 Research article
 Open Access
 Published:
DDmap: a MATLAB package for the double digest problem using multiple genetic operators
BMC Bioinformatics volume 20, Article number: 348 (2019)
Abstract
Background
In computational biology, the physical mapping of DNA is a key problem. We know that the double digest problem (DDP) is NPcomplete. Many algorithms have been proposed for solving the DDP, although it is still far from being resolved.
Results
We present DDmap, an opensource MATLAB package for solving the DDP, based on a newly designed genetic algorithm that combines six genetic operators in searching for optimal solutions. We test the performance of DDmap by using a typical DDP dataset, and we depict exact solutions to these DDP instances in an explicit manner. In addition, we propose an approximate method for solving some hard DDP scenarios via a scalingroundingadjusting process.
Conclusions
For typical DDP test instances, DDmap finds exact solutions within approximately 1 s. Based on our simulations on 1000 random DDP instances by using DDmap, we find that the maximum length of the combining fragments has observable effects towards genetic algorithms for solving the DDP problem. In addition, a Maple source code for illustrating DDP solutions as nested pie charts is also included.
Background
The physical mapping of DNA is a key problem in computational biology [5]. A large DNA molecule is a long string composed of four nucleotides, A, C, G and T. To understand the structure of DNA molecules, it is of interest to determine the occurrences of short substrings, such as GAATTC, on the DNA. Double digest experiments (DDE for short) are a standard approach for constructing physical DNA maps [2]. Given two restriction enzymes, denoted by \( \mathfrak{A} \) and \( \mathfrak{B} \), this approach cuts a target DNA sequence by using only enzyme \( \mathfrak{A} \), only enzyme \( \mathfrak{B} \), and both enzymes simultaneously, in three separate and parallel experiments [5]. As a result, we obtain three multisets of short DNA fragments. However, due to certain experimental limitations, only the length information (i.e., The number of nucleotides) of these short fragments can be measured with certain accuracy using certain mature biological techniques, such as gel electrophoresis. The objective of the double digest problem (DDP) is to reconstruct the original ordering of the fragments in the target DNA molecule.
Since the first successful reconstruction of restriction site mapping in the earlier 1970s [7, 11], the DDP problem has become an intensively studied issue that covers a variety of disciplines [6, 9]. Although the major concerns come from the community of bioinformation, the challenges related to this problem have also attracted attention from the artificial intelligence, algorithmic complexity, and optimization communities. We now know that DDP is strongly NPcomplete [1, 2], and many algorithms have been proposed for solving the DDP problem [3,4,5,6, 8,9,10, 12,13,14,15]. However, the DDP problem is still far from being resolved. All of the algorithms developed to address this problem have encountered significant difficulties as the number of restriction sites increases. Moreover, even for different DDP instances with the same size, the hardness for finding an exact solution might vary remarkably.
The main motivation of this work comes from three considerations: First, almost all existing formulations of the DDP problem use multiset as the basic data structure, while we find that it is even easier to model the DDP problem by using vectors. Second, some recently proposed genetic algorithms [3, 13] for addressing the DDP problem should be improved. Third, it is of interest to develop an opensource package for studying the DDP problem by using easily accessible engineering computation platforms, such as MATLAB.
Our main contributions are summarized as follows:

A vectorbased formulation of the DDP problem is presented and illustrated stepbystep.

A novel genetic algorithm for solving the DDP problem is proposed by combining six genetic operators, and a MATLAB package, DDmap, is implemented by integrating the proposed genetic algorithm and other necessary supporting and testing widgets. Then, by using DDmap, exact solutions for typical DDP test instances [13] are explicitly derived and depicted. (See the right column of Table 1.)

