm has an
ܱሺ݊ሻ average time complexity. Recently, Huang
et al. [7] studied the above approximate string
matching problem under non-overlapping
reversals by further restricting the number of the
used reversals not to exceed a given positive
integer ݇. They proposed a dynamic programming
algorithm to solve this problem in ܱሺ݊݉ଶ ሻ time
and ܱሺ݉ଶ ሻ space.
In this work, we are interested in the
computation of the mutation distance between two
strings of the same length under non-overlapping
inversions
and
transpositions
(i.e.,
non-overlapping inversion and transposition
distance), where the lengths of two adjacent and
non-overlapping fragments exchanged by a
transposition can be different. For this problem,
we devise an algorithm whose time and space
complexities are ܱሺ݊ଷ ሻ and ܱሺ݊ଶ ሻ, respectively,
where ݊ is the length of two input strings. The
rest of the paper is organized as follows. In
Section 2, we provide some notations that are
helpful when we present our algorithm later. Next,
Abstract
Given two strings of the same length ݊, the
non-overlapping inversion and transposition
distance (also called mutation distance) between
them is defined as the minimum number of
non-overlapping inversion and transposition
operations used to transform one string into the
other. In this study, we present an ܱሺ݊ଷ ሻ time and
ܱሺ݊ଶ ሻ space algorithm to compute the mutation
distance of two input strings.
1
Introduction
The dissimilarity of two strings is usually
measured by the so-called edit distance, which is
defined as the minimum number of edit operations
necessary to convert one string into the other. The
commonly used edit operations are character
insertions, deletions and substitutions. In
biological application, the aforementioned edit
operations correspond to point mutations of DNA
sequences (i.e., mutations at the level of individual
nucleotides). From evolutionary point of view,
however, DNA sequences may evolve by
large-scale mutations (also called rearrangements,
i.e., mutations at the level of sequence fragments)
[8], such as inversions (replacing a fragment of
DNA sequence by its reverse complement) and
transpositions (i.e., moving a fragment of DNA
sequence from one location to another or,
equivalently, exchanging two adjacent and
non-overlapping fragments on DNA sequence).
Note that a large-scale mutation that replaces a
fragment of DNA sequence only by its reverse
(without complement) is called a reversal. Based
on large-scale mutation operations, the
dissimilarity (or mutation distance) between two
strings can be defined to be the minimum number
of large-scale mutation operations used to
transform one string to the other. Cantone et al. [2]
introduced an ܱሺ݊݉ሻ time and ܱሺ݉ଶ ሻ space
algorithm to solve an approximate string matching
problem with non-overlapping reversals, which is
to find all locations of a given text that match a
55
The 32nd Workshop on Combinatorial Mathematics and Computation Theory
we develop the main algorithm and also analyze
its time and space complexities in Section 3.
Finally, we have a brief conclusion in Section 4.
mutations that converts ‫ ݔ‬into ‫ݕ‬, then ݉݀ሺ‫ݔ‬ǡ ‫ݕ‬ሻ
is infinite. Formally,
2
ቄ
݉݀ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ
Preliminaries
Let ‫ ݔ‬be a string of length ݊ over a finite
alphabet 6. A character at position ݅ of x is
represented with ‫ݔ‬௜ , where ͳ ൑ ݅ ൑ ݊ . A
substring of ‫ ݔ‬from position ݅ to ݆ is indicated
as ‫ݔ‬௜ǡ௝ , i.e., ‫ݔ‬௜ǡ௝ ൌ ‫ݔ‬௜ ‫ݔ‬௜ାଵ ǥ ‫ݔ‬௝ , for ͳ ൑ ݅ ൑ ݆ ൑ ݊.
In biological sequences, ȭ ൌ ሼ‫ܣ‬ǡ ‫ܥ‬ǡ ‫ܩ‬ǡ ܶሽ , in
which ‫ ܣ‬- ܶ and ‫ ܥ‬- ‫ ܩ‬are considered as
complementary base pairs. We use ߠሺ‫ݔ‬ሻ to
denote an inversion operation acting on a string ‫ݔ‬,
resulting in a reverse and complement of ‫ݔ‬. For
example, ߠሺ‫ܣ‬ሻ ൌ ܶ , ߠሺܶሻ ൌ ‫ ܣ‬, ߠሺ‫ܩ‬ሻ ൌ ‫ ܥ‬,
ߠሺ‫ܥ‬ሻ ൌ ‫ ܩ‬and ߠሺ‫ܣܩܥ‬ሻ ൌ ܶ‫ܩܥ‬. In addition, we
utilize ߬ሺ‫ݒݑ‬ሻ ൌ ‫ ݑݒ‬to represent a transposition
operation to exchange two non-empty strings ‫ݑ‬
and ‫ ݒ‬. Note that the lengths of ‫ ݑ‬and ‫ ݒ‬are
required to be identical in some previous works [3,
5, 6], but they may be different in this study. For
convenience, we call ߠ and ߬ as mutation
operations. We also let ߠ௜ǡ௝ ሺ‫ݔ‬ሻ ൌ ߠሺ‫ݔ‬௜ǡ௝ ሻ for ͳ ൑
݅ ൑ ݆ ൑ ݊ and ߬௜ǡ௝ǡ௞ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬௞ǡ௝ ‫ݔ‬௜ǡ௞ିଵ for ͳ ൑ ݅ ൏
݇ ൑ ݆ ൑ ݊, where ሾ݅ǡ ݆ሿ is called a mutation range
for ߠ௜ǡ௝ and ߬௜ǡ௝ǡ௞ and ݇ is called as a cut point
in ߬௜ǡ௝ǡ௞ .
For an integer ͳ ൑ ‫ ݐ‬൑ ݊ , we say that a
mutation operation ߠ௜ǡ௝ or ߬௜ǡ௝ǡ௞ covers ‫ ݐ‬if ݅ ൑
‫ ݐ‬൑ ݆. Given two mutation operations, they are
non-overlapping if the intersection of their
mutation ranges are empty. In this study, we are
only interested in non-overlapping mutation
operations. Consider a set 4 of non-overlapping
mutation operations and a string ‫ݔ‬, let 4ሺ‫ݔ‬ሻ be
the resulting string after consecutively applying
mutation operations in 4 on ‫ ݔ‬. For example,
suppose that 4 ൌ ൛߬ଵǡଷǡଶ ǡ ߠହǡହ ൟ and ‫ ݔ‬ൌ ܶ‫ܥܣܩܣ‬.
