CS201 Mid-term Examination

Michael P. Fourman

February 2, 2010

University of Western Australia
CS201 Mid-term Examination
Ross Lecture Theatre
Thursday 14th April, 1994
12.00 – 1.00pm



  1. Short Question 5 marks
    Give the responses of the ML system to the following sequence of declarations
    val a = 1;  
    val b = 2;  
    val c = 3;  
    fun f a = let val b = a + c in a + b end;  
    val b = 5;  
    f b;


  1. Long Question 10 marks
    The following datatype can be used to represent trees whose nodes can have an arbitrary number of children.
    datatype  ’a  Tree  = Tree  of ’a  * ’a  Tree  list
    1. What tree does the following expression denote (draw a picture):
      Tree (1, [Tree (2,  [ ]),  Tree (3, [Tree (4, [ ])] ),Tree (5,[  ])])
    2. Define a function to calculate the number of nodes in such a tree.
    3. We assign a level to each node in a tree as follows. The node at the root is at level 1. Its children are at level 2. Their children are at level 3 and so on.

      Define a function countLevel : int -> ’a Tree -> int that counts the number of nodes at a given level of a tree. The expression, countLevel n t, should return the number of nodes at level n in the tree t.


  2. Long Question 10 marks
    The EQueue signature is like the signature Queue, but is extended with an additional operation multiple enqueue, menq:(Item  list * Queue) -> Queue, intended to add a number of items to the queue in a single operation. The intention is that the items enqueued by a single menq operation may be dequeued in any order, but they must all be dequeued after any items entered in the queue by an earlier enq or menq operation, and before any items entered by any later operation.
    signature EQueue =  
        type Item  
        type Queue  
        val empty : Queue  
        val enq : (Item * Queue) -> Queue  
        val deq : Queue -> (Item * Queue)  
        val menq: (Item list * Queue) -> Queue  

    An implementation of a queue, including this operation, uses the type declaration

    type Queue = (Item list list) * (Item list list)

    the operations empty and menq are implemented as follows:

    val empty = ([],[])  
    fun menq(items, (enter, leave)) = (items :: enter, leave)

    1. Complete the following declarations of the functions enq and deq for this implementation
      fun enq(item, ([],leave))       =  
        | enq(item, ((h :: t),leave)) =  
      fun deq(enter,  (h :: t) :: r)  =  
        | deq(enter,  [] :: r      )  =  
        | deq(h :: t, []           )  =  
        | deq([],     []           )  =

    2. What is the complexity of the three operations
      1. enq,
      2. deq,
      3. menq

      for this implementation?


  3. Long Question 10 marks
    The PQueue signature is like the signature Queue, but is extended with an additional operation merge:(Queue * Queue) -> Queue, intended to merge together two queues.
    signature PQueue =  
        type Item  
        type Queue  
        val empty : Queue  
        val enq : (Item * Queue) -> Queue  
        val deq : Queue -> (Item * Queue)  
        val merge: (Queue * Queue) -> Queue  

    An implementation of a priority queue of integer priorities represents the queue by a list kept in order of decreasing priority:

    type Item  = int  
    type Queue = Item list

    Here is the function deq: Queue -> int * Queue from this implementation

    fun deq [] = raise Deq  
      | deq (h :: t) = (h, t)

    1. Give an implementation of the operation enq : (int*Queue) -> Queue, compatible with this representation
    2. Give an O(n) implementation of the operation merge: Queue * Queue -> Queue, compatible with this representation.
    3. Consider an alternative representation for a priority queue, using an unordered list to represent the queue. For this representation, the enq operation is simple
      fun enq (e, q) = e :: q

      Complete the following table giving the complexity of the operations for each representation. (You are not asked to implement all the operations.)


      enq O(1)

      deq O(1)



The End (C) Michael Fourman 1994-2006