moved appendix to methods; detailed splice aln
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Heng Li
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@ -179,3 +179,19 @@
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note = {doi:10.1101/130633},
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publisher = {Cold Spring Harbor Labs Journals},
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journal = {bioRxiv}}
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@article{Gotoh:1982aa,
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Author = {Gotoh, O},
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Journal = {J Mol Biol},
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Pages = {705-8},
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Title = {An improved algorithm for matching biological sequences},
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Volume = {162},
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Year = {1982}}
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@article{Altschul:1986aa,
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Author = {Altschul, S F and Erickson, B W},
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Journal = {Bull Math Biol},
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Pages = {603-16},
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Title = {Optimal sequence alignment using affine gap costs},
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Volume = {48},
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Year = {1986}}
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311
tex/minimap2.tex
311
tex/minimap2.tex
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@ -65,21 +65,21 @@ and the ability to produce detailed alignment.
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\subsection{Chaining}
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%\subsubsection{Chaining}
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\subsubsection{Chaining}
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An \emph{anchor} is a 3-tuple $(x,y,w)$, indicating interval $[x-w+1,x]$ on the
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reference matching interval $[y-w+1,y]$ on the query. Given a list of anchors
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sorted by ending reference position $x$, let $f(i)$ be the maximal chaining
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score up to the $i$-th anchor in the list. $f(i)$ can be calculated with
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dynamic programming (DP):
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\begin{equation}\label{eq:chain}
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f(i)=\max\big\{\max_{i>j\ge 1} \{ f(j)+d(j,i)-\beta_c(j,i) \},w_i\big\}
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f(i)=\max\big\{\max_{i>j\ge 1} \{ f(j)+\alpha(j,i)-\beta(j,i) \},w_i\big\}
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\end{equation}
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where $d(j,i)=\min\big\{\min\{y_i-y_j,x_i-x_j\},w_i\big\}$ is the number of
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matching bases between the two anchors. $\beta_c(j,i)>0$ is the gap cost. It
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where $\alpha(j,i)=\min\big\{\min\{y_i-y_j,x_i-x_j\},w_i\big\}$ is the number of
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matching bases between the two anchors. $\beta(j,i)>0$ is the gap cost. It
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equals $\infty$ if $y_j\ge y_i$ or $\max\{y_i-y_j,x_i-x_j\}>G$ (i.e. the
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distance between two anchors is too large); otherwise
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\begin{equation}\label{eq:chain-gap}
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\beta_c(j,i)=\gamma_c\big((y_i-y_j)-(x_i-x_j)\big)
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\beta(j,i)=\gamma_c\big((y_i-y_j)-(x_i-x_j)\big)
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\end{equation}
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In implementation, a gap of length $l$ costs $\gamma_c(l)=0.01\cdot \bar{w}\cdot
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|l|+0.5\log_2|l|$, where $\bar{w}$ is the average seed length. For $m$ anchors, directly computing all $f(\cdot)$ with
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@ -96,15 +96,15 @@ score after up to $h$ iterations. This approach reduces the average time to
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$O(h\cdot m)$. In practice, we can almost always find the optimal chain with
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$h=50$; even if the heuristic fails, the optimal chain is often close.
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%\subsubsection{Backtracking}
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For backtracking, let $P(i)$ be the index of the best predecessor of anchor
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$i$. It equals 0 if $f(i)=w_i$ or $\argmax_j\{f(j)+\eta(j,i)-\gamma(j,i)\}$
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otherwise. For each anchor $i$ in the descending order of $f(i)$, we apply
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$P(\cdot)$ repeatedly to find its predecessor and mark each visited $i$ as
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`used', until $P(i)=0$ or we reach an already `used' $i$. This way we find all
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chains with no anchors used in more than one chains.
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\subsubsection{Backtracking}
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Let $P(i)$ be the index of the best predecessor of anchor $i$. It equals 0 if
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$f(i)=w_i$ or $\argmax_j\{f(j)+\eta(j,i)-\gamma(j,i)\}$ otherwise. For each
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anchor $i$ in the descending order of $f(i)$, we apply $P(\cdot)$ repeatedly to
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find its predecessor and mark each visited $i$ as `used', until $P(i)=0$ or we
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reach an already `used' $i$. This way we find all chains with no anchors used
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in more than one chains.