Then we have 4ሺ‫ݔ‬ሻ ൌ ‫ܩܣܶܩܣ‬.
‹ሼȁ4ȁ ‫ ׷‬4ሺ‫ݔ‬ሻ ൌ ‫ݕ‬ሽ
f
‹ˆ4•—…Š–Šƒ–4ሺ‫ݔ‬ሻ ൌ ‫ݕ‬
‘–Š‡”™‹•‡
For example, consider ‫ ݔ‬ൌ ܶ‫ ܥܣܩܣ‬and ‫ ݕ‬ൌ
ܶ‫ ܩܥܣܣ‬. Then there are only two sets of
non-overlapping mutation operations 4ଵ ൌ
൛ߠଵǡଶ ǡ ߬ଷǡହǡସ ൟ and 4ଶ ൌ ൛߬ଷǡହǡସ ൟ such that 4ଵ ሺ‫ݔ‬ሻ ൌ
‫ ݕ‬and 4ଶ ሺ‫ݔ‬ሻ ൌ ‫ݕ‬. Thus, ݉݀ሺ‫ݔ‬ǡ ‫ݕ‬ሻ = ȁ4ଶ ȁ ൌ ͳ.
3
The algorithm
Basically, a transposition (respectively,
inversion) operation acting on a string ‫ ݔ‬can be
considered as a permutation of characters in ‫ݔ‬
(respectively, complement of ‫)ݔ‬. From this view
point, a mutation operation actually comprises
several sub-operations, called as mutation
fragments, each of which is denoted either by a
tuple ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ or ሺ݅ǡ ݆ǡ ߠሺ‫ݔ‬௝ ሻሻ . The mutation
fragment ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ (respectively, ሺ݅ǡ ݆ǡ ߠሺ‫ݔ‬௝ ሻሻ )
means that ‫ݔ‬௝ (respectively, complement of ‫ݔ‬௝ ) is
moved into the position ݅ in the resulting string
obtained when applying the mutation operation on
‫ ݔ‬. For convenience, we arrange all possible
mutation fragments in a three-dimensional table
‫ܯ‬௫ ሾʹǡ ݊ǡ ݊ሿ, which is called mutation table of ‫ݔ‬,
as follows.
x ‫ܯ‬௫ ሾͳǡ ݅ǡ ݆ሿ ൌ ሺ݅ǡ ݆ǡ ߠሺ‫ݔ‬௝ ሻሻfor ݅ǡ ݆ ൌ ͳǡ ʹǡ ǥ ǡ ݊.
x ‫ܯ‬௫ ሾʹǡ ݅ǡ ݆ሿ ൌ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ for ݅ǡ ݆ ൌ ͳǡ ʹǡ ǥ ǡ ݊.
For example, let ‫ ݔ‬ൌ ܶ‫ ܥܣܩܣ‬. Then its
mutation table is shown in Figure 1.
When an inversion operation ߠ௜ǡ௝ applies on a
string ‫ݔ‬, we can decompose it into ሺ݆ െ ݅ ൅ ͳሻ
mutation fragments ࣠ሺߠ௜ǡ௝ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ for ݅ ൑ ‫ ݐ‬൑ ݆ ,
where ࣠ሺߠ௜ǡ௝ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ሺ‫ݐ‬ǡ ݅ ൅ ݆ െ ‫ݐ‬, ߠሺ‫ݔ‬௜ା௝ି௧ ሻሻ .
Similarly, a transposition operation ߬௜ǡ௝ǡ௞ can be
also decomposed into ሺ݆ െ ݅ ൅ ͳሻ mutation
fragments ࣠ሺ߬௜ǡ௝ǡ௞ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ for ݅ ൑ ‫ ݐ‬൑ ݆ , where if
݅ ൑ ‫ ݐ‬൑ ݅ ൅ ݆ െ ݇ , then ࣠ሺ߬௜ǡ௝ǡ௞ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ሺ‫ݐ‬ǡ ݇ ൅
‫ ݐ‬െ ݅ǡ ‫ݔ‬௞ା௧ି௜ ሻ; otherwise (i.e., if ݅ ൅ ݆ െ ݇ ൏ ‫ ݐ‬൑
݆ ),
then
࣠ሺ߬௜ǡ௝ǡ௞ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ሺ‫ݐ‬ǡ ‫ ݐ‬െ ݆ ൅ ݇ െ
ͳ,‫ݔ‬௧ି௝ା௞ିଵ ሻ. For the sake of succinctness, let
߬௜ǡ௝ǡ௞ ሺ‫ݔ‬ǡ ͳሻ ൌ ሼሺ‫ݐ‬ǡ ݇ ൅ ‫ ݐ‬െ ݅ǡ ‫ݔ‬௞ା௧ି௜ ሻ ‫ ݅ ׷‬൑ ‫ ݐ‬൑ ݅ ൅
݆ െ ݇ሽ
and
߬௜ǡ௝ǡ௞ ሺ‫ݔ‬ǡ ʹሻ ൌ ൛ሺ‫ݐ‬ǡ ‫ ݐ‬െ ݆ ൅ ݇ െ
ͳ,‫ݔ‬௧ି௝ା௞ିଵ ሻ ‫ ݅ ׷‬൅ ݆ െ ݇ ൏ ‫ ݐ‬൑ ݆ൟ.
