moved aln-dp.tex to ksw2/tex (a new repo)

ksw2.c

@@ -302,6 +302,7 @@ int ksw_global2_sse(void *km, int qlen, const uint8_t *query, int tlen, const ui
 	w = (w + 1 + 15) / 16 * 16 - 1;
 	tlen16 = (tlen + 15) / 16 * 16;
 	n_col = w + 1 < tlen16? w + 1 : tlen16; // number of columns in the backtrack matrix
+	n_col += 16, tlen16 += 16; // leave enough space at the end
 
 	mem = (uint8_t*)kcalloc(km, tlen16 * 5 + 15, 1);
 	u = (int8_t*)(((size_t)mem + 15) >> 4 << 4); // 16-byte aligned (though not necessary)

tex/aln-dp.tex

@@ -1,171 +0,0 @@
-\documentclass[10pt]{article}
-
-\title{Alignment with dynamic programming}
-\author{Heng Li}
-
-\begin{document}
-
-\maketitle
-
-\section{General notations}
-
-Suppose we have two sequences: a \emph{target} sequence and a \emph{query}
-sequence. The length of the target sequence is $\ell_t$ with each residue
-indexed by $i$.  The length of query is $\ell_q$ with each residue indexed by
-$j$. Gaps on the target sequence are \emph{deletions} and gaps on the query are
-\emph{insertions}. Function $S(i,j)$ gives the score between two residues on
-the target and the query, respectively. $q>0$ is the gap open/initiation
-penalty and $e>0$ the gap extension penalty. A gap of length $k$ costs
-$q+k\cdot e$.
-
-\section{Global alignment with affine-gap penalties}
-
-\subsection{Durbin's formulation}
-
-The original Durbin's formulation is:
-\begin{eqnarray*}
-M_{ij}&=&\max\{M_{i-1,j-1}, E_{i-1,j-1}, F_{i-1,j-1}\} + S(i,j)\\
-E_{ij}&=&\max\{M_{i-1,j}-q, E_{i-1,j}\} - e\\
-F_{ij}&=&\max\{M_{i,j-1}-q, F_{i,j-1}\} - e
-\end{eqnarray*}
-This formulation disallows a deletion immediately followed an insertion, or
-vice versa. A more general form is:
-\begin{eqnarray*}
-M_{ij}&=&\max\{M_{i-1,j-1}, E_{i-1,j-1}, F_{i-1,j-1}\} + S(i,j)\\
-E_{ij}&=&\max\{M_{i-1,j}-q, E_{i-1,j}, F_{i-1,j}-q\} - e\\
-F_{ij}&=&\max\{M_{i,j-1}-q, E_{i,j-1}-q, F_{i,j-1}\} - e
-\end{eqnarray*}
-
-\subsection{Green's formulation}
-
-If we define:
-\[H_{ij}=\max\{M_{ij},E_{ij},F_{ij}\}\]
-the Durbin's formulation can be transformed to
-\begin{eqnarray*}
-E_{ij} &=& \max\{H_{i-1,j}-q, E_{i-1,j}\} - e \\
-F_{ij} &=& \max\{H_{i,j-1}-q, F_{i,j-1}\} - e \\
-H_{ij} &=& \max\{H_{i-1,j-1}+S(i,j), E_{ij}, F_{ij}\}
-\end{eqnarray*}
-I first saw this formulation in Phrap developed by Phil Green, though it may
-have been used earlier. If we further introduce
-\begin{eqnarray*}
-E'_{ij}&=&E_{i+1,j}\\
-F'_{ij}&=&F_{i,j+1}
-\end{eqnarray*}
-we have
-\begin{eqnarray*}
-H_{ij} &=& \max\{H_{i-1,j-1}+S(i,j),E'_{i-1,j},F'_{i,j-1}\}\\
-E'_{ij}&=& \max\{H_{ij}-q,E'_{i-1,j}\}-e\\
-F'_{ij}&=& \max\{H_{ij}-q,F'_{i,j-1}\}-e
-\end{eqnarray*}
-In fact, we more often use this set of equations in practical implementations.
-The initial conditions are
-\begin{eqnarray*}
-H_{-1,j}&=&
-  \left\{\begin{array}{ll}
-    0 & (j=-1)\\
-	-q-(j+1)\cdot e & (0\le j<\ell_q)
-  \end{array}\right.\\
-H_{i,-1}&=&
-  \left\{\begin{array}{ll}
-    0 & (i=-1)\\
-	-q-(i+1)\cdot e & (0\le i<\ell_t)
-  \end{array}\right.\\
-E'_{-1,j}&=&E_{0,j}=H_{-1,j}-q-e=-2q-(j+2)\cdot e\\
-F'_{i,-1}&=&F_{i,0}=-2q-(i+2)\cdot e