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%\subsubsection{Identifying primary chains}
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\subsubsection{Identifying primary chains}
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In the absence of copy number changes, each query segment should not be mapped
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to two places in the reference. However, chains found at the previous step may
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have significant or complete overlaps due to repeats in the reference.
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@ -117,32 +117,134 @@ add the chain to $Q$. In the end, $Q$ contains all the primary chains. We did
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not choose a more sophisticated data structure (e.g. range tree or k-d tree)
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because this step is not the performance bottleneck.
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\subsection{Alignment}
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\subsection{Aligning genomic DNA}
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Minimap2 performs global alignment between adjacent anchors in a chain. It
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adopted difference-based formulation to derive
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alignment~\citep{Wu:1996aa,Suzuki:2016}. When combined with SSE vectorization,
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this formulation has two advantages. First, because each score in the DP matrix
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is bounded by the gap cost and the maximum matching score, we can usually
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achieve 16-way SSE vectorization regardless of the peak score of the
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alignment. Second, filling the DP matrix along the diagonal, we can simplify
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banded alignment, which is critical to performance. In practice, our
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implementation is three times as fast as Parasail's 4-way
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vectorization~\citep{Daily:2016aa} for global alignment.
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Without banding, our implementation is slower than Edlib~\citep{Sosic:2017aa},
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but with a 1000bp band, it is considerably faster. When performing global
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alignment between anchors, we expect the alignment to stay close to the
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diagonal of the DP matrix. Banding is often applicable.
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\subsubsection{Alignment with 2-piece affine gap cost}
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Minimap2 uses a 2-piece affine gap cost~\citep{Gotoh:1990aa}:
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Minimap2 performs DP-based global alignment between adjacent anchors in a
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chain. It uses a 2-piece affine gap cost~\citep{Gotoh:1990aa}:
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\begin{equation}\label{eq:2-piece}
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\gamma_a(l)=\min\{q+|l|\cdot e,\tilde{q}+|l|\cdot\tilde{e}\}
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\end{equation}
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On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, this
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cost function is concave. It applies cost $q+|l|\cdot e$ to gaps shorter than
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$\lceil(\tilde{q}-q)/(e-\tilde{e})\rceil$ and applies
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$\tilde{q}+|l|\cdot\tilde{e}$ to longer gaps. This scheme helps to recover
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longer insertions and deletions~(INDEL).
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Without losing generality, we always assume $q+e<\tilde{q}+\tilde{e}$.
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On the condition that $e>\tilde{e}$, it applies cost $q+|l|\cdot e$ to gaps
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shorter than $\lceil(\tilde{q}-q)/(e-\tilde{e})\rceil$ and applies
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$\tilde{q}+|l|\cdot\tilde{e}$ to longer gaps. This scheme helps to recover
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longer insertions and deletions~(INDELs).
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The equation to compute the optimal alignment under $\gamma_a(\cdot)$ is
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\begin{equation}\label{eq:ae86}
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\left\{\begin{array}{l}
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H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\
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E_{i+1,j}= \max\{H_{ij}-q,E_{ij}\}-e\\
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F_{i,j+1}= \max\{H_{ij}-q,F_{ij}\}-e\\
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\tilde{E}_{i+1,j}= \max\{H_{ij}-\tilde{q},\tilde{E}_{ij}\}-\tilde{e}\\
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\tilde{F}_{i,j+1}= \max\{H_{ij}-\tilde{q},\tilde{F}_{ij}\}-\tilde{e}
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\end{array}\right.
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\end{equation}
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where $s(i,j)$ is the score between the $i$-th reference base and $j$-th query
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base. It is a natural extension to the algorithm under affine gap
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cost~\citep{Gotoh:1982aa,Altschul:1986aa}.