Definition 1. (Non-overlapping inversion and
transposition distance) Given two strings ‫ ݔ‬and
‫ ݕ‬of the same length, the non-overlapping
inversion and transposition distance (simply called
mutation distance) between ‫ ݔ‬and ‫ݕ‬, denoted by
݉݀ሺ‫ݔ‬ǡ ‫ݕ‬ሻ, is defined as the minimum number of
non-overlapping inversion and transposition
operations used to transform ‫ ݔ‬into ‫ݕ‬. If there
does not exist any set of non-overlapping
56
The 32nd Workshop on Combinatorial Mathematics and Computation Theory
‫ܯ‬௫ ሾͳǡ ݅ǡ ݆ሿ
݅ ൌ 1
2
3
4
5
݆ ൌ 1
ሺͳǡͳǡ ‫ܣ‬ሻ
ሺʹǡͳǡ ‫ܣ‬ሻ
ሺ͵ǡͳǡ ‫ܣ‬ሻ
ሺͶǡͳǡ ‫ܣ‬ሻ
ሺͷǡͳǡ ‫ܣ‬ሻ
2
ሺͳǡʹǡ ܶሻ
ሺʹǡʹǡ ܶሻ
ሺ͵ǡʹǡ ܶሻ
ሺͶǡʹǡ ܶሻ
ሺͷǡʹǡ ܶሻ
3
ሺͳǡ͵ǡ ‫ܥ‬ሻ
ሺʹǡ͵ǡ ‫ܥ‬ሻ
ሺ͵ǡ͵ǡ ‫ܥ‬ሻ
ሺͶǡ͵ǡ ‫ܥ‬ሻ
ሺͷǡ͵ǡ ‫ܥ‬ሻ
4
ሺͳǡͶǡ ܶሻ
ሺʹǡͶǡ ܶሻ
ሺ͵ǡͶǡ ܶሻ
ሺͶǡͶǡ ܶሻ
ሺͷǡͶǡ ܶሻ
5
ሺͳǡͷǡ ‫ܩ‬ሻ
ሺʹǡͷǡ ‫ܩ‬ሻ
ሺ͵ǡͷǡ ‫ܩ‬ሻ
ሺͶǡͷǡ ‫ܩ‬ሻ
ሺͷǡͷǡ ‫ܩ‬ሻ
‫ܯ‬௫ ሾʹǡ ݅ǡ ݆ሿ
݅ ൌ 1
2
3
4
5
݆ ൌ 1
ሺͳǡͳǡ ܶሻ
ሺʹǡͳǡ ܶሻ
ሺ͵ǡͳǡ ܶሻ
ሺͶǡͳǡ ܶሻ
ሺͷǡͳǡ ܶሻ
2
ሺͳǡʹǡ ‫ܣ‬ሻ
ሺʹǡʹǡ ‫ܣ‬ሻ
ሺ͵ǡʹǡ ‫ܣ‬ሻ
ሺͶǡʹǡ ‫ܣ‬ሻ
ሺͷǡʹǡ ‫ܣ‬ሻ
3
ሺͳǡ͵ǡ ‫ܩ‬ሻ
ሺʹǡ͵ǡ ‫ܩ‬ሻ
ሺ͵ǡ͵ǡ ‫ܩ‬ሻ
ሺͶǡ͵ǡ ‫ܩ‬ሻ
ሺͷǡ͵ǡ ‫ܩ‬ሻ
4
ሺͳǡͶǡ ‫ܣ‬ሻ
ሺʹǡͶǡ ‫ܣ‬ሻ
ሺ͵ǡͶǡ ‫ܣ‬ሻ
ሺͶǡͶǡ ‫ܣ‬ሻ
ሺͷǡͶǡ ‫ܣ‬ሻ
5
ሺͳǡͷǡ ‫ܥ‬ሻ
ሺʹǡͷǡ ‫ܥ‬ሻ
ሺ͵ǡͷǡ ‫ܥ‬ሻ
ሺͶǡͷǡ ‫ܥ‬ሻ
ሺͷǡͷǡ ‫ܥ‬ሻ
Figure 1. A mutation table ‫ܯ‬௫ for a given string ‫ ݔ‬ൌ ܶ‫ܥܣܩܣ‬, where the column is indexed by ݅ and the row
by ݆. Shaded cells respectively represent the inversion ߠଵǡଷ ሺ‫ݔ‬ሻ on ‫ܯ‬௫ ሾͳǡ ݅ǡ ݆ሿ and the transposition operation
߬ଵǡହǡସ ሺ‫ݔ‬ሻ on ‫ܯ‬௫ ሾʹǡ ݅ǡ ݆ሿǤ
κሺ‫ܨ‬ሻ ൌ ߪ . If a sequence ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ۄ‬
consists of ݉ mutation fragments ( ݉ is also
considered as the length of ܵ) with κሺ‫ܨ‬௜ ሻ ൌ ߪ௜
for ͳ ൑ ݅ ൑ ݉, then we say that ܵ yields a string
ߪଵ ߪଶ ǥ ߪ௠ , which is further written as κሺSሻ ൌ
ߪଵ ߪଶ ǥ ߪ௠ . A subsequence ܲ containing first ‫ݐ‬
elements of ܵ, i.e., ܲ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௧ ‫ۄ‬, is called a
prefix of ܵ and denoted by ܲ ٌ ܵ, where ͳ ൑
‫ ݐ‬൑ ݉.
The aforementioned decomposition can be
extended to apply on a set 4 of non-overlapping
mutation operations. That is, when 4 acts on a
string ‫ݔ‬, it can be decomposed into a sequence
࣠ሺ4ǡ ‫ݔ‬ሻ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௡ ‫ ۄ‬of mutation fragments
by the following formula: For ‫ ݐ‬ൌ ͳǡ ʹǡ ǥ ǡ ݊,
‹ˆߠ௜ǡ௝ ‫ א‬4–Šƒ–…‘˜‡”•‫ݐ‬
࣠ሺߠ௜ǡ௝ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ
‫ܨ‬௧ ൌ ቐ࣠ሺ߬௜ǡ௝ǡ௞ ǡ ‫ݔ‬ǡ ‫ݐ‬ሻ ‹ˆ߬௜ǡ௝ǡ௞ ‫ א‬4–Šƒ–…‘˜‡”•‫ݐ‬
ሺ‫ݐ‬ǡ ‫ݐ‬ǡ ‫ݔ‬௧ ሻ
‘–Š‡”™‹•‡
Definition 2. (Agreed sequence) A sequence ܵ ൌ
‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ ۄ‬of ݉ mutation fragments is an
agreed sequence if there exists a set 4 of
non-overlapping mutation operations such that
ܵ ٌ ࣠ሺ4ǡ ‫ݔ‬ሻ, where ݉ ൑ ݊.
