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exercise-sheet-2.Rmd
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exercise-sheet-2.Rmd
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---
title: "Exercise sheet 2: Edit operations and alignments"
---
---------------------------------
# Exercise 1 - Levenshtein Distance
Compute the minimal Levenshtein edit distance for the following pairs of sequences.
### 1a)
::: {.question data-latex=""}
\begin{align}
S_{1} = A\\
S_{2} = T
\end{align}
:::
#### {.tabset}
##### Hide
##### Hint
A `r knitr::asis_output("\U2192")` T
##### Solution
A `r knitr::asis_output("\U2192")` T = 1
#### {-}
### 1b)
::: {.question data-latex=""}
\begin{align}
S_{1} &= AGATATA\\
S_{2} &= TATATATA
\end{align}
:::
#### {.tabset}
##### Hide
##### Hint
AGATATA `r knitr::asis_output("\U2192")` ATATATA `r knitr::asis_output("\U2192")` ...
##### Solution
AGATATA `r knitr::asis_output("\U2192")` ATATATA `r knitr::asis_output("\U2192")` TATATATA = 2
#### {-}
### 1c)
::: {.question data-latex=""}
\begin{align}
S_{1} = AGTCCT\\
S_{2} = CGCTCA
\end{align}
:::
#### {.tabset}
##### Hide
##### Hint
AGTCCT `r knitr::asis_output("\U2192")` AGCTCA `r knitr::asis_output("\U2192")` ...
##### Solution
AGTCCT `r knitr::asis_output("\U2192")` CGTCCT `r knitr::asis_output("\U2192")` CGCCCT `r knitr::asis_output("\U2192")` CGCTCT `r knitr::asis_output("\U2192")` CGCTCA = 4
#### {-}
### 1d)
::: {.question data-latex=""}
\begin{align}
S_{1} = TGCATAT\\
S_{2} = ATCCGAT
\end{align}
:::
#### {.tabset}
##### Hide
##### Hint
TGCATAT `r knitr::asis_output("\U2192")` AGCATAT `r knitr::asis_output("\U2192")` ...
##### Solution
TGCATAT `r knitr::asis_output("\U2192")` AGCATAT `r knitr::asis_output("\U2192")` ATCATAT `r knitr::asis_output("\U2192")` ATCAGAT `r knitr::asis_output("\U2192")` ATCCGAT = 4
#### {-}
### 1e)
::: {.question data-latex=""}
\begin{align}
S_{1} = ACGTATATAGCCCCGCG\\
S_{2} = ACGTTATATAGCCGCGC
\end{align}
:::
#### {.tabset}
##### Hide
##### Hint
You need to use all the possible operations
ACGTATATAGCCCCGCG `r knitr::asis_output("\U2192")` ACGTTATATAGCCCCGCG `r knitr::asis_output("\U2192")` ...
##### Solution
ACGTATATAGCCCCGCG `r knitr::asis_output("\U2192")` ACGTTATATAGCCCCGCG `r knitr::asis_output("\U2192")` ACGTTATATAGCCGCGCG `r knitr::asis_output("\U2192")` ACGTTATATAGCCGCGC = 3
#### {-}
# Exercise 2 - Metric function
Check if the corresponding functions are metric.
#### {.tabset}
##### Hide
##### Formulae
:::: {#explaining .message-box }
::: {#note-exp .note-header}
```{r, include=knitr::is_html_output(), echo=FALSE,}
knitr::include_graphics("figures/infoicon.svg")
```
**Note**
:::
::: {#note-exp .note-body}
Definition Metric:
\begin{align}
w(x,y) &= 0 \leftrightarrow x = y\ &\text{(identity)}\\
w(x, y) &= w(y, x)\ &\text{(symmetric)}\\
w (x, z) &\leq w (x, y ) + w (y , z) &\text{(triangle inequality)}
\end{align}
:::
::::
#### {-}
### 2a)
\begin{align}
w(x,y) = x-y
\end{align}
#### {.tabset}
##### Hide
##### Hint
What if $x = 1$ and $y = 2$?
##### Solution
Not a metric, violates symmetry constraint.
$$
x = 1\\
y = 2\\
x - y = 1 - 2 = -1 \neq 1 = 2 - 1 = y - x
$$
#### {-}
### 2b)
\begin{align}
w(x,y) = |x-y|
\end{align}
#### {.tabset}
##### Hide
##### Hint
You need to check all the properties.
##### Solution
Metric
#### {-}
### 2c)
\begin{align}
w(x,y) = x+y
\end{align}
#### {.tabset}
##### Hide
##### Hint
What if $x = 1$ and $y = 1$?
##### Solution
Not metric, violates identity constraint:
$$
x = y = 1\\
x + y = x + x = 2 \neq 0
$$
#### {-}
### 2d)
\begin{align}
w(x,y) = \begin{cases} 1 \ \text{if}\ x \neq y
\\0\ \text{else}
\end{cases}
\end{align}
#### {.tabset}
##### Hide
##### Hint
You need to check all the properties.
##### Solution
Metric
#### {-}
#### {.tabset}
---------------------------------
# Exercise 3 - Programming assignment
Programming assignments are available via Github Classroom and contain automatic tests.
We recommend doing these assignments since they will help you to further understand this topic.
Access the Github Classroom link: [Programming Assignment: Sheet 02](https://classroom.github.com/a/e0m0I2OZ).
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