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\subsubsection{Suzuki's formulation}
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To efficiently align long sequences, minimap2 did not directly use
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Eq.~(\ref{eq:ae86}) for alignment. It instead adopted a difference-based
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formulation first proposed by \citet{Wu:1996aa} and later adapted by
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\citet{Suzuki:2016} for affine gap cost. In case of 2-piece affine gap cost in
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Eq.~(\ref{eq:2-piece}), define
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\[
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\left\{\begin{array}{ll}
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u_{ij}\triangleq H_{ij}-H_{i-1,j} & v_{ij}\triangleq H_{ij}-H_{i,j-1} \\
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x_{ij}\triangleq E_{i+1,j}-H_{ij} & \tilde{x}_{ij}\triangleq \tilde{E}_{i+1,j}-\tilde{H}_{ij} \\
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y_{ij}\triangleq F_{i,j+1}-H_{ij} & \tilde{y}_{ij}\triangleq \tilde{F}_{i,j+1}-\tilde{H}_{ij}
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\end{array}\right.
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\]
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We can transform Eq.~(\ref{eq:ae86}) to
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\begin{equation}\label{eq:suzuki}
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\left\{\begin{array}{lll}
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z_{ij}&=&\max\{s(i,j),x_{i-1,j}+v_{i-1,j},y_{i,j-1}+u_{i,j-1},\\
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&&\tilde{x}_{i-1,j}+v_{i-1,j},\tilde{y}_{i,j-1}+u_{i,j-1}\}\\
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u_{ij}&=&z_{ij}-v_{i-1,j}\\
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v_{ij}&=&z_{ij}-u_{i,j-1}\\
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x_{ij}&=&\max\{0,x_{i-1,j}+v_{i-1,j}-z_{ij}+q\}-q-e\\
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y_{ij}&=&\max\{0,y_{i,j-1}+u_{i,j-1}-z_{ij}+q\}-q-e\\
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\tilde{x}_{ij}&=&\max\{0,\tilde{x}_{i-1,j}+v_{i-1,j}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
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\tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}
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\end{array}\right.
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\end{equation}
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where $z_{ij}$ is a temporary variable that does not need to be stored.
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This is Suzuki's formulation for 2-piece affine gap cost. An important property
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of this formulation is that all values are bounded. To see that,
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\[
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x_{ij}=E_{i+1,j}-H_{ij}=\max\{-q,E_{ij}-H_{ij}\}-e
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\]
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With $E_{ij}\le H_{ij}$, we have
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\[
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-q-e\le x_{ij}\le\max\{-q,0\}-e=-e
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\]
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and similar inequations for $y_{ij}$, $\tilde{x}_{ij}$ and $\tilde{y}_{ij}$.
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In addition,
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\[
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u_{ij}=z_{ij}-v_{i-1,j}\ge\max\{x_{i-1,j},\tilde{x}_{i-1,j}\}\ge-q-e
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\]
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As the maximum value of $z_{ij}=H_{ij}-H_{i-1,j-1}$ is $M$, the maximal
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matching score, we can derive
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\[
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u_{ij}\le M-v_{i-1,j}\le M+q+e
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\]
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In conclusion, in Eq.~(\ref{eq:suzuki}), $x$ and $y$ are bounded by $[-q-e,-e]$,
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$\tilde{x}$ and $\tilde{y}$ by $[-\tilde{q}-\tilde{e},-\tilde{e}]$, and $u$ and
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$v$ by $[-q-e,M+q+e]$. When $-128\le-q-e<M+q+e\le127$, each of them can be stored as
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a 8-bit integer. This enables 16-way SSE vectorization regardless of the peak
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score of the alignment.
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For a more efficient SSE implementation, we transform the row-column coordinate
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to diagonal-anti-diagonal coordinate by letting $r\gets i+j$ and $t\gets i$.
