Volume 14 Supplement 2
Selected articles from ISCBAsia 2012
A highly accurate heuristic algorithm for the haplotype assembly problem
 Fei Deng^{1},
 Wenjuan Cui^{1} and
 Lusheng Wang^{1}Email author
https://doi.org/10.1186/1471216414S2S2
© Deng et al.; licensee BioMed Central Ltd. 2013
Published: 15 February 2013
Abstract
Background
Single nucleotide polymorphisms (SNPs) are the most common form of genetic variation in human DNA. The sequence of SNPs in each of the two copies of a given chromosome in a diploid organism is referred to as a haplotype. Haplotype information has many applications such as gene disease diagnoses, drug design, etc. The haplotype assembly problem is defined as follows: Given a set of fragments sequenced from the two copies of a chromosome of a single individual, and their locations in the chromosome, which can be predetermined by aligning the fragments to a reference DNA sequence, the goal here is to reconstruct two haplotypes (h_{1}, h_{2}) from the input fragments. Existing algorithms do not work well when the error rate of fragments is high. Here we design an algorithm that can give accurate solutions, even if the error rate of fragments is high.
Results
We first give a dynamic programming algorithm that can give exact solutions to the haplotype assembly problem. The time complexity of the algorithm is O(n × 2^{ t } × t), where n is the number of SNPs, and t is the maximum coverage of a SNP site. The algorithm is slow when t is large. To solve the problem when t is large, we further propose a heuristic algorithm on the basis of the dynamic programming algorithm. Experiments show that our heuristic algorithm can give very accurate solutions.
Conclusions
We have tested our algorithm on a set of benchmark datasets. Experiments show that our algorithm can give very accurate solutions. It outperforms most of the existing programs when the error rate of the input fragments is high.
Background
The recognition of genetic variations is an important topic in bioinformatics. Single nucleotide polymorphisms (SNPs) are the most common form of genetic variation in human DNA. Humans are diploid organisms. There are two copies of each chromosome (except the sex chromosomes), one from each parent. The sequence of SNPs in a given chromosome copy is referred to as a haplotype. Haplotype information is useful in many applications, such as gene disease diagnoses [1, 2], drug design, etc. Due to their essential importance in many biological analysis, haplotypes have been attracting great attention in recent years [3–7]. Since experimental methods for direct sequencing of haplotypes are both expensive and time consuming, computational methods are usually much more promising.
Currently, computational methods for computing haplotypes often fall into two categories: population haplotyping [8–11] and haplotype assembly (also known as single individual haplotyping) [12–15]. The former tries to compute haplotypes based on the genotype data from a sample of individuals in a population. Many software packages have been published in this field, e.g., PHASE [10]. An obvious drawback of population haplotyping lies in its weakness in recognizing rare and novel SNPs [16]. Contrary to population haplotyping, haplotype assembly is more efficient and has received more attention in recent years. The input to the haplotype assembly problem is a set of fragments sequenced from the two copies of a chromosome of a single individual, and their locations in the chromosome, which can be predetermined by aligning the fragments to a reference DNA sequence. The task here is to reconstruct two haplotypes from the input fragments. In this paper, we focus on the haplotype assembly problem.
The haplotype assembly problem was first introduced by Lancia et al. [17]. In [17], the authors proposed three optimization criteria for solving this problem, i.e. minimum fragment removal (MFR), minimum SNP removal (MSR) and longest haplotype reconstruction (LHR). Some polynomial time algorithms have been designed to solve some versions of such optimization problems [18, 19]. Lippert et al. [19] summarized the models in [17] and proposed some new models. Among these models, the most difficult and realistic one is minimum error correction (MEC), where we want to minimize the total number of conflicts (errors) between the fragments and the constructed haplotypes (h_{1}, h_{2}). The haplotype assembly problem with MEC is NPhard [5, 19] even for gapless fragments.
Levy et al. [20] designed a greedy heuristic algorithm that concatenates the fragments with minimum conflicts. The greedy heuristic algorithm is very fast but not very accurate when the error rate of fragments is high. Later, Bansal and Bafna [21] developed a software package HapCUT and the algorithm is based on the idea of building a graph from the sequenced fragments, in which each SNP site corresponds to a node in the graph and two nodes are connected by an edge if there exists a fragment that covers both SNP sites (which correspond to the two nodes). It then tries to minimize the MEC cost of the reconstructed haplotypes by iteratively finding maxcuts in the associated graph. Bansal et al. [22] designed a Markov Chain Monte Carlo (MCMC) algorithm, HASH. Both HASH and HapCUT have better performance than the greedy heuristic algorithm proposed in [20].
Recently, He et al. [16] gave a dynamic programming algorithm that can give the optimal solution to the haplotype assembly problem with MEC. The time complexity of the algorithm is O(m × 2^{ k } × n), where m is the number of fragments, n is the number of SNP sites, and k is the length of the longest fragments. This algorithm works well for k ≤ 15. However, it becomes impractical when k is large.
In this paper, we propose a heuristic algorithm for the haplotype assembly problem with MEC. It is worth mentioning that in HapCUT [21] and the dynamic programming algorithm proposed in [16], the authors assumed that the two constructed haplotypes are complementary with each other, i.e. there are only 2 choices at a SNP site in the reconstructed haplotypes. We drop this assumption in our heuristic algorithm. As a result, there are 4 choices at a SNP site in the reconstructed haplotypes. We have tested our algorithm on a set of benchmark datasets and compare it with several stateoftheart algorithms. Experiments show that our algorithm is highly accurate. It outperforms most of the existing programs when the error rate of input fragments is high.
Preliminaries
The input to the haplotype assembly problem is a set of fragments sequenced from the two copies of a chromosome of a single individual. Each fragment covers some SNP sites. We assume that all the fragments have been prealigned to a reference DNA sequence. As a result, we can organize the input fragments as an m × n matrix M (called fragment matrix), where m is the number of fragments and n is the number of SNP sites. Each row of M corresponds to a fragment that can be represented as a string on the alphabet ∑ = {a, c, g, t, }, where '' indicates a space when the SNP site is not covered by the fragment or the SNP value cannot be determined with enough confidence. The start (respectively, end) position of a fragment is defined as the first (respectively, last) position in the corresponding row that is not a ''. In the middle of a fragment, ''s are allowed due to data missing or pairedend fragments. Throughout the remainder of this paper, we will use the two notations, i.e. fragments and rows of M interchangeably when there is no ambiguity. Moreover, we will use columns and SNP sites interchangeably when there is no ambiguity.
We say that row i covers column j in M if M_{ i }_{,j}is not a '' or there are two integers p and q with p < j < q such that M_{ i }_{,p}≠  and M_{ i }_{,q}= . The number of rows covering column i in M is referred to as the coverage of column i.
Minimum error correction (MEC) is a commonly used model for the haplotype assembly problem. For the haplotype assembly problem with MEC, the input is a fragment matrix M, the task is to partition the fragments in M into two groups and construct two haplotypes (h_{1}, h_{2}), one from each group, such that the total number of conflicts (errors) between the fragments and the constructed haplotypes (h_{1}, h_{2}) is minimized.
Methods
In this section, we will describe the algorithms used to solve the problem. We first design a dynamic programming algorithm that gives an exact solution and runs in O(n × 2^{ t } × t) time, where n is the number of columns in M, and t is the maximum coverage of a column in M. The dynamic programming algorithm will be very slow when t is large. We then design a heuristic algorithm that first computes an initial pair of haplotypes by using the dynamic programming algorithm on only a subset of M. This initial pair of haplotypes can be viewed as an approximation to the optimal solution. To obtain a better solution, we further introduce some techniques to refine the initial solution.
A dynamic programming algorithm
Recall that the goal of the haplotype assembly problem is to partition the rows of the input fragment matrix M into two groups, each of which determining a haplotype. To obtain an optimal partition, a naive approach is to enumerate all possible partitions on the rows of M, among which we then choose the one minimizing MEC. For an instance with m rows, there are 2^{ m } total partitions, and thus the approach does not work in practice. Here we introduce a dynamic programming algorithm for the haplotype assembly problem with MEC that runs in O(n × 2^{ t } × t) time, where n is the number of columns in M, and t is the maximum coverage of a column in M.
Before we give the details of the dynamic programming algorithm, we first define some basic notations that will be used later:

R_{ i }: the set of rows covering column i in M.

P_{ j }(i): the jth partition on R_{ i }.

Q_{ j }(i): the jth partition on R_{ i }∩ R_{ i }_{+1}.

P_{ j }(i)_{ Ri }_{∩Ri+1}: the partition on R_{ i }∩ R_{ i }_{+1} obtained from P_{ j }(i) by restriction on the rows in R_{ i }∩ R_{ i }_{+1}.

QQ_{ j }(i): the set of partitions P_{ k }(i) such that P_{ k }(i)_{ Ri }_{∩Ri+1}= Q_{ j }(i).

C(P_{ j }(i)): the minimum number of corrections to be made in column i of M when the partition on R_{ i }is indicated by P_{ j }(i).

MEC(i, P_{ j }(i)): the optimal cost for the first i columns in M such that column i has a partition P_{ j }(i).
Based on P_{ j }(i + 1), we can get Q_{ k }(i) in O(t) time. Furthermore, we know the majority value (0 or 1) at column (i + 1) in each group. To compute C(P_{ j }(i + 1)), we can simply count the number of minorities in each group (at column (i + 1)) separately, and then add them up. Thus, it takes O(t) time to compute C(P_{ j }(i + 1)).
The optimal MEC cost for partitioning all the rows of M is the smallest MEC(n, P_{ j }(n)) over all possible P_{ j }(n), where n is the number of columns in M. A standard backtracking process can be used to obtain the optimal solution.
Let us look at the time complexity of the dynamic programming algorithm. To compute each MEC(i + 1, P_{ j }(i + 1)) in Equation (4), it requires O(t) time to compute C(P_{ j }(i + 1)). Thus, it takes O(n × 2^{ t } × t) time to compute all C(P_{ j }(i + 1))s for all the n columns in M. Now, let us look at the way to compute M E(i, Q_{ k }(i))s. For each partition P_{ j }(i) on R_{ i }, we can get Q_{ k }(i) = P_{ j }(i)_{ Ri }_{∩Ri+1}in O(t) time. We then update ME(i, Q_{ k }(i)) if the current value of ME(i, Q_{ k }(i)) is greater than MEC(i, P_{ j }(i)). There are at most 2^{ t } P_{ j }(i)s on R_{ i }. Thus, it takes O(t × 2^{ t }) time to compute all ME(i, Q_{ k }(i))s on R_{ i }. Since there are n columns in M, the total time required for computing all ME(i, Q_{ k }(i))s is O(n × 2^{ t } × t).
Theorem 1 Given a fragment matrix M, there is an O(n × 2^{ t } × t) time algorithm to compute an optimal solution for the haplotype assembly problem with MEC, where n is the number of columns in M, and t is the maximum coverage of a column in M.
Obtaining an initial solution via randomized sampling
The dynamic programming algorithm works well when t is relatively small. However, it will be very slow when t is large. To solve the problem when t is large, we look at each column i at a time, randomly select a fixed number of rows, say, boundOfCoverage, from the set of rows covering it and delete the characters in the remaining rows at all the columns after i  1. After that, the coverage of each column in the newly obtained submatrix is at most boundOfCoverage. We then run the dynamic programming algorithm on the submatrix. The resulting pair of haplotypes, which is referred to as the initial solution, can be viewed as an approximation to the optimal solution.
 1.
Compute the coverage c_{ i } for each column i in M.
 2.
For i = 1 to n, perform the following steps.
 3.
If c_{ i } ≤ boundOfCoverage, do nothing and goto the next column. Otherwise, goto step 4.
 4.
Randomly choose boundOfCoverage rows from the set of rows covering column i. Let $\stackrel{\u0304}{S}$ be the set of rows covering column i but are not chosen during this process.
 5.
For each row $r\in \stackrel{\u0304}{S}$, cut r from column i such that it no longer covers any column larger than i (including i). Accordingly, we need to reduce c_{ j } by 1 for each i ≤ j ≤ k, where k is the end position of r before being cut.
By employing this randomized sampling strategy, we can always make sure that the maximum coverage is bounded by the threshold boundOfCoverage in the selected submatrix. How to choose a proper value for boundOfCoverage? Actually, there is a tradeoff between the running time and the quality of the initial solution output by the dynamic programming algorithm. On one hand, reducing boundOfCoverage can reduce the running time of the algorithm. However, on the other hand, increasing boundOfCoverage can maintain more information from M. As a result, the initial solution output by the dynamic programming algorithm has a higher chance to be close to the optimal solution. In practice, boundOfCoverage is generally no larger than 15, which is feasible in terms of running time and is large enough to sample sufficient information from M. See Section Experiments for a detailed discussion on how the size of boundOfCoverage affects the initial solution.