-\end{eqnarray*}
-
-\subsection{Suzuki's formulation}
-
-\subsubsection{Standard coordinate}
-Now let
-\begin{eqnarray*}
-u'_{ij}&=&H_{ij}-H_{i-1,j}\\
-v'_{ij}&=&H_{ij}-H_{i,j-1}\\
-x'_{ij}&=&E'_{ij}-H_{ij}\\
-y'_{ij}&=&F'_{ij}-H_{ij}
-\end{eqnarray*}
-We have
-\begin{eqnarray}\label{eq:x}
-x'_{ij}&=&\max\{-q,E'_{i-1,j}-H_{i-1,j}+H_{i-1,j}-H_{ij}\}-e\\\nonumber
-&=&\max\{-q,x'_{i-1,j}-u'_{ij}\}-e
-\end{eqnarray}
-Similarly
-\begin{equation}
-y'_{ij}=\max\{-q,y'_{i,j-1}-v'_{ij}\}-e
-\end{equation}
-To derive the equation to compute $u'(i,j)$ and $v'(i,j)$, we note that
-\begin{eqnarray*}
-H_{ij}-H_{i-1,j-1}
-&=&\max\{S(i,j),E'_{i-1,j}-H_{i-1,j-1},F'_{i,j-1}-H_{i-1,j-1}\}\\
-&=&\max\{S(i,j),x'_{i-1,j}+v'_{i-1,j},y'_{i,j-1}+u'_{i,j-1}\}
-\end{eqnarray*}
-and
-\[H_{ij}-H_{i-1,j-1}=u'_{ij}+v'_{i-1,j}=v'_{ij}+u'_{i,j-1}\]
-We can derive the recursive equation for $u'_{ij}$ and $v'_{ij}$.
-
-From eq.~(\ref{eq:x}) we can infer that $x'_{ij}\ge-q-e$ and similarly
-$y'_{ij}\ge-q-e$. We further have:
-\[
-u'_{ij}=H_{ij}-H_{i-1,j-1}-v'_{i-1,j}\ge x'_{i-1,j}\ge-q-e
-\]
-Therefore, we have a lower bound $-q-e$ for $u'$, $v'$, $x'$ and $y'$.
-This motivates us to redefine the four variables as:
-\begin{eqnarray*}
-u''_{ij}&=&H_{ij}-H_{i-1,j}+q+e\\
-v''_{ij}&=&H_{ij}-H_{i,j-1}+q+e\\
-x''_{ij}&=&E'_{ij}-H_{ij}+q+e\\
-y''_{ij}&=&F'_{ij}-H_{ij}+q+e
-\end{eqnarray*}
-The recursion becomes
-\begin{eqnarray*}
-z''_{ij}&=&\max\{S(i,j)+2q+2e,x''_{i-1,j}+v''_{i-1,j},y''_{i,j-1}+u''_{i,j-1}\}\\
-u''_{ij}&=&z''_{ij}-v''_{i-1,j}\\
-v''_{ij}&=&z''_{ij}-u''_{i,j-1}\\
-x''_{ij}&=&\max\{0,x''_{i-1,j}-u''_{ij}+q\}=\max\{0,x''_{i-1,j}+v''_{i-1,j}-z''_{ij}+q\}\\
-y''_{ij}&=&\max\{0,y''_{i,j-1}-v''_{ij}+q\}=\max\{0,y''_{i,j-1}+u''_{i,j-1}-z''_{ij}+q\}
-\end{eqnarray*}
-Here $z_{ij}$ is a temporary variable. $u''$, $v''$, $x''$ and $y''$ are all
-non-negtive.
-
-\subsubsection{Rotated coordinate}
-
-We let
-\begin{eqnarray*}
-r&=&i+j\\
-t&=&i
-\end{eqnarray*}
-We have
-\begin{eqnarray*}
-z_{rt}&=&\max\{S(t,r-t)+2q+2e,x_{r-1,t-1}+v_{r-1,t-1},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\}\\
-y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}
-\end{eqnarray*}
-Due to the definition of $r$ and $t$, the following inequation must stand:
-\[0\le r-t \le\ell_q-1\]
-\[0\le t \le\ell_t-1\]
-where $\ell_t$ is the length of the sequence indexed by $i$ and $\ell_q$ the
-length indexed by $j$. In case of banded alignment with a fixed diagonal band
-of size $w$,
-\[-w\le j-i\le w\]
-In the $(r,t)$ coordinate, it is:
-\[\frac{r-w}{2}\le t\le \frac{r+w}{2}\]
-Putting these together:
-\[0\le r\le \ell_q+\ell_t-2\]
-\[\max\left\{0,r-\ell_q+1,\frac{r-w}{2}\right\}\le t\le\min\left\{\ell_t-1,r,\frac{r+w}{2}\right\}\]
-
-\subsubsection{Initial conditions}
-\[x_{r-1,-1}=x''_{-1,r}=E'_{-1,r}-H_{-1,r}+q+e=0\]
-\[y_{r-1,r}=y''_{r,-1}=0\]
-\[v_{r-1,-1}=v''_{-1,r}=H_{-1,r}-H_{-1,r-1}+q+e=\left\{\begin{array}{ll}
-  q & (r>0) \\
-  0 & (r=0)
-\end{array}\right.\]
-\[u_{r-1,r}=u''_{r,-1}=H_{r,-1}-H_{r-1,-1}+q+e=\left\{\begin{array}{ll}
-  q & (r>0) \\
-  0 & (r=0)
-\end{array}\right.\]
-
-\end{document}