A relation between the hardness of certain DDP instances and the maximum length of double digest sequences is revealed based on our simulations of 1000 random DDP instances. Meanwhile, an approximate approach for typical hard DDP instances is conceived based on this relation.
Results
To test the utility of DDmap, eight DDP instances, referred to as INS_{j}(j = 1⋯8), are taken from [13]. They are shown in the following Table 2:
First, the integrated effects of the six aforementioned genetic operators of DDmap are verified. For the instances INS_{1, 3, 4, 5, 7, 8}, DDmap performs considerably well, and the related results are collected in Table 1. For each instance, 100 trails were run using DDmap with respect to each combination of six genetic operators. Then, the average running time, the average evolution generation and the success rate of finding exact DDP solutions are counted. Two different exact solutions for the instances INS_{1, 3, 4, 5, 7, 8} are also depicted in the right column of Table 1. In addition, the average running time and the average evolution generations of finding exact DDP solutions are depicted in Fig. 1. From Table 1 and Fig. 1. We can see that the genetic operators combination of RWS + PCC performs best in running time, RWS + ALL performs best in evolving generation, while other combinations of different genetic operators perform similarly and equally effective. Moreover, the tendency of running time curve and evolving generation curve are very similar.
However, we find that DDmap performs very poorly for INS_{2} and INS_{6}. Upon further examination, we find that INS_{6}, is invalid, Simple calculation shows that as for INS_{6}, we have
because it violates the restriction condition of (5) (See Definition 1).
For INS_{2}, we run DDmap 100 trails and successfully obtain exact solutions of INS_{2} in 67 trails. But the average running time and evolution generations for reaching the exact solution of INS_{2} are 122 s and 3828, respectively, i.e., approximately 1000 times slower than the results of other test instances (see Table 1). Furthermore, we find that these 67 solutions are essentially the same: One solution is depicted in Fig. 2(a), and another solution is just to read out the sequences A, B and C of Fig. 2(a) in an reverse order. It seems that the solution to INS_{2} ’s solutions are very sparse, and thus, DDmap faces the difficulty of escaping from so many local optima.
We deal with the INS_{2} by using the scalingroundingadjusting approach. As expected, DDmap can find solutions towards INS_{2′} very efficiently. For each combination of six genetic operators, we run DDmap towards INS_{2′} 100 trials. The average running time is no more than 2 s, the evolution generation is no more than 80, and the success rate for finding exact DDP solutions is always 100%. The results are already contained in Table 1 and Fig. 1. Now, we directly take some INS_{2}, ’s solution, (μ, ν) ∈ S_{m} × S_{n}, as an approximate solution of INS_{2}. The resulted double digest pie charts are depicted in Fig. 2(b). Compared to the exact solution given in Fig. 2(a), we think this kind of approximation is an interesting result in the sense that the relative error, defined as the proportion of total length of gaps between two missaligned fragments, is merely 4.8%, calculated by
Next, via a number of simulations, we find that DDmap’s performance is tightly related to the maximum length of a piece in the sequence of C, denoted by ρ_{C} = max c_{i}. This is reasonable considering that for a fixed length of sequence C, denoted by L_{C} = C, the smaller ρ_{C} is, the denser the solutions, and thus, the easier for genetic algorithms, such as DDmap, to meet an exact solution during the evolution process. Based on our simulations towards 1000 random DDP instances with different ρ_{C}, the relationship between the success rate of finding exact DDP solutions with respect to ρ_{C} is depicted in Fig. 3.
Discussion

♦ Cases of k ≠ m + n − 1
Note that in both INS_{4} and INS_{5}, the given two enzymes cut the target DNA molecule at some of the same sites and lead to the case where k ≠ m + n − 1. At the beginning, DDmap performs very poorly on INS_{4} and INS_{5}. The performance of DDmap on INS_{4} and INS_{5} improves remarkably after we adopt the following simple preprocessing strategy:
• If k < m + n − 1, then introduce δ = (m + n − 1) − k fragments with length 0 into.
the sequence \( \overrightarrow{c} \);
• Otherwise, if k > m + n − 1, then introduce δ = k − (m + n − 1) fragments with length 0.
into the shorter sequence among \( \overrightarrow{a} \) and \( \overrightarrow{b} \);
• Otherwise, do nothing.
An interesting observation is that the newly introduced 0length fragments will explicitly appear in the pie charts of exact DDP solutions. For instance, Fig. 4(a) shows that a 0length fragment in sequence \( \overrightarrow{c} \) of INS_{4} appears at the fifteenth site, while Fig. 4(b) shows that two 0length fragments in sequence \( \overrightarrow{c} \) of INS_{5} appear at the sixth and eighth sites, respectively.
Here, we follow the convention of reading a pie chart from 0^{°} to 180^{°} or 360^{°}.

