 # Problem Set #2

### Objective

During this activity, students should be able to:

• Write functions of medium complexity using the Clojure programming language.

This activity helps students develop the following skills, values and attitudes: ability to analyze and synthesize, capacity for identifying and solving problems, and efficient use of computer systems.

## Activity Description

This activity can be developed individually or in pairs.

Solve the following programming problem set using Clojure. Place all your functions and unit tests in a file called problemset2.clj.

1. The function replic takes two arguments: a list lst and an integer number n, where n ≥ 0. It returns a new list that replicates n times each element contained in lst. Unit tests:

(deftest test-replic
(is (= () (replic 7 ())))
(is (= () (replic 0 '(a b c))))
(is (= '(a a a) (replic 3 '(a))))
(is (= '(1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4)
(replic 4 '(1 2 3 4)))))

2. The function expand takes a list lst as its argument. It returns a list where the first element of lst appears one time, the second elements appears two times, the third element appears three times, and so on. Unit test:

(deftest test-expand
(is (= () (expand ())))
(is (= '(a) (expand '(a))))
(is (= '(1 2 2 3 3 3 4 4 4 4) (expand '(1 2 3 4))))
(is (= '(a b b c c c d d d d e e e e e)
(expand '(a b c d e)))))

3. The function insert takes two arguments: a number n and a list of numbers lst in ascending order. It returns a new list with the same elements as lst but inserting n in its corresponding place. Unit tests:

(deftest test-insert
(is (= '(14) (insert 14 ())))
(is (= '(4 5 6 7 8) (insert 4 '(5 6 7 8))))
(is (= '(1 3 5 6 7 9 16) (insert 5 '(1 3 6 7 9 16))))
(is (= '(1 5 6 10) (insert 10 '(1 5 6)))))

4. The function my-sort takes an unordered list of numbers as an argument, and returns a new list with the same elements but in ascending order. You must use the insert function defined in the previous exercise to write my-sort. You must NOT use the predefined sort function. Unit tests:

(deftest test-my-sort
(is (= () (my-sort ())))
(is (= '(0 1 3 3 4 6 7 8 9) (my-sort '(4 3 6 8 3 0 9 1 7))))
(is (= '(1 2 3 4 5 6) (my-sort '(1 2 3 4 5 6))))
(is (= '(1 5 5 5 5 5 5) (my-sort '(5 5 5 1 5 5 5)))))

5. The function rotate-left takes two arguments: an integer number n and a list lst. It returns the list that results from rotating lst a total of n elements to the left. If n is negative, it rotates to the right. Unit tests:

(deftest test-rotate-left
(is (= () (rotate-left 5 ())))
(is (= '(a b c d e f g) (rotate-left 0 '(a b c d e f g))))
(is (= '(b c d e f g a) (rotate-left 1 '(a b c d e f g))))
(is (= '(g a b c d e f) (rotate-left -1 '(a b c d e f g))))
(is (= '(d e f g a b c) (rotate-left 3 '(a b c d e f g))))
(is (= '(e f g a b c d) (rotate-left -3 '(a b c d e f g))))
(is (= '(a b c d e f g) (rotate-left 7 '(a b c d e f g))))
(is (= '(a b c d e f g) (rotate-left -7 '(a b c d e f g))))
(is (= '(b c d e f g a) (rotate-left 8 '(a b c d e f g))))
(is (= '(g a b c d e f) (rotate-left -8 '(a b c d e f g))))
(is (= '(d e f g a b c) (rotate-left 45 '(a b c d e f g))))
(is (= '(e f g a b c d) (rotate-left -45 '(a b c d e f g)))))

6. The function binary takes an integer n as input (assume that n ≥ 0). If n is equal to zero, it returns an empty list. If n is greater than zero, it returns a list with a sequence of ones and zeros equivalent to the binary representation of n. Unit tests:

(deftest test-binary
(is (= () (binary 0)))
(is (= '(1 1 1 1 0) (binary 30)))
(is (= '(1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1) (binary 45123))))

7. The function prime-factors takes an integer n as input (assume that n > 0), and returns a list containing the prime factors of n in ascending order. The prime factors are the prime numbers that divide a number exactly. If you multiply all the prime factors you get the original number. Unit tests:

(deftest test-prime-factors
(is (= () (prime-factors 1)))
(is (= '(2 3) (prime-factors 6)))
(is (= '(2 2 2 2 2 3) (prime-factors 96)))
(is (= '(97) (prime-factors 97)))
(is (= '(2 3 3 37) (prime-factors 666))))

8. The function gcd takes two positive integer arguments a and b as arguments, where a > 0 and b > 0. It returns the greatest common divisor (GCD) of a and b. You must NOT use the predefined gcd function from the clojure.math.numeric-tower namespace.

NOTE: The GCD of two integers is the largest positive integer that divides both numbers exactly. For example, the GCD of 20 and 16 is 4.

