8. VALORACIÓN
8.2. Métodos de valoración por flujos de caja descontados
8.2.1. Método del Valor Presente Ajustado
In this section, we consider a different dissimilarity measure in algorithm SHR-TWO to improve its ability of error tolerance. We use the sum of the differences between ri and
every fragment r0 ∈ Mi, i = 1,2, to measure the dissimilarity of the fragments in Mi with
ri. The new algorithm SHR-Three is given in the following. We will present experimental
results in Section 6.7 to show that algorithm SHR-Three is reliable and robust in dealing with possible outliers in the data sets.
Algorithm SHR-Three
Input: M, an n×m matrix of SNP fragments.
u, a parameter to control the loop. Output: two haplotypes H1 and H2.
Let dmin =∞ and M=∅.
For (k = 1 to u) do { //the k-loop
Let M1 =M2 =∅ and d1 =d2 = 0.
Randomly select two fragments r1 =M[i1], r2 =M[i2] from M
For every fragment r0 fromM do{
If (diff(ri, r0) = min{diff(r1, r0),diff(r2, r0)} for i = 1 or 2, then put r0
into Mi. } Let di =Pr0∈M idiff(ri, r 0) fori= 1,2. Let d= max{d1, d2}.
If (d < dmin) then let M={M1, M2}and dmin =d. }
return H1 =vote(M1) and H2 =vote(M2). End of Algorithm
Theorem 6.8. Assume that α1, α2, β, and > 0 are small positive constants that satisfy
2α1 + 2α2 + 2 < 1, 0 < 2α1 +α2 − < 1, η > β(1−22(αα1−1+2α)2−2) with η = min(2nn1,n2), and β(1−2α1−2α2−2) > 2(α1 +). Let G1, G2 ∈ Σm1 be the two unknown haplotypes such
that diff(G1, G2)≥β. Let M be any given n×m matrix of SNP fragments such that M has ni fragments that are Fα1,α2(m, Gi) sequences, i= 1,2, and n1+n2 =n. Then, there exists
an O(umn) time algorithm that can find two haplotypes H1 and H2 with probability at least
1−(1−γ)u−4ne− 2βm 3 −2me− 2n1 2 −2me− 2n2
2 such that H1, H2 is a permutation of G1, G2,
where γ = n1n2
n(n−1) and u is an integer parameter.
Proof: LetNi be the set of the ni many Fα1,α2(m, Gi) sequences in M, fori= 1,2.
We first notice that both Claims 3 and 4 in the proof of Theorem 6.7 hold here following the same analysis. However, we need to prove the following claim with different analysis:
Claim 6. LetH1 =vote(M1) and H2 =vote(M2) be the two haplotypes returned by
the algorithm. With probability at most (1−γ)u+ 4ne−2βm
3 , M1, M2 is not a permutation
We assume that (1) diff(S1, S2) > β(1−2α1−2α2 −2) for every S1 from N1 and
every S2 fromN2; and (2)diff(Gi, S)≤α1+ and holes(S)≤(α2+)mfor each S fromNi
(i= 1,2).
We shall consider the two cases:
Case 1. At some iteration of the k-loop, both r1 and r2 are selected from the same Ni for i = 1 or 2. For each r0 ∈ Nj, j 6= i, by assumption (1) we have both diff(ri, r0) >
β(1−2α1 −2α2 −2), i = 1,2. Notice that r0 must be either in M1 or M2. So, at least
half of the fragments in Nj will be either inM1 orM2. Therefore, at this iteration, we have d≥ 1
2njβ(1−2α1−2α2−2)≥ηnβ(1−2α1−2α2−2) =ηβ(1−2α1−2α2−2)n >2(α1+)n.
Case 2. At some iteration of the k-loop, r1 and r2 are selected from different Ni for
i= 1 and 2. Without loss of generality,ri ∈Ni,i= 1,2. For eachr0 ∈N1, by assumption (2)
we have diff(r1, r0) ≤ 2(α1 +); by assumption (1) and the given condition of the theorem
we have diff(r2, r0) > β(1− 2α1 −2α2 − 2) > 2(α1 +). Similarly, for each r0 ∈ N2,
diff(r2, r0) ≤ 2(α1 +), and diff(r1, r0) > 2(α1 +). Therefore, at this iteration, we have M1 =N1 and M2 =N2, and d≤2(α1+)n1+ 2(α1+)n2 = 2(α1+)n.
The two cases implies that under the assumption of (1) and (2), if at any iteration of the k-loop, we have r1 ∈ N1 and r2 ∈N2, or r1 ∈ N2 and r2 ∈N1, then the final list of the
two setsM1, M2 is a permutation ofN1, N2. Hence, Claim 6 follows from Claims 3 and 4 in
the proof of Theorem 6.7, which are true here as we mentioned earlier.
Now, we assume that the final list of the two sets M1, M2 is a permutation of N1, N2. By Lemma 6.5, with probability at most 2m(e−
2n1
2 ) + 2m(e−
2n2
2 ), H1 =vote(M1), H2 =vote(M2) is not a permutation of G1, G2. This, together with Claim 6, completes the probabilistic claim of the theorem.
It is easy to see that the time complexity of the algorithm isO(umn), which is linear in the size of M.
6.7
Experimental Results
We design a MATLAB program to test both the accuracy and the speed of algorithm SHR-Three. Due to the difficulty of getting real data from the public domain, our experiment is based on the literature [9,93] which is considered as common practice. A random matrix of SNP fragments is created as follows: (1) Haplotype 1 is generated at random with length m
(m∈ {50,100,150}). (2) Haplotype 2 is generated by copying all the bits from haplotype 1 and flipping each bit with probabilityβ (β ∈ {0.1,0.2,0.3}). This simulates the dissimilarity rateβ between two haplotypes. (3) Each haplotype is copied n2 times so that the matrix has
m columns andn(n ∈ {10,20,30}) rows. (4) Set each bit in the matrix to - with probability
α2 (α2 ∈ {0.1,0.2,0.3}). This simulates the incompleteness error rate α2 in the matrix. (5)
Flip each non-empty bit with probabilityα1(α1 ∈ {0.01,0.02, ...,0.1}). This is the simulation
of the inconsistency error rate of α1.