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Eq.~(\ref{eq:suzuki}) becomes:
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\begin{equation*}
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\left\{\begin{array}{lll}
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z_{rt}&=&\max\{s(t,r-t),x_{r-1,t-1}+v_{r-1,t-1},y_{r-1,t}\\
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&&+u_{r-1,t},\tilde{x}_{r-1,t-1}+v_{r-1,t-1},\tilde{y}_{r-1,t}+u_{r-1,t}\}\\
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u_{rt}&=&z_{rt}-v_{r-1,t-1}\\
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v_{rt}&=&z_{rt}-u_{r-1,t}\\
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x_{rt}&=&\max\{0,x_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+q\}-q-e\\
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y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}-q-e\\
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\tilde{x}_{rt}&=&\max\{0,\tilde{x}_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
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\tilde{y}_{rt}&=&\max\{0,\tilde{y}_{r-1,t}+u_{r-1,t}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}
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\end{array}\right.
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\end{equation*}
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In this formulation, cells with the same diagonal index $r$ are independent of
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each other. This allows us to fully vectorize the computation of all cells on
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the same anti-diagonal in one inner loop.
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On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, the boundary
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condition of this equation in the diagonal-anti-diagonal coordinate is
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\[
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\left\{\begin{array}{l}
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x_{r-1,-1}=y_{r-1,r}=-q-e\\
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\tilde{x}_{r-1,-1}=\tilde{y}_{r-1,r}=-\tilde{q}-\tilde{e}\\
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u_{r-1,r}=v_{r-1,-1}=\eta(r)\\
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\end{array}\right.
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\]
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where
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\[
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\eta(r)=\left\{\begin{array}{ll}
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-q-e & (r=0) \\
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-e & (r<\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
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r\cdot(e-\tilde{e})-(\tilde{q}-q)-\tilde{e} & (r=\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
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-\tilde{e} & (r>\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil)
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\end{array}\right.
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\]
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In practice, our 16-way vectorized implementation of global alignment is three
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times as fast as Parasail's 4-way vectorization~\citep{Daily:2016aa}. Without
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banding, our implementation is slower than Edlib~\citep{Sosic:2017aa}, but with
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a 1000bp band, it is considerably faster. When performing global alignment
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between anchors, we expect the alignment to stay close to the diagonal of the
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DP matrix. Banding is applicable most of time.
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\subsubsection{The Z-drop heuristic}
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With global alignment, minimap2 may force to align unrelated sequences between
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two adjacent anchors. To avoid such an artifact, we compute accumulative
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@ -164,7 +266,7 @@ alignment between the two subsequences involved in the global alignment, but
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this time with the one subsequence reverse complemented. This additional
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alignment step may identify short inversions that are missed during chaining.
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\subsection{Spliced alignment}
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\subsection{Aligning spliced sequences}
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The algorithm described above can be adapted to spliced alignment. In this
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mode, the chaining gap cost distinguishes insertions to and deletions from the
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@ -183,10 +285,33 @@ q+l\cdot e & (l>0) \\
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\end{array}\right.
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\]
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In alignment, a deletion no shorter than $\lceil(\tilde{q}-q)/e\rceil$ is
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regarded as an intron, which pays no cost to gap extensions. With these
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modifications, minimap2 can retain multiple reasonably long introns in one
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alignment, rather than fragment the alignment into local hits, which often
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leads to the loss of small exons especially given noisy reads.
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regarded as an intron, which pays no cost to gap extensions.
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To pinpoint precise exon boundaries, minimap2 penalizes non-canonical splicing
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with the following equation
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\begin{equation}\label{eq:splice}
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\left\{\begin{array}{l}
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H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij}-a_i\}\\
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E_{i+1,j}= \max\{H_{ij}-q,E_{ij}\}-e\\
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F_{i,j+1}= \max\{H_{ij}-q,F_{ij}\}-e\\
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\tilde{E}_{i+1,j}= \max\{H_{ij}-d_i-\tilde{q},\tilde{E}_{ij}\}\\
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\end{array}\right.