Unit tests:

(deftest test-gcd
(is (= 1 (gcd 13 7919)))
(is (= 4 (gcd 20 16)))
(is (= 6 (gcd 54 24)))
(is (= 7 (gcd 6307 1995)))
(is (= 12 (gcd 48 180)))
(is (= 14 (gcd 42 56))))

9. The function insert-everywhere takes two arguments as input: an object x and a list lst. It returns a new list with all the possible ways in which x can be inserted into every position of lst. Unit tests:

(deftest test-insert-everywhere
(is (= '((1)) (insert-everywhere 1 ())))
(is (= '((1 a) (a 1)) (insert-everywhere 1 '(a))))
(is (= '((1 a b c) (a 1 b c) (a b 1 c) (a b c 1))
(insert-everywhere 1 '(a b c))))
(is (= '((1 a b c d e)
(a 1 b c d e)
(a b 1 c d e)
(a b c 1 d e)
(a b c d 1 e)
(a b c d e 1))
(insert-everywhere 1 '(a b c d e))))
(is (= '((x 1 2 3 4 5 6 7 8 9 10)
(1 x 2 3 4 5 6 7 8 9 10)
(1 2 x 3 4 5 6 7 8 9 10)
(1 2 3 x 4 5 6 7 8 9 10)
(1 2 3 4 x 5 6 7 8 9 10)
(1 2 3 4 5 x 6 7 8 9 10)
(1 2 3 4 5 6 x 7 8 9 10)
(1 2 3 4 5 6 7 x 8 9 10)
(1 2 3 4 5 6 7 8 x 9 10)
(1 2 3 4 5 6 7 8 9 x 10)
(1 2 3 4 5 6 7 8 9 10 x))
(insert-everywhere 'x '(1 2 3 4 5 6 7 8 9 10)))))

10. The function deep-reverse takes a list as its input. It returns a list with the same elements as its input but in reverse order. If there are any nested lists, these too should be reversed. Unit tests:

(deftest test-deep-reverse
(is (= () (deep-reverse ())))
(is (= '(3 (d c b) a) (deep-reverse '(a (b c d) 3))))
(is (= '(((6 5) 4) 3 (2 1))
(deep-reverse '((1 2) 3 (4 (5 6)))))))

11. The function pack takes a list lst as its argument. If lst contains consecutive repeated elements they should be placed in separate sublists. Unit tests:

(deftest test-pack
(is (= () (pack ())))
(is (= '((a a a a) (b) (c c) (a a) (d) (e e e e))
(pack '(a a a a b c c a a d e e e e))))
(is (= '((1) (2) (3) (4) (5)) (pack '(1 2 3 4 5))))
(is (= '((9 9 9 9 9 9 9 9 9)) (pack '(9 9 9 9 9 9 9 9 9)))))


TIP: This problem can be easily solved using the partition-by predefined function.

12. The function compress takes a list lst as its argument. If lst contains consecutive repeated elements, they should be replaced with a single copy of the element. The order of the elements should not be changed. Unit tests:

(deftest test-compress
(is (= () (compress ())))
(is (= '(a b c d) (compress '(a b c d))))
(is (= '(a b c a d e)
(compress '(a a a a b c c a a d e e e e))))
(is (= '(a) (compress '(a a a a a a a a a a)))))

13. The function encode takes a list lst as its argument. Consecutive duplicates of elements in lst are encoded as vectors [n e], where n is the number of duplicates of the element e. Unit tests:

(deftest test-encode
(is (= () (encode ())))
(is (= '([4 a] [1 b] [2 c] [2 a] [1 d] [4 e])
(encode '(a a a a b c c a a d e e e e))))
(is (= '([1 1] [1 2] [1 3] [1 4] [1 5])
(encode '(1 2 3 4 5))))
(is (= '([9 9]) (encode '(9 9 9 9 9 9 9 9 9)))))

14. The function encode-modified takes a list lst as its argument. It works the same as the previous problem, but if an element has no duplicates it is simply copied into the result list. Only elements with duplicates are converted to [n e] vectors. Unit tests:

(deftest test-encode-modified
(is (= () (encode-modified ())))
(is (= '([4 a] b [2 c] [2 a] d [4 e])
(encode-modified '(a a a a b c c a a d e e e e))))
(is (= '(1 2 3 4 5) (encode-modified '(1 2 3 4 5))))
(is (= '([9 9]) (encode-modified '(9 9 9 9 9 9 9 9 9)))))

15. The function decode takes as its argument an encoded list lst that has the same structure as the resulting list from the previous problem. It returns the decoded version of lst. Unit tests:

(deftest test-decode
(is (= () (decode ())))
(is (= '(a a a a b c c a a d e e e e)
(decode '([4 a] b [2 c] [2 a] d [4 e]))))
(is (= '(1 2 3 4 5) (decode '(1 2 3 4 5))))
(is (= '(9 9 9 9 9 9 9 9 9) (decode '([9 9])))))


## Deliverables

The program source file must include at the top the authors’ personal information (name and student id) within comments. For example:

;----------------------------------------------------------
; Problem Set #2
; Date: February 7, 2019.
; Authors:
;          A01166611 Pepper Pots
;          A01160611 Anthony Stark
;----------------------------------------------------------

Also, each function should include a documentation string (docstring) with a brief description of its behavior. For example:

(defn max2
"Returns the largest of the two numbers x and y."
[x y]
(if (> x y) x y))


To deliver the problemset1.clj file, please provide the following information: