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rephrasing a bit to avoid the apparent TeX bug 

Several hours sank into this. Damned...
This commit is contained in:
Heng Li 2017-08-03 20:19:00 -04:00
parent f57fd8c790
commit 95d3c30ae3
1 changed files with 5 additions and 4 deletions
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tex/minimap2.tex
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@ -274,8 +274,9 @@ y_{ij}&=&\max\{0,y_{i,j-1}+u_{i,j-1}-z_{ij}+q\}-q-e\\
\tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e} \tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}
\end{array}\right. \end{array}\right.
\end{equation} \end{equation}
where $z_{ij}$ is a temporary variable that does not need to be stored. We can where $z_{ij}$ is a temporary variable that does not need to be stored.
see that
All values in Eq.~(\ref{eq:suzuki}) are bounded. To see that,
\[ \[
x_{ij}=E_{i+1,j}-H_{ij}=\max\{-q,E_{ij}-H_{ij}\}-e x_{ij}=E_{i+1,j}-H_{ij}=\max\{-q,E_{ij}-H_{ij}\}-e
\] \]
@ -293,8 +294,7 @@ matching score, we can derive
\[ \[
u_{ij}\le M-v_{i-1,j}\le M+q+e u_{ij}\le M-v_{i-1,j}\le M+q+e
\] \]
In conclusion, all values in Eq.~(\ref{eq:suzuki}) are bounded: $x$ and $y$ by In conclusion, $x$ and $y$ by $[-q-e,-e]$ and $\tilde{x}$, $\tilde{y}$ by
$[-q-e,-e]$ and $\tilde{x}$, $\tilde{y}$ by
$[-\tilde{q}-\tilde{e},-\tilde{e}]$, and $u$ and $v$ by $[-q-e,M+q+e]$. When $[-\tilde{q}-\tilde{e},-\tilde{e}]$, 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 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 integer. This enables 16-way SSE vectorization regardless of the peak score of
@ -318,6 +318,7 @@ y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}-q-e\\
In this formulation, cells with the same row index $r$ are independent of each In this formulation, cells with the same row index $r$ are independent of each
other. This allows us to fully vectorize the computation of all cells on the other. This allows us to fully vectorize the computation of all cells on the
same anti-diagonal in one inner loop. same anti-diagonal in one inner loop.
On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, the boundary 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 condition of this equation in the diagonal-anti-diagonal coordinate is
\[ \[