1. Algorithm design

In many important cases we have a vector x with growing values. That is, each value is bigger or equal to the previous one, so

x[i+1] >= x[i]

for all values of the index i. It is easy to see that the position of the minimum value has to be 1. We also know that the position of the maximum value is the last position. What about the position of the half value?

The half value is the average of the minimum and the maximum. For example if x is the vector c(1, 4, 4, 6, 10, 15) then the half value is (1+15)/2, that is 8.

The position of the half value of the vector x is the index of the first value that is equal or bigger than the half value of x. In the example the position of the half value is 5, since x[5] is the smallest value that is bigger or equal than 8.

Please write a function called position_of_half(), with one input called x. The function must return a single number, which is the index of the smallest value in x that is bigger than or equal to the average of minimum and maximum of x.

You can test your functions with the following code.

x <- 1:9
position_of_half(x)
position_of_half(x + 20)
position_of_half(x * x)
position_of_half(sqrt(x))

The answers should be 5, 5, 7, 4, respectively.

2. Merge two sorted vectors

Please write a function called vector_merge(x, y) that receives two sorted vectors x and y and returns a new vector with the elements of x and y together sorted. The output vector has size length(x)+length(y).

You must assume that each of the input vectors is already sorted.

in your code you have to use three indices: i, j, and k; to point into x, y and the output vector answer, respectively. On each step you have to compare x[i] and y[j]. If x[i] < y[j] then you make answer[k] <- x[i], otherwise make answer[k] <- y[j].

You have to increment i or j, and k carefully. To test your function, you can use this code:

x <- c("a", "d", "e", "h", "i", "k", "m", "s", "t", "u", "v", "w", "z")
y <- c("b", "c", "f", "g", "j", "l", "n", "o", "p", "q", "r", "x", "y")
vector_merge(x, y)

The output must be a sorted alphabet.

"a" "b" "c" "d" "e" "f" "g" "h" "i" "j" "k" "l" "m"
"n" "o" "p" "q" "r" "s" "t" "u" "v" "w" "x" "y" "z"

3. Transcription

Write a function called transcribe(), that takes a DNA sequence (a vector of character) and returns the corresponding RNA sequence.

In other words, if you have

dna <- c("T","C","A","G","A","T","T","A","C")

then we transcribe it with transcribe(dna). The result should be

"U" "C" "A" "G" "A" "U" "U" "A" "C"

4. Codons

Write a function called codons(), that takes a RNA sequence (a vector of character) and returns a list of vectors. Each vector in the list must have only 3 letters, representing the codon.

5. Translation

Write a function called translate_codon(), that takes a codon (a vector with 3 letters), and returns a single letter representing one amino acid. For your convenience, here you have the correspondence between codons and amino acids.

6. Reverse complement

Write a function called reverse_complement(), that takes a DNA sequence (a vector of character) and returns the DNA sequence of the opposite strand. Both sequences are represented from 5’ to 3’.

In other words, if you have

dna <- c("T","C","A","G","A","T","T","A","C")

then applying reverse_complement(dna) we should getThe complement of “A” is “T”, the complement of “C” is “G”, and vice-versa.

"G" "T" "A" "A" "T" "C" "T" "G" "A"

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