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\end{equation}
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Let $T$ be the reference sequence. $d_i$ is the cost of a non-canonical donor
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site, which takes 0 if $T[i+1,i+2]={\tt GT}$, or a postive number $p$
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otherwise. Similarly, $a_i$ is the cost of a non-canonical acceptor site, which
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takes 0 if $T[i-1,i]={\tt AG}$, or $p$ otherwise. When the read strand relative
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to the underlying transcript is unknown, minimap2 aligns each chain twice, first
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assuming ${\tt GT}$--${\tt AG}$ as the splice signal and then assuming ${\tt
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CT}$--${\tt AC}$, the reverse complement of ${\tt GT}$--${\tt AG}$, as the
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splice signal. The alignment with a higher score is taken as the final
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alignment. This procedure also infers the relative strand of reads spanning
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canonical splicing sites.
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In the spliced alignment mode, minimap2 further increases the density of
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minimizers and disables banded alignment. Together with the two-round DP-based
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alignment, spliced alignment is several times slower than DNA sequence
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alignment.
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\end{methods}
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@ -258,112 +383,4 @@ issues.
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\bibliography{minimap2}
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\appendix
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\begin{methods}
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\section*{Appendix}
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A 2-piece gap cost function is
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\[
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\gamma(l)=\min\{q+l\cdot e,\tilde{q}+l\cdot\tilde{e}\}
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\]
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Without losing generality, we assume $q+e\le\tilde{q}+\tilde{e}$. The equation
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to compute the optimal alignment under such a gap cost is~\citep{Gotoh:1990aa}
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\begin{equation}\label{eq:ae86}
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\left\{\begin{array}{l}
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H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\
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E_{i+1,j}= \max\{H_{ij}-q,E_{ij}\}-e\\
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F_{i,j+1}= \max\{H_{ij}-q,F_{ij}\}-e\\
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\tilde{E}_{i+1,j}= \max\{H_{ij}-\tilde{q},\tilde{E}_{ij}\}-\tilde{e}\\
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\tilde{F}_{i,j+1}= \max\{H_{ij}-\tilde{q},\tilde{F}_{ij}\}-\tilde{e}
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\end{array}\right.
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\end{equation}
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where $s(i,j)$ is the score between the $i$-th reference base and $j$-th query
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base. If we define
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\[
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\left\{\begin{array}{ll}
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u_{ij}\triangleq H_{ij}-H_{i-1,j} & v_{ij}\triangleq H_{ij}-H_{i,j-1} \\
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x_{ij}\triangleq E_{i+1,j}-H_{ij} & \tilde{x}_{ij}\triangleq \tilde{E}_{i+1,j}-\tilde{H}_{ij} \\
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y_{ij}\triangleq F_{i,j+1}-H_{ij} & \tilde{y}_{ij}\triangleq \tilde{F}_{i,j+1}-\tilde{H}_{ij}
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\end{array}\right.
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\]
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we can transform Eq.~(\ref{eq:ae86}) to
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\begin{equation}\label{eq:suzuki}
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\left\{\begin{array}{lll}
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z_{ij}&=&\max\{s(i,j),x_{i-1,j}+v_{i-1,j},y_{i,j-1}+u_{i,j-1},\\
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&&\tilde{x}_{i-1,j}+v_{i-1,j},\tilde{y}_{i,j-1}+u_{i,j-1}\}\\
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u_{ij}&=&z_{ij}-v_{i-1,j}\\
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v_{ij}&=&z_{ij}-u_{i,j-1}\\
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x_{ij}&=&\max\{0,x_{i-1,j}+v_{i-1,j}-z_{ij}+q\}-q-e\\
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y_{ij}&=&\max\{0,y_{i,j-1}+u_{i,j-1}-z_{ij}+q\}-q-e\\
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\tilde{x}_{ij}&=&\max\{0,\tilde{x}_{i-1,j}+v_{i-1,j}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
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\tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}
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\end{array}\right.
|
||||
\end{equation}
|
||||
where $z_{ij}$ is a temporary variable that does not need to be stored.