Given a mutation operation ߠ௜ǡ௝ or ߬௜ǡ௝ǡ௞ , we
call ݅ and ݆ as the left end point and right end
point of the mutation operation, respectively. We
also say that this mutation operation (i.e., ߠ௜ǡ௝ or
߬௜ǡ௝ǡ௞ ) intersects with a range ሾܽǡ ܾሿ if the
intersection of ሾ݅ǡ ݆ሿ and ሾܽǡ ܾሿ is non-empty.
The number of mutation operations in 4 that
intersects with the range ሾͳǡ ݉ሿ where ݉ ൑ ݊ is
denoted as ߩሺ4ǡ ݉ሻ . For example, if 4 ൌ
൛ߠଵǡଶ ǡ ߬ସǡ଺ǡହ ǡ ߠ଻ǡଽ ൟ, then ߩሺ4ǡ Ͷሻ ൌ ʹ. Suppose that
a mutation fragment ‫ ܨ‬ൌ ሺ݅ǡ ݆ǡ ߪሻ is in ࣠ሺ4ǡ ‫ݔ‬ሻ
for some set 4 of non-overlapping mutation
operations. Then ‫ ܨ‬is said to be covered by a
mutation operation ߠ௟ǡ௥ or ߬௟ǡ௥ǡ௞ in 4 if ‫ ܨ‬ൌ
࣠ሺߠ௟ǡ௥ ǡ ‫ݔ‬ǡ ݅ሻ or ‫ ܨ‬ൌ ࣠ሺ߬௟ǡ௥ǡ௞ ǡ ‫ݔ‬ǡ ݅ሻ . If ‫ ܨ‬is not
For instance, given 4 ൌ ൛߬ଵǡଶǡଶ ǡ ߠସǡହ ൟ and ‫ ݔ‬ൌ
ܶ‫ ܥܣܩܣ‬, we have ࣠ሺ4ǡ ‫ݔ‬ሻ ൌ ‫ۦ‬ሺͳǡʹǡ ‫ܣ‬ሻǡ ሺʹǡͳǡ ܶሻǡġ
ሺ͵ǡ͵ǡ ‫ܩ‬ሻǡ ሺͶǡͷǡ ‫ܩ‬ሻǡ ሺͷǡͶǡ ܶሻۧį
Observation 1. Given a string x and its mutation
table ‫ܯ‬௫ , the result of an inversion ߠ௜ǡ௝ ሺ‫ݔ‬ሻ can
be obtained by concatenating ሺ݆ െ ݅ ൅ ͳሻ
mutation fragments starting at ‫ܯ‬௫ ሾͳǡ ݅ǡ ݆ሿ and
continuing to move in the anti-diagonal direction
to ‫ܯ‬௫ ሾͳǡ ݆ǡ ݅ሿ . Moreover, the result of a
is
obtained
by
transposition
߬௜ǡ௝ǡ௞ ሺ‫ݔ‬ሻ
concatenating the mutation fragments on the
following two paths, one starting at ‫ܯ‬௫ ሾʹǡ ݅ǡ ݇ሿ
and continuing to move in the diagonal direction
to ‫ܯ‬௫ ሾʹǡ ݅ ൅ ݆ െ ݇ǡ ݆ሿ and the other beginning at
‫ܯ‬௫ ሾʹǡ ݅ ൅ ݆ െ ݇ ൅ ͳǡ ݅ሿ and also continuing to
move in the diagonal direction to ‫ܯ‬௫ ሾʹǡ ݆ǡ ݇ െ ͳሿ.
For a mutation fragment ‫ ܨ‬ൌ ሺ݅ǡ ݆ǡ ߪሻ, we say
that ‫ ܨ‬yields the character ߪ , denoted by
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The 32nd Workshop on Combinatorial Mathematics and Computation Theory
covered by any mutation operation in 4, that is
‫ ܨ‬ൌ ሺ݅ǡ ݅ǡ ‫ݔ‬௜ ሻ, then we can consider that ‫ ܨ‬is still
covered by a virtual mutation operation. We use
ࣦሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ and ࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ to denote the left and
right end points of a mutation operation in 4
respectively that covers ‫ܨ‬. Formally,
length ݅ ൅ ͳ , which can be obtained from all
previously created legal sequences of length ݅. At
the end, if there exists a legal sequence of length
݊, then the algorithm returns the minimum number
of the used mutation operations among all legal
sequences of length ݊. Otherwise, the algorithm
returns infinity.
To distinguish all the legal sequences of length
݅ , we define three sets of extended mutation
fragments ‫ܫ‬௫௜ , ܸ௫௜ and ܶ௫௜ of x as follows:
x ‫ܫ‬௫௜ ൌ ቄ൬݃ǡ ݀ǡ ቀ݅ǡ ݆ǡ ߠ൫‫ݔ‬௝ ൯ቁ൰ቅ for Ͳ ൑ ݃ ൑ ݊ െ ͳǡ
Ͳ ൑ ݀ ൑ ݊ and ݆ ൌ ͳǡ ʹǡ ǥ ǡ ݊.
x ܸ௫௜ ൌ ൛൫Ͳǡ ݀ǡ ሺ݅ǡ ݅ǡ ‫ݔ‬௜ ሻ൯ൟ for Ͳ ൑ ݀ ൑ ݊.
x ܶ௫௜ ൌ ൛൫‫ݏ‬ǡ ݀ǡ1ǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯ൟ ‰ ൛൫݃ǡ ݀ǡ ʹǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯ൟ
for ͳ ൑ ‫ ݏ‬൑ ݊ െ ͳ, Ͳ ൑ ݃ ൑ ݊ െ ͳ, Ͳ ൑ ݀ ൑ ݊
and ݆ ൌ ͳǡ ʹǡ ǥ ǡ ݊.
Actually, each extended mutation fragment
contains the last mutation fragment ‫ ܨ‬of some
agreed sequence ܵ of length ݅ with extra guiding
information ݃ǡ ݀ and ‫ݏ‬. The value of ݃ is the
number of steps that are required to move towards
the anti-diagonal or diagonal direction to complete
a mutation operation and ݀ indicates the number
of the currently used mutation operations in ܵ. If
the last mutation fragment ‫ ܨ‬is covered by a
transposition operation, say ݉௫ , and ‫א ܨ‬
݉௫ ሺ‫ݔ‬ǡ ͳሻ, then ‫ ݏ‬denotes the left end point of ݉௫ .
An extended mutation fragment is further called
an ending fragment (respectively, continuing
fragment) if the value of its first component is
zero (respectively, non-zero).
Next, we represent each legal sequence by an
extended fragment as follows. Given a legal
sequence ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௜ ‫ ۄ‬that consists of ݅
mutation fragments of ‫ݔ‬, let 4 be any set of
non-overlapping mutation operations such that
ܵ ٌ ࣠ሺ4ǡ ‫ݔ‬ሻ. We further let ݉௫ be the mutation
operation in 4 that covers ‫ܨ‬௜ . Note that ݉௫ may
be a virtual mutation operation. We create an
extended fragment ݁ by the following cases (each
case is also called as a type of extended mutation
fragment):
Case 1 (type ‫)ܫ‬: ݉௫ is an inversion operation.
Then ݁ ൌ ሺ࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬௜ ሻ െ ݅ǡ ߩሺ4ǡ ݅ሻǡ ‫ܨ‬௜ ሻ.
Case 2 (type ܸ ): ݉௫ is a virtual mutation
operation. Then ݁ ൌ ሺͲǡ ߩሺ4ǡ ݅ሻǡ ‫ܨ‬௜ ሻ.
Case 3 (type ‫ܨ‬ሻǣ ݉௫ is a transposition operation
and ‫ܨ‬௜ ‫݉ א‬௫ ሺ‫ݔ‬ǡ ͳሻǤ Then ݁ ൌ ሺࣦሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬௜ ሻǡ
ߩሺ4ǡ ݅ሻǡ ͳǡ ‫ܨ‬௜ ሻ.
Case 4 (type ܵሻ: ݉௫ is a transposition operation
and ‫ܨ‬௜ ‫݉ א‬௫ ሺ‫ݔ‬ǡ ʹሻǤ Then ݁ ൌ ሺ࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬௜ ሻ െ
݅ǡ ߩሺ4ǡ ݅ሻǡ ʹǡ ‫ܨ‬௜ ሻ.
Now, we briefly describe how to produce all
possible legal sequences of length ݅ ൅ ͳ from all
previously created legal sequences of length ݅. Let
‫ܮ‬௜ be the set of all extended mutation fragments at
column ݅, i.e., each element in ‫ܮ‬௜ corresponds to
some legal sequence of length ݅. Suppose that ‫ܮ‬௜
x ࣦሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ ൌ
݈
ቐ
݅
‹ˆߠ௟ǡ௥ ‘”߬௟ǡ௥ǡ௞ ‫ א‬4•Ǥ –Ǥ ‫ ܨ‬ൌ ࣠൫ߠ௟ǡ௥ ǡ ‫ݔ‬ǡ ݅൯
‘”‫ ܨ‬ൌ ࣠൫߬௟ǡ௥ǡ௞ ǡ ‫ݔ‬ǡ ݅൯
‘–Š‡”™‹•‡
x ࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ ൌ
‫ݎ‬
ቐ
݅
‹ˆߠ௟ǡ௥ ‘”߬௟ǡ௥ǡ௞ ‫ א‬4•Ǥ –Ǥ‫ ܨ‬ൌ ࣠൫ߠ௟ǡ௥ ǡ ‫ݔ‬ǡ ݅൯
‘”‫ ܨ‬ൌ ࣠൫߬௟ǡ௥ǡ௞ ǡ ‫ݔ‬ǡ ݅൯
‘–Š‡”™‹•‡
For example, suppose that 4 ൌ ൛߬ଵǡଷǡଶ ǡ ߠସǡହ ൟ ,
‫ ݔ‬ൌ ܶ‫ ܥܣܩܣ‬and ࣠ሺ4ǡ ‫ݔ‬ሻ ൌ ‫ۦ‬ሺͳǡʹǡ ‫ܣ‬ሻǡ ሺʹǡ͵ǡ ‫ܩ‬ሻǡġ
ሺ͵ǡͳǡ ܶሻǡ ሺͶǡͷǡ ‫ܩ‬ሻǡ ሺͷǡͶǡ ܶሻۧǤ For ‫ ܨ‬ൌ ሺʹǡ͵ǡ ‫ܩ‬ሻ, we
have ࣦሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ ൌ ͳ and ࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬ሻ ൌ ͵.
For an integer ݅, let ‫ܣ‬௜௫ ൌ ൛൫݅ ൅ ͳǡ ‫ݐ‬ǡ ߠሺ‫ݔ‬௧ ሻ൯ ‫׷‬
݅ ൅ ͳ ൏ ‫ ݐ‬൑ ݊ൟ and ‫ܤ‬௫௜ ൌ ሼሺ݅ ൅ ͳǡ ‫ݐ‬ǡ ‫ݔ‬௧ ሻ ‫ ݅ ׷‬൅ ͳ ൏
‫ ݐ‬൑ ݊ሽ , which are sets of mutation fragments
covered by inversion and transposition operations
respectively acting on ‫ ݔ‬whose left end points are
all ݅ ൅ ͳ, where Ͳ ൑ ݅ ൏ ݊.
Definition 3. Let ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ ۄ‬be an
agreed sequence of ݉ mutation fragments of ‫ݔ‬,
where ݉ ൑ ݊, and 4 be a set of non-overlapping
mutation operations such that ܵ ٌ ࣠ሺ4ǡ ‫ݔ‬ሻ .
Sequence ܵ is called a complete sequence if
࣬ሺ4ǡ ‫ݔ‬ǡ ‫ܨ‬௠ ሻ ൌ ݉.
Observation 2. Suppose that ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ۄ‬
is a complete sequence, where ݉ ൏ ݊. Then ܵ ᇱ ൌ
‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ǡ ‫ܨ‬௠ାଵ ‫ ۄ‬is an agreed sequence if
௠
‰
‫ܨ‬௠ାଵ ‫ܣ א‬௠
௫ ‰ ‫ܯ‬௫ ሾͳǡ ݉ ൅ ͳǡ ݉ ൅ ͳሿ ‰ ‫ܤ‬௫
‫ܯ‬௫ ሾʹǡ ݉ ൅ ͳǡ ݉ ൅ ͳሿ.