Figure 6.1 shows the performance of algorithm SHR-Three with different parameter settings in accordance with those in [9]. See Tables A.5 to A.8 in the appendix section A.4 for result details. The typical parameters used in [9] are m = 100, n = 20, β = 0.2, α2 = 0.2 and 0.01 ≤ α1 ≤ 0.05. These are considered to be close to the real situations. In Figure
95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 R eco ns tru ction Rat e (% ) α1(%)
Results for m = 100, β= 20%
and α
2= 20%
n = 10 n = 20 n = 30 0 2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 R ec o n struct ion Time (s) α1(%)Results for m = 100, β= 20%
and α
2= 20%
n = 10 n = 20 n = 30 95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 Reco nst ru ction Ra te (%) α1(%)Results for m = 100, n = 20 and
α
2= 20%
β = 10% β = 20% β = 30% 0 2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Reco nst ru ction Time (s) α1(%)Results for m = 100, n = 20 and
α
2= 20%
β = 10% β = 20% β = 30% 95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 R ec o n struct ion Rate (% ) α1(%)Results for m = 100, n = 20 and
β= 20%
α2 = 10% α2 = 20% α2 = 30% 0 2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 R eco ns tru ction Time (s) α1(%)Results for m = 100, n = 20 and
β= 20%
α2 = 10% α2 = 20% α2 = 30%
algorithm SHR-Three. A new random matrix is used for each execution. The reconstruction rate is defined as the ratio of the total number of correctly reconstructed bits to the total number of bits in two haplotypes. The computing environment is a PC machine with a typical configuration of 1.6GHz AMD Turion 64X2 CPUs and 1GB memory.
The software of our algorithms is available for public access and for real-time on-line demonstration at http://fpsa.cs.uno.edu/HapRec/HapRec.html. Thank Liqiang Wang for implementing the programs in Java and setting up this web site.
Summary We have developed three linear time random algorithms for the singular hap- lotype reconstruction problem with provable high probability in reconstruction rate. Our experimental results conform with the theoretical efficiency and accuracy of the algorithms. It should be pointed out that our work can be extended to reconstruct multiple haplotypes from a set of fragments. Our approach also opens the door to develop probabilistic methods for other variants of the haplotyping problem involving both inconsistency and incompleteness errors.
Chapter 7
Non-Breaking Similarity of Genomes with Gene
Repetitions
7.1
Introduction
Until a few years ago, in genome rearrangement research, it is always assumed that every gene appears in a genome exactly once. Under this assumption, the genome rear- rangement problem is in essence the problem of comparing and sorting signed/unsigned permutations [51, 77]. In the case of breakpoint distance, given two perfect genomes (ev- ery gene appears exactly once, i.e., there is no gene repetition) it is easy to compute their breakpoint distance in linear time.
However, perfect genomes are hard to obtain and so far they can only be obtained in several small virus genomes. For example, perfect genomes do not occur on eukaryotic genomes where paralogous genes are common [76, 84]. On the one hand, it is important in practice to compute genomic distances, e.g., Hannenhalli and Pevzner’s method [51], when no gene duplication arises; on the other hand, one might have to handle this gene duplication problem as well. In 1999, Sankoff proposed a way to select, from the duplicated copies of genes, the common ancestor gene such that the distance between the reduced genomes (exemplar genomes) is minimized [84]. A general branch-and-bound algorithm was also implemented in [84]. Recently, Nguyen, Tay and Zhang proposed to use a divide-and- conquer method to compute the exemplar breakpoint distance empirically [76].
For the theoretical part of research, it was shown that computing the signed reversals and breakpoint distances between exemplar genomes are both NP-complete [15]. Two years ago, Blin and Rizzi further proved that computing the conserved interval distance between exemplar genomes is NP-complete [14]; moreover, it is NP-complete to compute the min- imum conserved interval matching (i.e., without deleting the duplicated copies of genes). In [22, 25] it was shown that the exemplar genomic distance problem does not admit any ap- proximation (regardless of the approximation factor) unless P=NP, as long as G=H implies that d(G,H)=0. This implies that for the exemplar breakpoint distance problem, there does not exist any approximation.
In [17] three new kinds of genomic similarities were considered. These similarity measures do not satisfy the condition that G=H implies that d(G,H)=0. Among them, the common interval distance problem seems to be the most interesting one. When gene duplications are allowed, Chauve et al. proved that the problem is NP-complete.
In this chapter6 , we define a new similarity measure called non-breaking similarity.
Intuitively, this is the complement of the traditional breakpoint distance measure. Compared with the problem of computing exemplar breakpoint distance, which is a minimization prob- lem, we want to maximize the number of non-breaking points for the exemplar non-breaking similarity problem. Unfortunately, we show that Independent Set can be reduced to ENbS, and this reduction implies that ENbS is W[1]-complete (and ENbS does not have a factor-n
approximation). This reduction works even when one of the two genomes is given exemplar. While the W[1]-completeness [32] and the recent lower bound results [20] implies that if k is the optimal solution value, unless an unlikely collapse occurs in parameterized complexity theory, ENbS is not solvable in time f(k)no(k), for any function f. We show that for several practically interesting cases of the problem, there are polynomial time al- gorithms. This is done by parameterize some quantities in the input genomes, followed by some traditional algorithmic techniques.