SML Practice Problems A

credit: Charilaos Skiadas

Coursera Week 2 Extra Practice Problems

Code To Implement

file:	src/main/sml/practice1/practice1.sml
functions:	alternate min_max cumsum greeting repeat addOpt addAllOpt any all ...

alternate

Write a function alternate : int list -> int that takes a list of numbers and adds them with alternating sign. For example alternate [1,2,3,4] = 1 - 2 + 3 - 4 = -2.

min_max

Write a function min_max : int list -> int * int that takes a non-empty list of numbers, and returns a pair (min, max) of the minimum and maximum of the numbers in the list.

cumsum

Write a function cumsum : int list -> int list that takes a list of numbers and returns a list of the partial sums of those numbers. For example cumsum [1,4,20] = [1,5,25].

greeting

Write a function greeting : string option -> string that given a string option \verb|SOME|SOME name returns the string "Hello there, ...!" where the dots would be replaced by name. Note that the name is given as an option, so if it is NONE then replace the dots with "you".

repeat

Write a function repeat : int list * int list -> int list that given a list of integers and another list of nonnegative integers, repeats the integers in the first list according to the numbers indicated by the second list. For example: repeat ([1,2,3], [4,0,3]) = [1,1,1,1,3,3,3].

addOpt

Write a function addOpt : int option * int option -> int option that given two "optional" integers, adds them if they are both present (returning SOME of their sum), or returns NONE if at least one of the two arguments is NONE.

addAllOpt

Write a function addAllOpt : int option list -> int option that given a list of "optional" integers, adds those integers that are there (i.e. adds all the SOME i). For example: addAllOpt ([SOME 1, NONE, SOME 3]) = SOME 4. If the list does not contain any SOME i is in it, i.e. they are all NONE or the list is empty, the function should return NONE.

any

Write a function any : bool list -> bool that given a list of booleans returns true if there is at least one of them that is true, otherwise returns false. (If the list is empty it should return false because there is no true.)

all

Write a function all : bool list -> bool that given a list of booleans returns \verb|true|true if all of them \verb|true|true, otherwise returns false. (If the list is empty it should return true because there is no false.)

zip

Write a function zip : int list * int list -> int * int that given two lists of integers creates consecutive pairs, and stops when one of the lists is empty. For example: zip ([1,2,3], [4, 6]) = [(1,4), (2,6)].

zipRecycle

Challenge: Write a version zipRecycle of zip, where when one list is empty it starts recycling from its start until the other list completes. For example: \verb|zipRecycle ([1,2,3], [1, 2, 3, 4, 5, 6, 7]) =|zipRecycle ([1,2,3], [1, 2, 3, 4, 5, 6, 7]) = \verb|[(1,1), (2,2), (3, 3), (1,4), (2,5), (3,6), (1,7)]|[(1,1), (2,2), (3, 3), (1,4), (2,5), (3,6), (1,7)].

zipOpt

Lesser challenge: Write a version zipOpt of zip with return type \verb|(int * int) list option|(int * int) list option. This version should return \verb|SOME|SOME of a list when the original lists have the same length, and \verb|NONE|NONE if they do not. Write a function \verb|lookup : (string * int) list * string -> int option|lookup : (string * int) list * string -> int option that takes a list of pairs \verb|(s, i)|(s, i) and also a string \verb|s2|s2 to look up. It then goes through the list of pairs looking for the string \verb|s2|s2 in the first component. If it finds a match with corresponding number \verb|i|i, then it returns \verb|SOME i|SOME i. If it does not, it returns \verb|NONE|NONE.

splitup

Write a function splitup : int list -> int list * int list that given a list of integers creates two lists of integers, one containing the non-negative entries, the other containing the negative entries. Relative order must be preserved: All non-negative entries must appear in the same order in which they were on the original list, and similarly for the negative entries.

splitAt

Write a version splitAt : int list * int -> int list * int list of the previous function that takes an extra "threshold" parameter, and uses that instead of 0 as the separating point for the two resulting lists.

isSorted

Write a function isSorted : int list -> boolean that given a list of integers determines whether the list is sorted in increasing order.