Definition 4. (Legal sequence) An agreed
sequence ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ ۄ‬of ݉ mutation
fragments of ‫ ݔ‬is legal if κሺSሻ ൌ ‫ݕ‬ଵǡ௠ , where
݉ ൑ ݊.
Observation 3. Assume that ܵ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ ‫ۄ‬
is a legal sequence. Then ܵԢ ൌ ‫ܨۃ‬ଵ ǡ ‫ܨ‬ଶ ǡ ǥ ǡ ‫ܨ‬௠ିଵ ‫ۄ‬
is also a legal sequence and κሺ‫ܨ‬௠ ሻ ൌ ‫ݕ‬௠ , where
ͳ ൏ ݉ ൑ ݊.
The main idea of our algorithm is as follows.
First, the algorithm constructs all legal sequences
of length ͳ by examining all mutation fragments
in ‫ܣ‬଴௫ ‰ ‫ܯ‬௫ ሾͳǡͳǡͳሿ ‰ ‫ܤ‬௫଴ ‰ ‫ܯ‬௫ ሾʹǡͳǡͳሿ. A legal
sequence is then created if the mutation fragment
yields ‫ݕ‬ଵ . Next, for each ͳ ൑ ݅ ൏ ݊ , the
algorithm produces all possible legal sequences of
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The 32nd Workshop on Combinatorial Mathematics and Computation Theory
is already known. Below, we construct ‫ܮ‬௜ାଵ by
examining all extended mutation fragments in ‫ܮ‬௜ .
For each extended mutation fragment ݁ ‫ܮ א‬௜ , we
identify all candidate mutation fragments at
column ݅ ൅ ͳ of the mutation tables ‫ܯ‬௫ based
on the guiding information contained in ݁. Let
ܰ௫௜ ሺ݁ሻ be set of the candidate mutation fragments
of ‫ ݔ‬at column ݅ ൅ ͳ. Then we compute ܰ௫௜ ሺ݁ሻ
as follows:
Case 1: ݁ is a continuing fragment. Let ‫ܨ‬௜ ൌ
ሺ݅ǡ ݆ǡ ߪሻ be the mutation fragment contained in ݁.
Then we consider three sub-cases to compute
ܰ௫௜ ሺ݁ሻ: (1) Suppose that ݁ is of type ‫ܫ‬. Then
ܰ௫௜ ሺ݁ሻ ൌ ൛ሺ݅ ൅ ͳǡ ݆ െ ͳǡ ߠሺ‫ݔ‬௝ିଵ ሻሻൟǤ (2) Suppose
that ݁ is of type ‫ܨ‬, i.e., ݁ ൌ ሺ‫ݏ‬ǡ ݀ǡ ͳǡ ‫ܨ‬௜ ሻ. If ݆ ൏
݊ ,
then
ܰ௫௜ ሺ݁ሻ ൌ ൛ሺ݅ ൅ ͳǡ ݆ ൅ ͳǡ ‫ݔ‬௝ାଵ ሻǡ ሺ݅ ൅
ͳǡ ‫ݏ‬ǡ ‫ݔ‬௦ ሻൟ ; otherwise (i.e., ݆ ൌ ݊ ), ܰ௫௜ ሺ݁ሻ ൌ
ሼሺ݅ ൅ ͳǡ ‫ݏ‬ǡ ‫ݔ‬௦ ሻሽ. (3) Suppose that ݁ is of type ܵ,
i.e., ݁ ൌ ሺ݃ǡ ݀ǡ ʹǡ ‫ܨ‬௜ ሻ. Then ܰ௫௜ ሺ݁ሻ ൌ ൛ሺ݅ ൅ ͳǡ ݆ ൅
ͳǡ ‫ݔ‬௝ାଵ ሻൟ.
Case 2: ݁ is an ending fragment. We set
ܰ௫௜ ሺ݁ሻ ൌ ‫ܣ‬௜௫ ‰ ‫ܯ‬௫ ሾͳǡ ݅ ൅ ͳǡ ݅ ൅ ͳሿ ‰ ‫ܤ‬௫௜ ‰
‫ܯ‬௫ ሾʹǡ ݅ ൅ ͳǡ ݅ ൅ ͳሿ.
For any ‫ܨ‬௜ାଵ ‫ܰ א‬௫௜ ሺ݁ሻ with κሺ‫ܨ‬௜ାଵ ሻ ൌ ‫ݕ‬௜ାଵ , an
extended mutation fragment ݁Ԣ is created to
contain ‫ܨ‬௜ାଵ . As to the values of the guiding
information in ݁Ԣ, they can easily be obtained
from those in ݁ based on Observations 1 and 2.
Finally, we add ݁Ԣ into ‫ܮ‬௜ାଵ .
for convenience. Based on Lemma 1, we have the
following corollary.
Corollary 1. Let ݁ଵ and ݁ଶ be two ending
fragments in ‫ܮ‬௜ . If ݀ሺ݁ଵ ሻ ൏ ݀ሺ݁ଶ ሻ, then we can
safely discard ݁ଶ from ‫ܮ‬௜ when computing the
mutation distance between ‫ ݔ‬and ‫ݕ‬.
Lemma 2. Let ݁ଵ and ݁ଶ be two continuing
fragments ݁ଵ and ݁ଶ of the same type ܵ in ‫ܮ‬௜
and
݁ଶ ൌ
with
݁ଵ ൌ ൫݃ǡ ݀ଵ ǡ ʹǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯
൫݃ǡ ݀ଶ ǡ ʹǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯. If ݀ሺ݁ଵ ሻ ൏ ݀ሺ݁ଶ ሻ (i.e., ݀ଵ ൏
݀ଶ ), then we can safely remove ݁ଶ from ‫ܮ‬௜
without affecting the computation of the mutation
distance between ‫ ݔ‬and ‫ݕ‬.
