A Boolean-Matrix-based Filtering Algorithm

Generally speaking, any subsequence [li1, ..., lik]of the reference genome wherek≥m could be a

valid candidate sequence of[r1, ..., rm]. However, calculating the posterior likelihoods with respect to all these potential candidates is often prohibitive with the limited computational resources. In this section, we proposed a fast filtering algorithm that can eliminate out most of the candidates that are unlikely to generate the given observed read.

The filtering algorithm starts with a boolean matrix (matrix with only1’s and 0’s as its entries) B∈_Rm×N _{such that} Bij =      1, if √ ri_√−√lj lj ∈[y(1), y(K)]. 0, otherwise.

In words, Bij = 1 if and only if it is feasible that lj to map into ri. If no drops are allowed, a sequence[li1, .., lim]is a feasible potential original sequence if and only if Π

m

j=1Bj,ij = 1. Under the semi-parametric distribution, a fragment lij can be dropped if and only if

√ T−p lij p lij ∈[y₍₁₎, y_(K)]. (3.13)

Therefore we can pre-calculate a vector v∈_RN _{such that}

vj =      1, if (3.13) is satisfied. 0, otherwise. (3.14)

Vector v stores the set of fragment indexes that can be dropped.

Definition 3.3.1. A sequence [li1, ..., lik] is a feasible candidate with respect to read [r1, ..., rm] if

and only if: there exists a subset{s1, s2, ..., sm} of [i1, .., ik] such that

Πm_j=1Bj,sj ·Πi∈[i1,..,ik]\{s1,...,im}vi = 1.

We call the number of drops allowed as the dropping threshold. The filtering algorithm takes one read and a dropping thresholddas the input, and outputs all the feasible candidates with dimension

k=m+d. The detailed steps of the filtering algorithm are listed in Algorithm 2. Please note that in Step3,j is not a positive integer, instead it is a tuple that contains one possible combination of dropping positions. For example, when k=m+ 1, the dropping position could happen at any position between 2andm (we do not consider the first dimension nor the last dimension, since it essentially degenerates to the casek=m), soj is a list that contains just one positive integer; when

input: The reference genome L, the observed read[r1, ..., rm], the dropping threshold d, the corresponding boolean matrixB, and the dropping index vectorv.

output: The matrixCthat stores the starting and ending positions for all feasible candidates.

1. CalculateJ as the set that contains all the combinations of possible positions of drops, so the cardinality ofJ is m+_dd−2 . 2. forj∈ J do for i= 1 toN −m−d+ 1do [j1, ..., jm] :={i, i+ 1, ..., i+m+d−1} \(i+j) if Πm s=1Bs,jsΠt∈jvt== 1 then C= [C;i, i+m+d−1] end end end

Algorithm 2:Boolean Matrix Based Filtering

Pre-filtering of reference genome One drawback of Algorithm 2 is, the dropping thresholdd

is prefixed. This limits our searching space in the whole reference genome. One solution to this issue is to try differentd and merge the outputs. While it is relatively cheap to run Algorithm 2 with smalld, the cardinality ofJ in Algorithm 2 makes largedprohibitive. In this section, we discuss the pre-filtering of reference genome, which can significantly increase the searching space of Algorithm 2.

To pre-filter the reference genome, we remove all the fragments in the reference genome with lengths smaller than a thresholdc(T). For any specific original sequence, we can divide its fragments into three categories: (1) fragments that map into the observable reads, hence the number of fragments in this category equals to the dimension of observed read, (2) fragments that are shorter thanc(T)and map into something un-observable, (3) fragments that are longer than c(T)and map into something un-observable.

In Algorithm 2, we allow exact ddrops happen in the candidate sequences, hence dis the total number of fragments in the second and third category above. However, when c(T) is small enough, the probability that something below c(T) maps into a fragment shorter thanT is almost 1(we can calculate the value ofc(T) based on the semi-parametric distribution), hence as an approximation we may just remove all the fragments with lengths shorter than c(T). By doing so the numberdis used to bound the number of sequences in category (3) above, and hence we implicitly allow more

Table 3.1: Statistics for different cutters.

# of Fragments Mean Median Max Min Cutter_1 693849 436 2114 18164240 6 Cutter_2 192920 15711 8869 19848848 6 Cutter_3 186644 16240 9948 18171066 7 drops and add flexibility to the algorithm. Definec(T)p as a constant such that

P " x≤ p c(T)p− √ T p c(T)p # ≤1−p. (3.15)

Therefore c(T)p is monotonically increasing inp. WriteLp as the vector after removing all the frag- ments{li}i∈[N],li<c(T) in reference genomeL, we callLp the “filtered genome”. Applying Algorithm 2 on Lp with different pyields faster calculation since we only need to build boolean matrixB on a shorter vector.

To sum up. In order to avoid narrowing down the searching space of Algorithm 2, we recommend applying Algorithm 2 onLp with different choices of(p, d) and merge all the candidate sequences output by different parameters.