|
||||
|
||||
All values in Eq.~(\ref{eq:suzuki}) are bounded. To see that,
|
||||
\[
|
||||
x_{ij}=E_{i+1,j}-H_{ij}=\max\{-q,E_{ij}-H_{ij}\}-e
|
||||
\]
|
||||
With $E_{ij}\le H_{ij}$, we have
|
||||
\[
|
||||
-q-e\le x_{ij}\le\max\{-q,0\}-e=-e
|
||||
\]
|
||||
and similar inequations for $y_{ij}$, $\tilde{x}_{ij}$ and $\tilde{y}_{ij}$.
|
||||
In addition,
|
||||
\[
|
||||
u_{ij}=z_{ij}-v_{i-1,j}\ge\max\{x_{i-1,j},\tilde{x}_{i-1,j}\}\ge-q-e
|
||||
\]
|
||||
As the maximum value of $z_{ij}=H_{ij}-H_{i-1,j-1}$ is $M$, the maximal
|
||||
matching score, we can derive
|
||||
\[
|
||||
u_{ij}\le M-v_{i-1,j}\le M+q+e
|
||||
\]
|
||||
In conclusion, $x$ and $y$ are bounded by $[-q-e,-e]$, $\tilde{x}$ and $\tilde{y}$ by
|
||||
$[-\tilde{q}-\tilde{e},-\tilde{e}]$, and $u$ and $v$ by $[-q-e,M+q+e]$. When
|
||||
matching score and gap cost are small, each of them can be stored as a 8-bit
|
||||
integer. This enables 16-way SSE vectorization regardless of the peak score of
|
||||
the alignment.
|
||||
|
||||
For a more efficient SSE implementation, we transform the row-column coordinate
|
||||
to diagonal-anti-diagonal coordinate by letting $r\gets i+j$ and $t\gets i$.
|
||||
Eq.~(\ref{eq:suzuki}) becomes:
|
||||
\begin{equation*}
|
||||
\left\{\begin{array}{lll}
|
||||
z_{rt}&=&\max\{s(t,r-t),x_{r-1,t-1}+v_{r-1,t-1},y_{r-1,t}+u_{r-1,t},\\
|
||||
&&\tilde{x}_{r-1,t-1}+v_{r-1,t-1},\tilde{y}_{r-1,t}+u_{r-1,t}\}\\
|
||||
u_{rt}&=&z_{rt}-v_{r-1,t-1}\\
|
||||
v_{rt}&=&z_{rt}-u_{r-1,t}\\
|
||||
x_{rt}&=&\max\{0,x_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+q\}-q-e\\
|
||||
y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}-q-e\\
|
||||
\tilde{x}_{rt}&=&\max\{0,\tilde{x}_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
|
||||
\tilde{y}_{rt}&=&\max\{0,\tilde{y}_{r-1,t}+u_{r-1,t}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}
|
||||
\end{array}\right.
|
||||
\end{equation*}
|
||||
In this formulation, cells with the same diagonal index $r$ are independent of
|
||||
each other. This allows us to fully vectorize the computation of all cells on
|
||||
the same anti-diagonal in one inner loop.
|
||||
|
||||
On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, the boundary
|
||||
condition of this equation in the diagonal-anti-diagonal coordinate is
|
||||
\[
|
||||
\left\{\begin{array}{l}
|
||||
x_{r-1,-1}=y_{r-1,r}=-q-e\\
|
||||
\tilde{x}_{r-1,-1}=\tilde{y}_{r-1,r}=-\tilde{q}-\tilde{e}\\
|
||||
u_{r-1,r}=v_{r-1,-1}=\eta(r)\\
|
||||
\end{array}\right.
|
||||
\]
|
||||
where
|
||||
\[
|
||||
\eta(r)=\left\{\begin{array}{ll}
|
||||
-q-e & (r=0) \\
|
||||
-e & (r<\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
|
||||
r\cdot(e-\tilde{e})-(\tilde{q}-q)-\tilde{e} & (r=\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
|
||||
-\tilde{e} & (r>\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil)
|
||||
\end{array}\right.
|
||||
\]
|
||||
|
||||
\citet{Suzuki:2016} first derived a similar set of equations under affine gap
|
||||
cost but with different notations.
|
||||
\end{methods}
|
||||
\end{document}
|
||||
|
|
|
|||
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Reference in New Issue