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minor modifications 

I will submit this version arXiv. There is definitely room for improvement, but
as a semi-formal technical report, it should be good for now.
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Heng Li 2017-08-03 12:47:36 -04:00
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tex/minimap2.tex
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@ -60,8 +60,8 @@ to develop minimap2 towards higher accuracy and more practical functionality.
\section{Methods} \section{Methods}
Minimap2 is the successor of minimap~\citep{Li:2016aa}. It uses similar Minimap2 is the successor of minimap~\citep{Li:2016aa}. It uses similar
indexing and seeding algorithms, and furthers it with a more accurate chaining indexing and seeding algorithms, and furthers it with more accurate chaining
algorithm and adds the ability to produce detailed alignment. and the ability to produce detailed alignment.
\subsection{Chaining} \subsection{Chaining}
@ -89,13 +89,12 @@ are inapplicable to generic gap cost, complex to implement and usually
associated with a large constant. We introduced a simple heuristic to associated with a large constant. We introduced a simple heuristic to
accelerate chaining. accelerate chaining.
We note that if anchor $i$ is appended to $j$, appending $i$ to a predecessor We note that if anchor $i$ is chained to $j$, chaining $i$ to a predecessor
of $j$ is likely to yield a lower score. When evaluating Eq.~(\ref{eq:chain}), of $j$ is likely to yield a lower score. When evaluating Eq.~(\ref{eq:chain}),
we start from anchor $i-1$ and stop the evaluation if we cannot find a better we start from anchor $i-1$ and stop the evaluation if we cannot find a better
score after up to $h$ iterations. This approach reduces the average time to score after up to $h$ iterations. This approach reduces the average time to
$O(h\cdot m)$. In practice, we can almost always find the optimal chain with $O(h\cdot m)$. In practice, we can almost always find the optimal chain with
$h=50$; even if the heuristic fails, the optimal chain is often not $h=50$; even if the heuristic fails, the optimal chain is often close.
trustworthy, either.
\subsubsection{Backtracking} \subsubsection{Backtracking}
Let $P(i)$ be the index of the best predecessor of anchor $i$. It equals 0 if Let $P(i)$ be the index of the best predecessor of anchor $i$. It equals 0 if
@ -141,7 +140,7 @@ On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, this
cost function is concave. It applies cost $q+l\cdot e$ to gaps shorter than cost function is concave. It applies cost $q+l\cdot e$ to gaps shorter than
$\lceil(\tilde{q}-q)/(e-\tilde{e})\rceil$ and applies $\lceil(\tilde{q}-q)/(e-\tilde{e})\rceil$ and applies
$\tilde{q}+l\cdot\tilde{e}$ to longer gaps. This scheme helps to recover $\tilde{q}+l\cdot\tilde{e}$ to longer gaps. This scheme helps to recover
longer insertions and deletions~(INDEL; \citealp{Gotoh:1990aa}); longer insertions and deletions~(INDEL; \citealp{Gotoh:1990aa}).
With global alignment, minimap2 may force to align unrelated sequences between With global alignment, minimap2 may force to align unrelated sequences between
two adjacent anchors. To avoid such an artifact, we compute accumulative two adjacent anchors. To avoid such an artifact, we compute accumulative
@ -153,7 +152,7 @@ $j'<j$, such that
\[ \[
S(i',j')-S(i,j)>Z+e\cdot(\max\{i-i',j-j'\}-\min\{i-i',j-j'\}) S(i',j')-S(i,j)>Z+e\cdot(\max\{i-i',j-j'\}-\min\{i-i',j-j'\})
\] \]
where $e$ is the gap extension penalty and $Z$ is an arbitrary threshold. where $e$ is the gap extension cost and $Z$ is an arbitrary threshold.
This strategy is similar to X-drop employed in BLAST~\citep{Altschul:1997vn}. This strategy is similar to X-drop employed in BLAST~\citep{Altschul:1997vn}.