Proof. Since two continuing fragments ݁ଵ and ݁ଶ
of type ܵ contain the same mutation fragment
and guiding information ݃, the right end point of
the mutation operation covering ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ in ݁ଵ is
equal to that of the mutation operation covering
ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ in ݁ଶ . Denote by ‫ ݐ‬the above right end
point. Therefore, the candidate mutation fragments
from column ݅ to column ‫ ݐ‬on the mutation
table ‫ܯ‬௫ are exactly equivalent. Then we can
safely discard ݁ଶ from ‫ܮ‬௜ if ݀ଵ ൏ ݀ଶ according
to Corollary 1. Based on Corollary 1, we can also verify that
there does not exist two continuing fragments of
same type ‫ ܫ‬or ‫ ܨ‬in ‫ܮ‬௜ such that the values of
all their components, excluding the second
components (i.e., the number of used mutation
operations), are equal. Now, the details of our
algorithm for computing non-overlapping
inversion and transposition distance between two
strings is described in Algorithm 1.
Lemma 1. Let ܲଵ and ܲଶ be two complete
sequences of length ݉ of ‫ ݔ‬with κሺܲଵ ሻ ൌ κሺܲଶ ሻ.
Moreover, let ܵଵ be an agreed sequence of ‫ݔ‬
with ܲଵ ٌ ܵଵ and ܵଶ be the resulting sequence
obtained by replacing ܲଵ with ܲଶ in ܵଵ . Then,
ܵଶ is also an agreed sequence of ‫ ݔ‬and κሺܵଵ ሻ ൌ
κሺܵଶ ሻ.
Proof. Since ܵଵ and ܲଶ are agreed sequences of
‫ݔ‬, there exist sets of non-overlapping mutation
operations 4ଵ and 4ଶ such that ܵଵ ٌ ࣠ሺ4ଵ ǡ ‫ݔ‬ሻ
and ܲଶ ٌ ࣠ሺ4ଶ ǡ ‫ݔ‬ሻ . Moreover, ܲଵ ٌ ࣠ሺ4ଵ ǡ ‫ݔ‬ሻ
since ܲଵ ٌ ܵଵ . We construct a set 4ଷ of
non-overlapping mutation operations from 4ଵ as
follows. First, let 4ଷ be the resulting set obtained
by removing all mutation operations in 4ଵ that
intersect with the range ሾͳǡ ݉ሿ . Note that the
ranges of these removed mutation operations are
completely contained in the range ሾͳǡ ݉ሿ because
ܲଵ is a complete sequence of length ݉. Next, we
insert those mutation operations in 4ଶ whose
ranges entirely lie in the range ሾͳǡ ݉ሿ into 4ଷ . It
can be verified that 4ଷ is a set of
non-overlapping mutation operations and as a
result, ܵଶ ٌ ࣠ሺ4ଷ ǡ ‫ݔ‬ሻ and κሺܵଵ ሻ ൌ κሺܵଶ ሻ. Given an extended mutation fragment݁, we use
݀ሺ݁ሻ to indicate the value of its second component
(i.e., the number of the used mutation operations)
Algorithm
1:
Computing
non-overlapping
inversion
and
transposition distance between two
strings
Input: Two strings ‫ ݔ‬and ‫ ݕ‬of the
same length ݊.
Output: Mutation distance ݉݀ሺ‫ݔ‬ǡ ‫ݕ‬ሻ.
1) Let ݅ ൌ Ͳ, ݀ ൌ Ͳ and ‫ܮ‬ଵ ൌ ‡;
2) Construct ‫ܮ‬ଵ by calling
EMF_Creationሺ݅ǡ ݀ሻ;
3) if ‫ܮ‬ଵ ൌ ‡ then return f;
4) for ݅ ൌ ͳ to ݊ െ ͳ do
5)
‫ܮ‬௜ାଵ ൌ EMF_Extensionሺ‫ܮ‬௜ ሻ;
6)
if ‫ܮ‬௜ାଵ ൌ ‡ then return f;
7) end for
8) return ‹ሼ݀ሺ݁ሻ ‫ܮ א ݁ ׷‬௡ ሽ;
Procedure 1: EMF_Creationሺ݅ǡ ݀ሻ
Input: Column ݅ and current number
of mutation operations ݀.
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The 32nd Workshop on Combinatorial Mathematics and Computation Theory
19)
case 3: ݁ is of type ‫ܫ‬, let
݁ ൌ ቀ݃ǡ ݀ǡ ൫݅ǡ ݆ǡ ߠሺ‫ݔ‬௝ ሻ൯ቁ.
20)
if ݃ ൐ Ͳ then
21)
if ߠ൫‫ݔ‬௝ିଵ ൯ ൌ ‫ݕ‬௜ାଵ then
22)
Add ቀ݃ െ ͳǡ ݀ǡ ൫݅ ൅ ͳǡ ݆ െ
ͳǡ ߠሺ‫ݔ‬௝ିଵ ሻ൯ቁ of type ‫ ܫ‬to ‫ܮ‬௜ାଵ ;
23)
end if
24)
else
//݃ ൌ Ͳ
25)
Add ݁ to ‫ܧ‬௜ based on
Corollary 1;
26)
end if
27)
case 4: ݁ is of type ܸ.
28)
Add ݁ to ‫ܧ‬௜ based on
Corollary 1;
29) end for
30) if ‫ܧ‬௜ ് ‡ then
31)
݀ ൌ ‹ሼ݀ሺ݁ሻ ‫ܧ א ݁ ׷‬௜ ሽ;
32)
EMF_Creationሺ݅ǡ ݀ሻ;
33) end if
34) return ‫ܮ‬௜ାଵ ;
Output: Extended mutation fragments
to be added to ‫ܮ‬௜ାଵ when there is an
ending fragment at column ݅.
