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\section{Sequence Alignment}  Sequence alignment refers to the computational task of trying to find common subsequences within two different sequences with minimal gaps. As a simplified example, consider trying to align the following words:  % -atte  % || |  % pat-e  \section{Smith-Waterman Algorithm}  The Smith-Waterman algorithm is used to find optimal local alignments between two sequences based on a cost matrix for symbols of the alphabet being used and defined gap costs.  A matrix $H$ is constructed in the following manner:  \[ H(i,0) = 0, 0 \leq i \leq m \]  \[ H(0,j) = 0, 0 \leq j \leq n \]  \[ H(i,j) = \hbox{max} \begin{cases} 0 \\ H(i-1,j-1)+s(a_i,b_j) \\ \hbox{max}_{k \geq 1} \left\{ H(i-k,j)+W_k \right\} \\ \hbox{max}_{l \geq 1}\left\{ H(i, j-l)+W_l) \right\} \end{cases} \]  diff --git a/layout.md b/layout.md  index af7468e..62dd3b2 100644  --- a/layout.md  +++ b/layout.md 
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DistanceFunction.tex  CImplementation.tex  Introduction1.tex  SmithWaterman.tex  SmithDemonstration.tex  Visualization.tex  figures/Screen Shot 2015-02-06 at 1.48.55 PM/Screen Shot 2015-02-06 at 1.48.55 PM.png