♦ Comparison
Figure 5(a) and (b) are the comparison of the average running time between DDmap and the algorithm in 2005 [13] and 2012 [3]. Operator 1–5 are the crossover and mutation operator in DDmap. Because the crossover operator in [13] is the same as our operator 2 and the two mutation operators in [3] are similar to our operators op4 and op5, so we only implement the mutation operator op6 in [13] and crossover operator op7 in [3]. Eight instances are from the paper [13]. In the comparison experiment, each instance is run 100 times for operators op1–7 respectively, and then we got the average running time and the success rate of finding the exact DDP solution.
Through the experimental data, we found the data of op6 is much larger than that of the other six operators, the data of the other six operators will be neglected in the rectangular coordinate system, so we choose the logarithmic coordinate system. Figure 5(a) is the comparison between DDmap and the algorithm in 2005 [13], the blue line is the average running time of op6, it is higher than the other six lines, our algorithm has a significant time advantage over the [3]‘s algorithm. As can be seen from Fig. 5(b), the six lines have little difference, however, the op7’s line is always at the top, so our algorithm has a slight advantage over that of [3].
The comparison of success rate is shown in Fig. 5(c). The success rate of operators 1, 2, 3, 4, 5, 7 is 100%, they are all effective for these instances. Operator 6 runs very irregularly and the results are not very good.
Instance 2 and 6 does not appear in Fig. 5. In fact, INS_{6} is invalid. As aforementioned, INS_{2} is very complex, so we analyze it separately. To reset the maximum evolution generation as large as 100,000, running each operator 10 times towards INS_{2}, the average running time and the success rate is shown in Fig. 6(a) and (b), respectively. We can see that the running time of op6 is about 10 times longer than other operators, while the running time of op7 is about twice longer than our operators op1–5. The success rates of our five operators are all 100%, however, op7’s success rate is 90%, but op6 does not produce the exact DDP solution.
In conclusion, DDmap is much better than the algorithm in [13] and it is slightly better than [3]’s algorithm.
Conclusions
An opensource MATLAB package DDmap based on a newly designed genetic algorithm that combines six genetic operators is designed for solving the double digest problem. This algorithm finds exact solutions within approximately 1 s for typical DDP test instances. For some hard DDP instances, DDmap performs very well via a scalingroundingadjusting process. The experimental results of our algorithm confirm its efficiency.
Methods
Problem formulation
Let S_{m} denote the symmetric group on m indices {1, 2, ⋯, m}. Then, for a given permutation π ∈ S_{m} and a given vector \( \overrightarrow{a}=\left({a}_1,\cdots, {a}_m\right) \), the action of π on \( \overrightarrow{\mathrm{a}} \) derives a vector \( {\overrightarrow{a}}^{\pi }=\left({a}_{\pi (i)},\cdots, {a}_{\pi (m)}\right) \), reassembling of the order of entries of \( \overrightarrow{\mathrm{a}} \) according to π. Further, let us define the accumulative sum vector of \( \overrightarrow{\mathrm{a}} \), denoted by \( \mathrm{AS}\left(\overrightarrow{\mathrm{a}}\right) \), and the step difference vector of \( \overrightarrow{\mathrm{a}} \), denoted by \( \mathrm{AS}\left(\overrightarrow{\mathrm{a}}\right) \), as follows:
and
where \( \Sigma \left(\overrightarrow{a},\mathrm{j}\right)={\sum}_{i=1}^j{a}_i\left(j=1,\cdots, m\right) \) indicates the partial sum of \( \overrightarrow{\mathrm{a}} \).
Now, the double digest problem (DDP) can be formulated by the following steps:

Given two vectors \( \overrightarrow{a}=\left({a}_1,\cdots, {a}_m\right) \) and \( \overrightarrow{b}=\left({b}_1,\cdots, {b}_n\right) \) with the restriction \( \Sigma \left(\overrightarrow{a},\mathrm{m}\right)=\Sigma \left(\overrightarrow{b},\mathrm{n}\right) \), we define the combining sequence of \( \overrightarrow{\mathrm{a}} \) and \( \overrightarrow{\mathrm{b}} \), denoted by \( \coprod \left(\overrightarrow{a},\overrightarrow{b}\right) \), as the concatenation of vectors \( \mathrm{AS}\left(\overrightarrow{\mathrm{a}}\right) \) and \( \mathrm{AS}\left(\overrightarrow{\mathrm{b}}\right) \) and removing the tail entry. That is,

The sequence \( \coprod \left(\overrightarrow{a},\overrightarrow{b}\right) \) can be reassembled to obtain a new sequence according to the nondecreasing order, denoted by \( \hat{\bigsqcup}\left(\overrightarrow{a},\overrightarrow{b}\right) \).