Refining the initial solution with all fragments
In the newly obtained submatrix, it is possible that (1) some columns are not covered by any rows, thus leaving the haplotype values at these SNP sites undetermined in the initial solution, (2) the haplotype values at some SNP sites in the initial solution are wrongly determined due to the lack of sufficient information sampled from M during the randomized sampling process. In this subsection, we try to refine the initial solution with all input fragments, aiming to fill haplotype values that are left undetermined and correct those that are wrongly determined.
The refining procedure contains several iterations. In each iteration, we take two haplotypes as its input and output a new pair of haplotypes. Initially, the two haplotypes in the initial solution are used as the input to the first iteration. The haplotypes output in an iteration are then used as the input to the subsequent iteration. In each iteration, we try to reassign the rows of M into two groups based on the two input haplotypes. More specifically, for each row r of M, we first compute the generalized hamming distance between r and the two haplotypes. Then, we assign r to the group associated with the haplotype that has the smaller (generalized hamming) distance with r. After reassigning all rows of M into two groups, we can compute a haplotype from each of the two groups by majority rule. At the same time, we can also obtain the corresponding MEC cost.
The refining procedure stops when, at the end of some iteration, the obtained haplotypes no longer change, or when a certain number of iterations have been finished. The two haplotypes output in the last iteration are the output of the refining procedure.
Voting procedure
To further reduce the effect of randomness caused by the randomized sampling process, we try to obtain several different submatrices from M by repeating the randomized sampling process several times. Accordingly, we can obtain several initial solutions, one derived from each submatrix. Furthermore, we can refine these initial solutions with all fragments. Given a set of solutions, each of which containing a pair of haplotypes, the goal here is to compute a single pair of haplotypes by adopting a voting procedure.
In the voting procedure, the two haplotypes are computed separately. We next see how to compute one of the two haplotypes. The other case is similar. Let S be the set of solutions used for voting. First, we find a set of haplotypes (denoted by S_{1}), one from each solution in S, such that the haplotypes in S_{1} all correspond to the same copy of a chromosome. With S_{1}, we can then compute a haplotype by majority rule. Simply speaking, at each SNP site, we count the number of 0s and 1s at the given SNP site over the haplotypes in S_{1}. If we have more 0s, the resulting haplotype takes 0 at the SNP site, otherwise, it takes 1.
How to find S_{1}? First, we need to clarify that the two haplotypes in each solution in S are unordered. That is, given a solution H = (h_{1}, h_{2}), we do not know which chromosome copy h_{1} (or h_{2}) corresponds to. So, we should first find the correspondence between the haplotypes in different solutions. Let ${H}_{1}=\left({h}_{1}^{1},{h}_{2}^{1}\right),.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.,{H}_{y}=\left({h}_{1}^{y},{h}_{2}^{y}\right)$ be the set of solutions in S. Without loss of generality, assume that the MEC cost associated with H_{1} is the smallest among all the y solutions. We use H_{1} as our reference and try to find the correspondence between haplotypes in H_{1} and other solutions. For each i (1 < i ≤ y), we first compute two generalized hamming distances $D\left({h}_{1}^{1},{h}_{1}^{i}\right)$ and $D\left({h}_{1}^{1},{h}_{2}^{i}\right)$. If $D\left({h}_{1}^{1},{h}_{1}^{i}\right)<D\left({h}_{1}^{1},{h}_{2}^{i}\right)$, we claim that ${h}_{1}^{i}$ corresponds to the same chromosome copy as ${h}_{1}^{1}$. Otherwise, ${h}_{2}^{i}$ corresponds to the same chromosome copy as ${h}_{1}^{1}$. As a result, the set of haplotypes in S that correspond to the same chromosome copy as ${h}_{1}^{1}$ is the S_{1} we want to find.
Assume that at the beginning of this procedure, we obtain x solutions by repeating the randomized sampling process along with the refining procedure x times. It is worth mentioning that in the voting procedure, we only use part of the solutions, say, the first y (y ≤ x) solutions with the highest quality. Given two solutions A and B, we say that A has higher quality than B if the MEC cost associated with A is smaller than that of B. In this case, we assume that A is much closer to the optimal solution and contains less noises than B. To reduce the sideeffect of noises and improve the quality of the solution output by the voting procedure, it is helpful to use only solutions with high quality in the voting procedure.