isAnySorted

Write a function isAnySorted : int list -> boolean, that given a list of integers determines whether the list is sorted in either increasing or decreasing order.

sortedMerge

Write a function sortedMerge : int list * int list -> int list that takes two lists of integers that are each sorted from smallest to largest, and merges them into one sorted list. For example: \verb|sortedMerge ([1,4,7], [5,8,9]) = [1,4,5,7,8,9]|sortedMerge ([1,4,7], [5,8,9]) = [1,4,5,7,8,9].

qsort

Write a sorting function qsort : int list -> int list that works as follows: Takes the first element out, and uses it as the "threshold" for \verb|splitAt|splitAt. It then recursively sorts the two lists produced by \verb|splitAt|splitAt. Finally it brings the two lists together. (Don't forget that element you took out, it needs to get back in at some point). You could use \verb|sortedMerge|sortedMerge for the "bring together" part, but you do not need to as all the numbers in one list are less than all the numbers in the other.)

divide

Write a function divide : int list -> int list * int list that takes a list of integers and produces two lists by alternating elements between the two lists. For example: \verb|divide ([1,2,3,4,5,6,7]) = ([1,3,5,7], [2,4,6])|divide ([1,2,3,4,5,6,7]) = ([1,3,5,7], [2,4,6]).

Write another sorting function not_so_quick_sort : int list -> int list that works as follows: Given the initial list of integers, splits it in two lists using divide, then recursively sorts those two lists, then merges them together with \verb|sortedMerge|sortedMerge.

fullDivide

Write a function fullDivide : int * int -> int * int that given two numbers \verb|k|k and \verb|n|n it attempts to evenly divide \verb|k|k into \verb|n|n as many times as possible, and returns a pair \verb|(d, n2)|(d, n2) where \verb|d|d is the number of times while \verb|n2|n2 is the resulting \verb|n|n after all those divisions. Examples: \verb|fullDivide (2, 40) = (3, 5)|fullDivide (2, 40) = (3, 5) because \verb|2*2*2*5 = 40|2*2*2*5 = 40 and \verb|fullDivide((3,10)) = (0, 10) |fullDivide((3,10)) = (0, 10) because \verb|3|3 does not divide \verb|10|10.

factorize

Using fullDivide, write a function factorize : int -> (int * int) list that given a number \verb|n|n returns a list of pairs \verb|(d, k)|(d, k) where \verb|d|d is a prime number dividing \verb|n|n and \verb|k|k is the number of times it fits. The pairs should be in increasing order of prime factor, and the process should stop when the divisor considered surpasses the square root of \verb|n|n. If you make sure to use the reduced number \verb|n2|n2 given by \verb|fullDivide|fullDivide for each next step, you should not need to test if the divisors are prime: If a number divides into \verb|n|n, it must be prime (if it had prime factors, they would have been earlier prime factors of \verb|n|n and thus reduced earlier). Examples: \verb|factorize(20) = [(2,2), (5,1)]|factorize(20) = [(2,2), (5,1)]; \verb|factorize(36) = [(2,2), (3,2)]|factorize(36) = [(2,2), (3,2)]; \verb|factorize(1) = []|factorize(1) = [].

multiply

Write a function multiply : (int * int) list -> int that given a factorization of a number \verb|n|n as described in the previous problem computes back the number \verb|n|n. So this should do the opposite of \verb|factorize|factorize.

all_products

Challenge (hard): Write a function all_products : (int * int) list -> int list that given a factorization list result from \verb|factorize|factorize creates a list all of possible products produced from using some or all of those prime factors no more than the number of times they are available. This should end up being a list of all the divisors of the number \verb|n|n that gave rise to the list. Example: \verb|all_products([(2,2), (5,1)]) = [1,2,4,5,10,20]|all_products([(2,2), (5,1)]) = [1,2,4,5,10,20]. For extra challenge, your recursive process should return the numbers in this order, as opposed to sorting them afterwards.

Test

file:	unit_test_practice1.sml
source folder:	src/test/sml/practice1