The double digest sequence of \( \overrightarrow{\mathrm{a}} \) and \( \overrightarrow{\mathrm{b}} \), denoted by \( \mathrm{DDS}\left(\overrightarrow{a},\overrightarrow{b}\right) \), can be defined as the step difference vector of \( \hat{\bigsqcup}\left(\overrightarrow{a},\overrightarrow{b}\right) \). That is,

Now, we introduce the following definition:
Definition 1
A double digest problem (DDP) instance is specified by three vectors \( \overrightarrow{a}=\left({a}_1,\cdots, {a}_m\right),\overrightarrow{b}=\left({b}_1,\cdots, {b}_n\right)\ and\;\overrightarrow{c}=\left({c}_1,\cdots, {c}_k\right) \) with the restriction of
and the objective is to find a pair permutations (μ, ν) ∈ S_{m} × S_{n} such that.
Remark 1
If two enzymes cut a target DNA molecule at disjoint sites, then we have the condition k = m + n − 1. It was previously suspected that this case might lead to easier reconstruction problems [2]. (However, our simulation does support this conjecture, and details are given in the supplementary part). However, due to some unavoidable experimental errors, this condition does not always hold. Thus, in DDmap, we employ a very simple strategy to address the cases of k = m + n − 1: Introducing 0length fragments in sequence A,B, or C if necessary. Our simulation results show that this method is considerably robust.
Remark 2
If we take into consideration possible partial cleavage errors, then the optimization goal (6) should be updated to
where symbol ⊕ indicates the set exclusive operation, and the two operands \( DDS\left({\overrightarrow{a}}^{\mu },{\overrightarrow{b}}^{\nu}\right) \) and\( \overrightarrow{c} \) should be regarded as unordered multisets. By doing so, the searching space of the DDP solution is reduced to S_{m} × S_{n}, instead of S_{m} × S_{n} × S_{k}. In fact,π can be easily extracted from any valid solution (μ, ν). A simple method for obtaining π is to at first sort \( DDS\left({\overrightarrow{a}}^{\mu },{\overrightarrow{b}}^{\nu}\right) \) to obtain a nondecreasing sequence and then let π be the permutation specified by the reverse index of the sorting subscripts. Apparently, this step can be performed within the complexity Ο(klogk).
Example 1
For given three vectors \( \overrightarrow{a}=\left(1,2,3,5\right) \) , \( \overrightarrow{b}=\left(2,2,3,4\right) \) and \( \overrightarrow{c}=\left(1,1,1,2,2,2,2\right) \) as well as two permutations.
\( \mu =\left(\begin{array}{c}1\ 2\ 3\ 4\\ {}2\ 4\ 3\ 1\end{array}\right) \) and \( \nu =\left(\begin{array}{c}1\ 2\ 3\ 4\\ {}3\ 1\ 2\ 4\end{array}\right) \), we can verify that (μ, ν) is a valid solution for the DDP instance specified by \( \left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right) \). The pie charts of a solution and the corresponding calculation steps and complexities are depicted in Table 3 .
The proposed genetic operators
Recall that the basic idea of a genetic algorithm consists of the following concepts: an individual is totally specified by a chromosome; a chromosome is the carrier of a gene, and the position of a gene in a chromosome is called a locus; the gene composition of an individual is called the genotype; and the fitness value, called phenotype, is the result of mutual effects of genotype and external environments. Thus, to design a genetic algorithm for a given optimization problem, we need to specify how to represent a chromosome, evaluate the fitness value, design genetic operators, and determine evolution strategies such as the population size, the maximum evolution generation, the elitism keeping method, the probabilities for each genetic operator, etc.
First, for a given DDP instance \( \left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right) \), we directly use a random pair of permutations (μ, ν) ∈ S_{m} × S_{n} to represent a chromosome, and its fitness value is given by
Second, the following 6 genetic operators are employed in this work:

RWS. This is a natural selection operator defined as the wellknown roulette wheel algorithm.

PCC. This is a crossing operator defined as a combination of two permutations. Given two chromosomes (μ^{(1)}, ν^{(1)}) and (μ^{(2)}, v^{(2)}), this operator produces two new offspring
and
respectively.

RSC. This is a crossing operator defined as the socalled referencing sorting (RS). Given a target sequence \( \overrightarrow{a} \) and a reference sequence \( \overrightarrow{b} \), assuming both are defined over the same alphabet. Then, during the sorting process, the swapping operation of two elements in \( \overrightarrow{a} \) is performed only if they are in the reverse order in the referencing sequence \( \overrightarrow{b} \). RS is a generalization of ordinary sorting in the sense that any two elements can be compared even if they do not come from a complete order. RS is inspired by operator precedence grammars. More details about RS and RSC are given in the supplementary section. In fact, RSC is called order preserving weighted crossover in [13].

P4X. This is a fourpoint mutating operator defined as follows: Given a chromosome (μ, ν), randomly exchange two elements of μ and two elements of ν.

FLP. This is a fragment mutating operator defined as flipping of the given fragment. By flipping a fragment (2, 5, 4, 1), we obtain (1, 4, 5, 2).

CSH. This is a fragment mutating operator defined as cyclic shifting of the given fragment. By cyclically shifting a fragment (2, 5, 4, 1), we obtain (5, 4, 1, 2).
More details about the referenced sorting crossover (RSC) genetic operator.
RSC is in fact the order preserving weighted crossover given in [13]. Suppose two parent chromosomes are
and the crossover point is 3. Then, the producing of the offspring is given below:

(1)
p_{1} is split into two pieces: p_{11} = (1, 3, 2) and p_{12} = (1, 3, 4, 2, 2), and p_{2} is split into two pieces: p_{21} = (1, 2, 2) and p_{22} = (2, 4, 3, 3, 1).

(2)
The piece p_{12} is sorted by taking p_{2} as the referenced sequence. Since in p_{2} there exists a chain 2 − 2 − 4 − 3 − 1 this leads to p ' _{12} = (2, 2, 4, 3, 1).

(3)
Similarly, p_{22} is sorted by taking p_{1} as the referenced sequence. This time, we obtain p '_{ 22} = (3, 1, 3, 4, 2) since there exists a chain 3 − 1 − 3 − 4 − 2 in p_{1}.

(4)
Two offspring chromosomes are
Among the above 6 genetic operators, RWS is widely used in most genetic algorithms, and RSC was first used in [13] to solve the DDP problem. Four other genetic operators, although being easily conceived, are new to DDPoriented genetic algorithms, as far as we know.
Third, the evolution strategies in this work refer to [13]. That is, the population size and maximum evolution generation are set to 50 and 10,000, respectively. Elitists in each generation are kept, and the crossing probability is set to 0.85. The linearly adaptive mutation probability in [13] is also used in our work, but with a slight modification to ensure the cyclic increment of mutation probability is nonnegative. The details are as follows:
We follow the suggestion given in [13] by letting the mutation probability vary linearly in cycles of 200 iterations. However, in the original paper, this cycle varies from \( \frac{2}{m+n} \) to 0.45, while in our work, the cycle varies from \( \frac{2}{m+n} \) to 0.55, considering that in the case of m = n = 2, the start point would be 0.5, which is larger than 0.45.
Scalingroundingadjusting approach
Based on the above observation, we try to deal with the instance INS_{2} in another way. A new test instance, INS_{2}, is derived by using a scalingroundingadjusting process on INS_{2}. The details of this process are as follows:

Scaling and rounding. Because the minimum length of pieces in sequence \( \overrightarrow{c} \) of INS_{2} is 1120, we take 0.001 as the scaling factor. That is, we multiply the sequences \( \overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}} \) by 0.001 and then round them. By doing so, we obtain