Summarization of the algorithm
Generally speaking, given an input fragment matrix M, our heuristic algorithm can be summarized as the following four steps.
Step 1: We first perform a preprocessing on M to detect possible errors in it. After removing errors from M, we further convert it into ${M}^{\prime}$ in which each entry is encoded by a character from the alphabet ${\sum}^{\prime}=\left\{0,1,\right\}$. See Section Preliminaries for more details. ${M}^{\prime}$ is used as the input to the following steps.
Step 2: We compute an initial solution by running the dynamic programming algorithm on a subset of ${M}^{\prime}$. The submatrix is computed by using the randomized sampling approach.
Step 3: Refine the initial solution with all the fragments in ${M}^{\prime}$, instead of the submatrix that is used to generate the initial solution in Step 2.
Step 4: To further reduce the effect of randomness caused by the randomized sampling process, we repeat Step 2 and Step 3 several times. Each repeat ends with a solution, from which we then compute a single pair of haplotypes by adopting the voting procedure. The resulting pair of haplotypes is the output of our algorithm.
Results
We have tested our algorithm on a set of benchmark datasets and compare its performance with several other algorithms. The main purpose here is to evaluate how accurately our algorithm can reconstruct haplotypes from input fragments. All the tests have been done on a WindowsXP (32 bits) desktop PC with 3.16 GHz CPU and 4GB RAM.
The benchmark we use was created by Geraci in [23]. It was generated by using real human haplotype data from the HapMap project [4]. There are three parameters associated with the benchmark, i.e haplotype length, error rate and coverage rate, denoted by l, e, c, respectively. Each parameter has several different values, l = 100, 350, and 700, e = 0.0, 0.1, 0.2 and 0.3, c = 3, 5, 8 and 10. Note that unlike the "coverage" defined in Section Preliminaries, the coverage rate c defined in this benchmark refers to the number of times each of the two haplotypes replicates when generating the dataset. In other words, given an instance in the benchmark, i.e. a fragment matrix, there are up to 2c rows which take non '' value at each column in the matrix. For each combination of the three parameters, there are 100 instances in the benchmark. As for the details on how to generate the benchmark, the reader is referred to [23].
Intuitively speaking, the reconstruction rate measures the ability of an algorithm to reconstruct the correct haplotypes.
Evaluation of how the size of boundOfCoverage affects the initial solution.
c = 3  c = 5  c = 8  c = 10  

10  0.708(< 0.1)  0.753(< 0.1)  0.764(0.14)  0.774(0.18) 
12  0.728(< 0.1)  0.785(< 0.1)  0.794(0.15)  0.797(0.21) 
15  0.776(0.30)  0.837(0.33)  0.841(0.36)  0.857(0.45) 
From Table 1, we can see that for a fixed coverage rate c, when increasing the size of boundOfCoverage, the reconstruction rate of the obtained initial solution gets higher, and the running time increases accordingly. Since the reconstruction rate in the case where boundOfCoverage = 12 is relatively high, and the running time is feasible, we will set boundOfCoverage to be 12 in the following experiments.
Comparisons of the algorithms when l = 100.
e  c  SpeedHap  Fast Hare  2dmec  HapCUT  MLF  SHRthree  DGS  Ours 

0.0  3  0.999  0.999  0.990  1.000  0.973  0.816  1.000  1.000 
5  1.000  0.999  0.997  1.000  0.992  0.861  1.000  1.000  
8  1.000  1.000  1.000  1.000  0.997  0.912  1.000  1.000  
10  1.000  1.000  1.000  1.000  0.998  0.944  1.000  1.000  
0.1  3  0.895  0.919  0.912  0.929  0.889  0.696  0.930  0.973 
5  0.967  0.965  0.951  0.920  0.970  0.738  0.985  0.996  
8  0.989  0.993  0.983  0.901  0.985  0.758  0.989  0.999  
10  0.990  0.998  0.988  0.892  0.995  0.762  0.997  1.000  
0.2  3  0.623  0.715  0.738  0.782  0.725  0.615  0.725  0.903 
5  0.799  0.797  0.793  0.838  0.836  0.655  0.813  0.963  
8  0.852  0.881  0.873  0.864  0.918  0.681  0.878  0.990  
10  0.865  0.915  0.894  0.871  0.938  0.699  0.917  0.996  
0.3  3  0.480  0.617  0.623  0.602  0.618  0.557  0.611  0.776 
5  0.637  0.639  0.640  0.629  0.653  0.599  0.647  0.874  
8  0.667  0.661  0.675  0.673  0.697  0.632  0.663  0.950  
10  0.676  0.675  0.678  0.709  0.715  0.632  0.688  0.972 
Comparisons of the algorithms when l = 350.
e  c  SpeedHap  Fast Hare  2dmec  HapCUT  MLF  SHRthree  DGS  Ours 

0.0  3  0.999  0.990  0.965  1.000  0.864  0.830  0.999  1.000 
5  1.000  0.999  0.993  1.000  0.929  0.829  1.000  1.000  
8  1.000  1.000  0.998  1.000  0.969  0.895  1.000  1.000  
10  1.000  0.999  0.999  1.000  0.981  0.878  1.000  1.000  
0.1  3  0.819  0.871  0.837  0.930  0.752  0.682  0.926  0.970 
5  0.959  0.945  0.913  0.913  0.858  0.724  0.978  0.993  
8  0.984  0.985  0.964  0.896  0.933  0.742  0.996  0.999  
10  0.984  0.995  0.978  0.888  0.962  0.728  0.998  1.000  
0.2  3  0.439  0.684  0.675  0.771  0.642  0.591  0.691  0.877 
5  0.729  0.746  0.728  0.831  0.728  0.632  0.769  0.953  
8  0.825  0.853  0.791  0.862  0.798  0.670  0.842  0.988  
10  0.855  0.877  0.817  0.867  0.831  0.668  0.878  0.994  
0.3  3  0.251  0.590  0.593  0.565  0.581  0.548  0.578  0.725 
5  0.578  0.602  0.606  0.582  0.606  0.557  0.609  0.833  
8  0.629  0.626  0.623  0.621  0.634  0.604  0.628  0.922  
10  0.638  0.644  0.634  0.664  0.641  0.619  0.641  0.951 
Comparisons of the algorithms when l = 700.
e  c  SpeedHap  Fast Hare  2dmec  HapCUT  MLF  SHRthree  DGS  Ours 

0.0  3  0.999  0.988  0.946  1.000  0.787  0.781  0.999  0.997 
5  1.000  0.999  0.976  1.000  0.854  0.832  1.000  0.999  
8  1.000  1.000  0.992  1.000  0.919  0.868  1.000  1.000  
10  1.000  0.999  0.997  1.000  0.933  0.898  1.000  1.000  
0.1  3  0.705  0.829  0.786  0.927  0.698  0.668  0.931  0.951 
5  0.947  0.949  0.880  0.916  0.809  0.716  0.977  0.989  
8  0.985  0.986  0.948  0.896  0.863  0.743  0.987  0.997  
10  0.986  0.995  0.965  0.889  0.884  0.726  0.997  0.998  
0.2  3  0.199  0.652  0.647  0.753  0.624  0.591  0.669  0.837 
5  0.681  0.712  0.697  0.825  0.682  0.617  0.741  0.927  
8  0.801  0.808  0.751  0.856  0.747  0.653  0.818  0.974  
10  0.813  0.872  0.778  0.861  0.765  0.675  0.861  0.982  
0.3  3  0.095  0.581  0.583  0.552  0.570  0.536  0.573  0.676 
5  0.523  0.591  0.596  0.555  0.594  0.562  0.595  0.777  
8  0.616  0.615  0.613  0.597  0.614  0.611  0.614  0.876  
10  0.627  0.616  0.622  0.645  0.625  0.625  0.622  0.909 
Take a close look at the three tables, we can see that (1) each of the seven algorithms studied in [23] only works well in some cases, e.g., SpeedHap works well when the error rate is low (≤ 0.1), while MLF works well when the error rate is high (≥ 0.2); (2) the reconstruction rates of all the seven algorithms are relatively low when the error rate of input fragments is high. For example, in the case where l = 700, e = 0.3 and c = 10, the best reconstruction rate of the seven other algorithm is 0.645. Compared with its competitors, our algorithm can give solutions with high reconstruction rate. It outperforms its competitors in almost all cases, especially in cases in which the error rate of fragments is high (≥ 0.2). We also notice that when the error rate is 0, our algorithm may introduce some errors in the output solution, e.g., in the case where l = 700, c = 3, e = 0.0. However, even in this case, the reconstruction rate can still reach up to 0.997.
Discussion
In the first step of our algorithm, we perform a preprocessing on the input fragment matrix. This allows us to detect errors in the input. For example, for the benchmark datasets with l = 350 and e = 0.3, Step 1 of our algorithm can identify about 47%, 55%, 59% and 61% of the total errors for the cases c = 3, 5, 8 and 10, respectively. Thus, Step 1 has significant importance to the following steps of our algorithm.
From Figure 2(a), we can see that the two versions of our algorithm have almost the same reconstruction rate. However, when e = 0.3, the final version of our algorithm has higher reconstruction rate than the one without the voting procedure, see Figure 2(b). Thus, the voting procedure is essential for our algorithm.
Conclusion
In this paper, we propose a heuristic algorithm for the haplotype assembly problem. Experiments show that our algorithm is highly accurate. It outperforms most of the existing programs when the error rate of input fragments is high.
Declarations
The publication costs for this article were funded by the corresponding author's institution.
Declarations
Acknowledgements
The authors would like to thank Filippo Geraci for kindly providing us with the set of benchmark datasets. This work is fully supported by a grant from City University of Hong Kong (Project No. 7002728).
This article has been published as part of BMC Genomics Volume 14 Supplement 2, 2013: Selected articles from ISCBAsia 2012. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/14/S2.
Authors’ Affiliations
References
 Hoehe M, Köpke K, Wendel B, Rohde K, Flachmeier C, Kidd K, Berrettini W, Church G: Sequence variability and candidate gene analysis in complex disease: association of μ opioid receptor gene variation with substance dependence. Human Molecular Genetics. 2000, 9 (19): 28952908. 10.1093/hmg/9.19.2895.View ArticlePubMedGoogle Scholar
 Schaid D: Evaluating associations of haplotypes with traits. Genetic epidemiology. 2004, 27 (4): 348364. 10.1002/gepi.20037.View ArticlePubMedGoogle Scholar
 Bonizzoni P, Della Vedova G, Dondi R, Li J: The haplotyping problem: an overview of computational models and solutions. Journal of Computer Science and Technology. 2003, 18 (6): 675688. 10.1007/BF02945456.View ArticleGoogle Scholar
 Altshuler D, Wing T et al: A haplotype map of the human genome. Nature. 2005, 437 (7063): 12991320. 10.1038/nature04226.View ArticleGoogle Scholar
 Cilibrasi R, Van Iersel L, Kelk S, Tromp J: On the complexity of several haplotyping problems. Algorithms in Bioinformatics. 2005, 128139.View ArticleGoogle Scholar
 Schwartz R: Theory and algorithms for the haplotype assembly problem. Communications in Information and Systems. 2010, 10: 2338.View ArticleGoogle Scholar
 Zhang X, Wang R, Wu L, Chen L: Models and algorithms for haplotyping problem. Current Bioinformatics. 2006, 1: 104114.View ArticleGoogle Scholar
 Gusfield D: Haplotyping as perfect phylogeny: conceptual framework and efficient solutions. Proceedings of the Sixth Annual International Conference on Computational Biology. 2002, ACM, 166175.Google Scholar
 Wang L, Xu Y: Haplotype inference by maximum parsimony. Bioinformatics. 2003, 19 (14): 17731780. 10.1093/bioinformatics/btg239.View ArticlePubMedGoogle Scholar
 Stephens M, Smith N, Donnelly P: A new statistical method for haplotype reconstruction from population data. The American Journal of Human Genetics. 2001, 68 (4): 978989. 10.1086/319501.View ArticlePubMedGoogle Scholar
 Halperin E, Eskin E: Haplotype reconstruction from genotype data using imperfect phylogeny. Bioinformatics. 2004, 20 (12): 18421849. 10.1093/bioinformatics/bth149.View ArticlePubMedGoogle Scholar
 Duitama J, Huebsch T, McEwen G, Suk E, Hoehe M: ReFHap: A reliable and fast algorithm for single individual haplotyping. Proceedings of the First ACM International Conference on Bioinformatics and Computational Biology. 2010, 160169. ACMView ArticleGoogle Scholar
 Li L, Kim J, Waterman M: Haplotype reconstruction from SNP alignment. Journal of Computational Biology. 2004, 11 (23): 505516. 10.1089/1066527041410454.View ArticlePubMedGoogle Scholar
 Xie M, Wang J, Chen J: A model of higher accuracy for the individual haplotyping problem based on weighted SNP fragments and genotype with errors. Bioinformatics. 2008, 24 (13): i105i113. 10.1093/bioinformatics/btn147.PubMed CentralView ArticlePubMedGoogle Scholar
 Wang R, Wu L, Li Z, Zhang X: Haplotype reconstruction from SNP fragments by minimum error correction. Bioinformatics. 2005, 21 (10): 24562462. 10.1093/bioinformatics/bti352.View ArticlePubMedGoogle Scholar
 He D, Choi A, Pipatsrisawat K, Darwiche A, Eskin E: Optimal algorithms for haplotype assembly from wholegenome sequence data. Bioinformatics. 2010, 26 (12): i183i190. 10.1093/bioinformatics/btq215.PubMed CentralView ArticlePubMedGoogle Scholar
 Lancia G, Bafna V, Istrail S, Lippert R, Schwartz R: SNPs Problems, Complexity, and Algorithms. Proceedings of the 9th Annual European Symposium on Algorithms. 2001, SpringerVerlag, 182193.Google Scholar
 Rizzi R, Bafna V, Istrail S, Lancia G: Practical Algorithms and FixedParameter Tractability for the Single Individual SNP Haplotyping Problem. Workshop on Algorithms in Bioinformatics. 2002, Springer, 2943.View ArticleGoogle Scholar
 Lippert R, Schwartz R, Lancia G, Istrail S: Algorithmic strategies for the single nucleotide polymorphism haplotype assembly problem. Briefings in bioinformatics. 2002, 3: 2331. 10.1093/bib/3.1.23.View ArticlePubMedGoogle Scholar
 Levy S, Sutton G, Ng P, Feuk L, Halpern A, Walenz B, Axelrod N, Huang J, Kirkness E, Denisov G et al: The diploid genome sequence of an individual human. PLoS biology. 2007, 5 (10): e25410.1371/journal.pbio.0050254.PubMed CentralView ArticlePubMedGoogle Scholar
 Bansal V, Bafna V: HapCUT: an efficient and accurate algorithm for the haplotype assembly problem. Bioinformatics. 2008, 24 (16): i153i159. 10.1093/bioinformatics/btn298.View ArticlePubMedGoogle Scholar
 Bansal V, Halpern A, Axelrod N, Bafna V: An MCMC algorithm for haplotype assembly from wholegenome sequence data. Genome research. 2008, 18 (8): 13361346. 10.1101/gr.077065.108.PubMed CentralView ArticlePubMedGoogle Scholar
 Geraci F: A comparison of several algorithms for the single individual SNP haplotyping reconstruction problem. Bioinformatics. 2010, 26 (18): 22172225. 10.1093/bioinformatics/btq411.PubMed CentralView ArticlePubMedGoogle Scholar
 Genovese L, Geraci F, Pellegrini M: SpeedHap: an accurate heuristic for the single individual SNP haplotyping problem with many gaps, high reading error rate and low coverage. IEEE/ACM Transactions on Computational Biology and Bioinformatics. 2008, 5 (4): 492502.View ArticlePubMedGoogle Scholar
 Panconesi A, Sozio M: Fast hare: A fast heuristic for single individual SNP haplotype reconstruction. Algorithms in Bioinformatics. 2004, 266277.View ArticleGoogle Scholar
 Wang Y, Feng E, Wang R: A clustering algorithm based on two distance functions for MEC model. Computational biology and chemistry. 2007, 31 (2): 148150. 10.1016/j.compbiolchem.2007.02.001.View ArticlePubMedGoogle Scholar
 Zhao Y, Wu L, Zhang J, Wang R, Zhang X: Haplotype assembly from aligned weighted SNP fragments. Computational Biology and Chemistry. 2005, 29 (4): 281287. 10.1016/j.compbiolchem.2005.05.001.View ArticlePubMedGoogle Scholar
 Chen Z, Fu B, Schweller R, Yang B, Zhao Z, Zhu B: Linear time probabilistic algorithms for the singular haplotype reconstruction problem from SNP fragments. Journal of Computational Biology. 2008, 15 (5): 535546. 10.1089/cmb.2008.0003.View ArticlePubMedGoogle Scholar
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