1) for each ൫݅ ൅ ͳǡ ‫ݐ‬ǡ ߠሺ‫ݔ‬௧ ሻ൯ ‫ܣ א‬௜௫ do
2)
if ߠሺ‫ݔ‬௧ ሻ ൌ ‫ݕ‬௜ାଵ then
3)
Add ൫‫ ݐ‬െ ݅ െ ͳǡ ݀ ൅ ͳǡ ሺ݅ ൅
ͳǡ ‫ݐ‬ǡ ߠሺ‫ݔ‬௧ ሻሻ൯ of type ‫ ܫ‬to ‫ܮ‬௜ାଵ ;
4)
end if
5) end for
6) if ߠሺ‫ݔ‬௜ାଵ ሻ ൌ ‫ݕ‬௜ାଵ then
7)
Add ending fragment ൫Ͳǡ ݀ ൅
ͳǡ ሺ݅ ൅ ͳǡ ݅ ൅ ͳǡ ߠሺ‫ݔ‬௜ାଵ ሻሻ൯ of type ‫ ܫ‬to ‫ܮ‬௜ାଵ
based on Corollary 1;
8) end if
9) for each ሺ݅ ൅ ͳǡ ‫ݐ‬ǡ ‫ݔ‬௧ ሻ ‫ܤ א‬௫௜ do
10)
if ‫ݔ‬௧ ൌ ‫ݕ‬௜ାଵ then
11)
Add ൫݅ ൅ ͳǡ ݀ ൅ ͳǡͳǡ ሺ݅ ൅ ͳǡ ‫ݐ‬ǡ ‫ݔ‬௧ ሻ൯
of type ‫ ܨ‬to ‫ܮ‬௜ାଵ ;
12)
end if
13) end for
14) if ‫ݔ‬௜ାଵ ൌ ‫ݕ‬௜ାଵ then
15)
Add ending fragment ൫Ͳǡ ݀ǡ ሺ݅ ൅
ͳǡ ݅ ൅ ͳǡ ‫ݔ‬௜ାଵ ሻ൯ of type ܸ to ‫ܮ‬௜ାଵ based on
Corollary 1;
16) end if
Theorem 1. Algorithm 1 computes mutation
distance of two input strings in Oሺ݊ଷ ሻ time and
Oሺ݊ଶ ሻ space.
Proof. The correctness of Algorithm 1 can be
verified according to the discussion we mentioned
before. Below, we analyze the complexities of
Algorithm 1. For convenience, we use ȁ‫ܪ‬ȁ to
denote the number of extended mutation
fragments of type ‫ ܪ‬in ‫ܮ‬௜ . By Corollary 1 and
Lemma 2, the numbers of extended mutation
fragments of type ‫ܫ‬, ‫ ܨ‬and ܵ in ‫ܮ‬௜ are ܱሺ݊ଶ ሻ.
Therefore, Procedure 2 can be done in ܱሺȁ‫ܫ‬ȁ ൅
ȁܵȁ ൅ ȁ‫ܨ‬ȁሻ ൌ ܱሺ݊ଶ ሻ time. It is not hard to see that
Procedure 1 costs ܱሺ݊ሻ time in the worst case.
Moreover, Procedure 2 is called at most ܱሺ݊ሻ
times in Algorithm 1. As a result, the total time
complexity of Algorithm 1 is ܱሺ݊ଷ ሻ. In addition,
the space complexity of Algorithm 1 is ܱሺȁ‫ܫ‬ȁ ൅
ȁ‫ܨ‬ȁ ൅ ȁܵȁሻ ൌ ܱሺ݊ଶ ሻ. Procedure 2: EMF_Extensionሺ‫ܮ‬௜ ሻ
Input: ‫ܮ‬௜ .
Output:‫ܮ‬௜ାଵ .
1) Let ‫ܧ‬௜ be a set of ending
fragments at column ݅;
2) ‫ܮ‬௜ାଵ m ‡; ‫ܧ‬௜ m ‡;
3) for each extended mutation
fragment ݁ in ‫ܮ‬௜ do
4)
case 1: ݁ is of type ‫ܨ‬, let ݁ ൌ
൫‫ݏ‬ǡ ݀ǡ ͳǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯.
5)
if ݆ ൏ ݊ and ‫ݔ‬௝ାଵ ൌ ‫ݕ‬௜ାଵ then
6)
Add ൫‫ݏ‬ǡ ݀ǡ ͳǡ ሺ݅ ൅ ͳǡ ݆ ൅ ͳǡ ‫ݔ‬௝ାଵ ሻ൯
of type ‫ ܨ‬to ‫ܮ‬௜ାଵ ;
7)
end if
8)
if ‫ݔ‬௦ ൌ ‫ݕ‬௜ାଵ then
9)
Add ൫݆ െ ݅ െ ͳǡ ݀ǡ ʹǡ ሺ݅ ൅
ͳǡ ‫ݏ‬ǡ ‫ݔ‬௦ ሻ൯ of type ܵ to ‫ܮ‬௜ାଵ , based on
Lemma 2;
10)
end if
11)
case 2: ݁ is of type ܵ, let ݁ ൌ
൫݃ǡ ݀ǡ ʹǡ ሺ݅ǡ ݆ǡ ‫ݔ‬௝ ሻ൯.
12)
if ݃ ൐ Ͳ then
13)
if ‫ݔ‬௝ାଵ ൌ ‫ݕ‬௜ାଵ then
14)
Add ൫݃ െ ͳǡ ݀ǡ ʹǡ ሺ݅ ൅ ͳǡ ݆ ൅
ͳǡ ‫ݔ‬௝ାଵ ሻ൯ of type ܵ to ‫ܮ‬௜ାଵ based on
Lemma 2;
15)
end if
16)
else
// ݃ ൌ Ͳ
17)
Add ݁ to ‫ܧ‬௜ based on
Corollary 1;
18)
end if
4
Conclusions
In this work, we studied the computation of
non-overlapping inversion and transposition
distance between two strings, which can be useful
applications especially in computational biology
(e.g., evolutionary tree reconstruction of biological
sequences). As a result, we presented an Oሺ݊ଷ ሻ
time and Oሺ݊ଶ ሻ space algorithm to compute this
mutation distance. Note that the mutation
operations we considered allow both inversion and
transposition and the lengths of two substrings
exchanged by a transposition are not necessarily
identical. It is worth mentioning that the algorithm
we proposed in this study can be used to solve the
approximate string matching problem under
60
The 32nd Workshop on Combinatorial Mathematics and Computation Theory
non-overlapping inversion and/or transposition
distance.
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