Description
Q1. [25 points]
Write a function common that takes a pair of lists and forms a new list containing a unique
copy of each element that occurs in both lists. Here is the type, as echoed by the interpreter,
and an example.
val common : ’a list * ’a list -> ’a list =
# let l1 = [1;3;2;4;1;5;6;3];;
val l1 : int list = [1; 3; 2; 4; 1; 5; 6; 3]
# let l2 = [3;9;8;2;11;21;3];;
val l2 : int list = [3; 9; 8; 2; 11; 21; 3]
# common (l1,l2);;
– : int list = [3; 2]
It does not matter if the final list is sorted or not. The input lists are not necessarily
sorted.
The following three questions are about merge sort. The mergesort algorithm is a recursive
algorithm for sorting lists which runs in time O(n log n). The items in the list must have an
order relation defined on them, otherwise sorting does not make sense of course.
The idea is as follows: the given list l is split into two equal (if the length of l is odd then
one of the “halves” is one item longer than the other) lists l1 and l2. These lists are sorted
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recursively and then the results are merged back to give a single sorted list. Code this
in OCaml. Your algorithm can use < as a comparison operator. Your code must have a
function split that produces a pair of lists, a function merge that merges sorted lists and a
function mergesort that implements the overall algorithm.
Q2. [25 points]
In this question you will implement split with the following type:
val split : ’a list -> ’a list * ’a list =
# split [1;3;2;4;5;6;9;11;17;13;12];;
– : int list * int list = ([1; 2; 5; 9; 17; 12], [3; 4; 6; 11; 13])
Q3. [25 points] In this question you will implement the merge algorithm. The inputs are
sorted lists.
val merge : ’a list * ’a list -> ’a list =
# let l3 = [1; 3; 5; 7; 9];;
val l3 : int list = [1; 3; 5; 7; 9]
# let l4 = [2; 4; 6; 8];;
val l4 : int list = [2; 4; 6; 8]
# merge (l3,l4);;
– : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9]
Q4.[25 points] Finally we complete the mergesort algorithm. Here is the type and an example.
val mergesort : ’a list -> ’a list =
# mergesort [10;2;8;5;1;4;3;9;7;6];;
– : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]
# mergesort [1;3;2;4;1;2;5;3];;
– : int list = [1; 1; 2; 2; 3; 3; 4; 5]
Q5. [0 points] This question is for your spiritual growth only. Do not think it will give you
extra credit or help you learn the material better. It will however stretch your brain in other
directions. Do not attempt it if you have not yet finished the required homework. Do not
submit a solution; please talk to me if you have solved it. Do not worry about it if you don’t
understand the question.
How many comparisons are required to find the largest member of a list of n elements? How
many are required to find the largest and the smallest. Here “required” means that in the
worst case you will have to do that many comparisons. Can you find the smallest and the
largest with fewer than 2n − 3 comparisons? Note that we are interested in exact counts;
not just in O(·) estimates.
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