Adjusting. Next, we find that
That is, \( \left(\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}\right) \) is an invalid DDP instance. Intuitively, this occurs because the round operation, round(·), introduces more errors. Thus, we try to adjust the rounding operation in the previous step according to the socalled roundingup and roundingdown strategies:
 Roundingup: round(x) is replaced by x ' = round(x + 0.1), and we obtain
This DDP instance is again invalid since
 Roundingdown: round(x) is replaced by x ' = round(x − 0.1)
Now, the DDP instance may be valid since
Note that the constant 0.1 in roundingup/roundingdown is a value defined by experience. A reasonable domain of this constant would be the interval [0.0001, 0.4999].
• Now, the newly derived DDP instance INS_{2}, is given by the three vectors \( \left(\overrightarrow{{\mathrm{a}}^{{\prime\prime\prime} }},\overrightarrow{{\mathrm{b}}^{{\prime\prime\prime} }},\overrightarrow{{\mathrm{c}}^{{\prime\prime\prime} }}\right) \).
Finally, we would like to mention that all simulations in this work are conducted on a X1 Carbon laptop with Windows(TM) 8, Intel(R)Core(TM)i5 − 4300U CPU @ 1.90 GHz/2.49 GHz and 8GB RAM. The complete genetic algorithm for solving the DDP problem is implemented as a MATLAB package, DDmap, and a Maple source code for drawing DDP solutions as nested pie charts is also included in this package.
The package DDmap consists of

13 MATLAB algorithms:

permGA.m, the MATLAB genetic algorithm for solving the DDP problem. This is the main algorithm, and its flowchart is depicted in Fig. 7. Note that this file also contains the definitions of five genetic operators — RWS, PCC, P4X, FLP, CSH and related MATLAB functions for calculating the fitness values.

referIndexSort.m, the MATLAB algorithm for implementing the socalled referenced sorting (based on index).

opPermCross.m, the MATLAB algorithm for implementing the RSC genetic operator.

getInstance.m, the auxiliary MATLAB algorithm for outputting test DDP instances in [SurKolay S. et al., 2005].

randDDPinstance.m, the auxiliary MATLAB algorithm for producing a valid DDP instance according the given parameters.

strABC.m, the auxiliary MATLAB algorithm for producing Maple commands for reading data before calling the Maple algorithm DDdraw.mws.

simu1004.m, simu1004plots.m, simu1007.m, and simu1008.m, the auxiliary MATLAB algorithms for organizing simulations and producing the related figures.

Trans.m, the MATLAB algorithm for implementing Scalingroundingadjusting approach for Cases of INS_{2.}

Plot1.m, the auxiliary MATLAB algorithms for comparison of DDmap and algorithm in [3, 13].

Plot2.m, the auxiliary MATLAB algorithms for comparison of DDmap and other algorithms under the condition of INS_{2.}

1 Maple algorithm, DDdraw.mws, is used for drawing the DDP solution in nested pie charts, with inputs A, B and C that are assigned by using Maple commands produced by strABC.m.

43 Data files: 42 of them are named as INS_{xx − ggg}. TEX, where xx ∈ {01, 03, 04, 05, 07, 08, 10} and ggg ∈ {pcc, rsc, p4x, flp, csh, all}, and the last is named as INS_{02 − rs − 1008}. TEX. These data files are in fact the running records of our simulations towards the 7 valid DDP test instances given in [13]. In these running records, many exact DDP solutions are provided.
Abbreviations
 CSH:

Cyclic shifting
 DDE:

Double digest experiments
 DDP:

Double digest problem
 FLP:

Flipping
 P4X:

Fourpoint exchange
 PCC:

Permutations combination crossing
 RSC:

Referencing sorting crossing
 RWS:

Roulette wheel selection
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Acknowledgements
We thank the anonymous reviewers for giving us valuable suggestions.
Funding
This work was supported by the National Key R&D Program of China (Grant No. 2016YFB0800602), the Shandong provincial Key R&D Program of China (Grant No. 2018CXGC0701), and the 111 Project (No. B08004). The experimental environments and the editing services are supported by the Shandong provincial Key R&D Program of China (Grant No. 2018CXGC0701) and the 111 Project (No. B08004), respectively. The publication cost is funded by National Key R&D Program of China (Grant No. 2016YFB0800602). The funders had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Availability of data and materials
DDmap is open source and available at https://github.com/wanglc2012/DDmap.
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LW proposed the main concept of the presented technique, implemented the main framework. JS finished the main experimental study, as well as analysis on related work. YP conceived the main idea about this work and implemented the algorithms about different genetic operators. LL verified the main method, checked the correctness of the presented technique. All authors reviewed the manuscript and approved the final manuscript.
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Wang, L., Suo, J., Pan, Y. et al. DDmap: a MATLAB package for the double digest problem using multiple genetic operators. BMC Bioinformatics 20, 348 (2019). https://doi.org/10.1186/s128590192862x
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