However, unlike X-drop, it would not break the alignment in the presence of a However, unlike X-drop, it would not break the alignment in the presence of a
single long gap. single long gap.
@ -187,7 +186,7 @@ GraphMap~\citep{Sovic:2016aa},
minialign~\citep{Suzuki:2016} and minialign~\citep{Suzuki:2016} and
NGMLR~\citep{Sedlazeck169557}. We excluded rHAT~\citep{Liu:2016ab}, NGMLR~\citep{Sedlazeck169557}. We excluded rHAT~\citep{Liu:2016ab},
LAMSA~\citep{Liu:2017aa} and Kart~\citep{Lin:2017aa} because they either LAMSA~\citep{Liu:2017aa} and Kart~\citep{Lin:2017aa} because they either
crashed or produced malformatted SAM. In this evaluation, Minimap2 has a crashed or produced malformatted output. In this evaluation, Minimap2 has a
higher power to distinguish unique and repetitive hits, and achieves overall higher power to distinguish unique and repetitive hits, and achieves overall
higher mapping accuracy (Fig.~\ref{fig:eval}a). It is still the most accurate higher mapping accuracy (Fig.~\ref{fig:eval}a). It is still the most accurate
even if we skip DP-based alignment (data not shown), suggesting chaining alone even if we skip DP-based alignment (data not shown), suggesting chaining alone
@ -218,11 +217,11 @@ further accelerate minimap2 with a few other tweaks such as adaptive
banding~\citep{Suzuki130633} or incremental banding. banding~\citep{Suzuki130633} or incremental banding.
In addition to reference-based read mapping, minimap2 inherits minimap's In addition to reference-based read mapping, minimap2 inherits minimap's
ability to search against huge multi-species data and to find read overlaps. On ability to search against huge multi-species databases and to find read
a few test data sets, minimap2 appears to yield slightly better miniasm overlaps. On a few test data sets, minimap2 appears to yield slightly better
assembly. Minimap2 can also align long assemblies or closely related genomes, miniasm assembly. Minimap2 can also align long assemblies or closely related
though more thorough evaluations are needed. Genome alignment is an intricate genomes, though more thorough evaluations are needed. Genome alignment is an
topic. intricate topic.
\section*{Acknowledgements} \section*{Acknowledgements}
We owe a debt of gratitude to Hajime Suzuki for releasing his masterpiece and We owe a debt of gratitude to Hajime Suzuki for releasing his masterpiece and
@ -242,7 +241,7 @@ A 2-piece gap cost function is
\gamma(l)=\min\{q+l\cdot e,\tilde{q}+l\cdot\tilde{e}\} \gamma(l)=\min\{q+l\cdot e,\tilde{q}+l\cdot\tilde{e}\}
\] \]
Without losing generality, we assume $q+e\le\tilde{q}+\tilde{e}$. The equation Without losing generality, we assume $q+e\le\tilde{q}+\tilde{e}$. The equation
to compute the optimal alignment under such a gap cost is to compute the optimal alignment under such a gap cost is~\citep{Gotoh:1990aa}
\begin{equation}\label{eq:ae86} \begin{equation}\label{eq:ae86}
\left\{\begin{array}{l} \left\{\begin{array}{l}
H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\ H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\
@ -288,8 +287,8 @@ In addition,
\[ \[
u_{ij}=z_{ij}-v_{i-1,j}\ge\max\{x_{i-1,j},\tilde{x}_{i-1,j}\}\ge-q-e u_{ij}=z_{ij}-v_{i-1,j}\ge\max\{x_{i-1,j},\tilde{x}_{i-1,j}\}\ge-q-e
\] \]
We also note that the maximum possible $z_{ij}=H_{ij}-H_{i-1,j-1}$ is $M$, the As the maximum value of $z_{ij}=H_{ij}-H_{i-1,j-1}$ is $M$, the maximal
maximal matching score. As a result, matching score, we can derive
\[ \[
u_{ij}\le M-v_{i-1,j}\le M+q+e u_{ij}\le M-v_{i-1,j}\le M+q+e
\] \]