In this homework, you will implement a binary search tree, that you will use to gather stats about words from a particular text. Generally, in a binary search tree:

• each non-empty node contains exactly one value and two children
• all values from the left sub-tree are smaller or equal to that of the current node
• all values from the right sub-tree are larger or equal to that of the current node

In your project, the value of each node will be represented by Token objects. The class Token is already implemented for you:

case class Token(word: String, freq: Int)

A token stores:

• the number of occurrences, or frequency freq of a string word, in a text.

Your binary search tree will use frequencies as an ordering criterion. For instance, the text: All for one and one for one, may be represented by the tree:

      for (2)
      /   \
 and (1)  one (3)
  /            
all (1)            

Notice that there are multiple possible BS trees to represent one text, however you do not need to take this into account in this homework. Our tree is called WTree, and is implemented by the following case classes:

case object Empty extends WTree
case class Node(word: Token, left: WTree, right: WTree) extends WTree 

WTree implements the following trait:

trait WTreeInterface {
  def isEmpty: Boolean
  def filter(pred: Token => Boolean): WTree
  def ins(w: Token): WTree
  def contains(s:String): Boolean
  def size: Int
}

The method ins is already implemented, but the rest must be implemented by you. The project has two parts:

• building a WTree from a text, and
• using a WTree, to gather info about that particular text.

In the next section you will find implementation details about each of the above.

1. Write a function which splits a text using the single whitespace character as a separator. Multiple whitespaces should be treated as a single separator. If the list contains only whitespaces, split should return the empty list. (Hints: Your implementation must be recursive, but do not try to make it tail-recursive. It will make your code unnecessarily complicated. Several patterns over lists, in the proper order will make the implementation cleaner.)

/*  split(List('h','i',' ','t','h','e','r','e')) = List(List('h','i'), List('t','h','e','r','e'))
*/
def split(text: List[Char]): List[List[Char]] = ???

2. Write a function which computes a list of Token from a list of strings. Recall that Tokens keep track of the string frequency. Use an auxiliary function insWord which inserts a new string in a list of Tokens. If the string is already a token, its frequency is incremented, otherwise it is added as a new token. (Hint: the cleanest way to implement aux is to use one of the two folds).

def computeTokens(words: List[String]): List[Token] = {
    /* insert a new string in a list of tokens */
    def insWord(s: String, acc: List[Token]): List[Token] = ???
    def aux(rest: List[String], acc: List[Token]): List[Token] = ???
    ???
  }

3. Write a function tokensToTree which creates a WTree from a list of tokens. Use the insertion function ins which is already implemented. (Hint: you can implement it as a single fold call, but you have to choose the right one)

def tokensToTree(tokens: List[Token]): WTree = ??

4. Write a function makeTree which takes a string and builds a WTree. makeTree relies on all the previous functions you implemented. You should use _.toList, which converts a String to List[Char]. You can also use andThen, which allows writing a concise and clear implementation. andThen is explained in detail in the next section.

def makeTree(s:String): WTree = ???

5. Implement the member method size, which must return the number of non-empty nodes in the tree.

6. Implement the member method contains, which must check if a string is a member of the tree (no matter its frequency).

7. Implement the filter method in the abstract class WTree. Filter will rely on the tail-recursive filterAux method, which must be implemented in the case classes Empty and Node.

8. In the code template you will find a string: scalaDescription.

Compute the number of occurrences of the keyword “Scala” in scalaDescription. Use word-trees and any of the previous functions you have defined.

def scalaFreq: Int = ??? 

9. Find how many programming languages are referenced in the same text. You may consider that a programming language is any keyword which starts with an uppercase character. To reference character i in a string s, use s(i). You can also use the method _.isUpper.

def progLang: Int = ???

10. Find how many words which are not prepositions or conjunctions appear in the same text. You may consider that a preposition or conjunction is any word whose size is less or equal to 3.

def wordCount : Int = ???

Note: In order to be graded, exercises 5 to 9 must rely on a correct implementation of the previous parts of the homework.

Suppose you want to apply a sequence of transformations over an object o. Some of them may be functions (f, g) while other may be member functions (m1,m2). Instead of defining expressions such as: g(f(o).m1).m2 which reflects the sequence: f, m1, g, m2 of transformations on object o, you can instead use andThen:

val sequence = 
   (x => f(x))
      andThen (_.m1)
      andThen (x => g(x))
      andThen(_.m2)

which is more legible and easy to debug.

Sets are unordered collections of unique elements. There are several ways to store sets. One of them relies on characteristic functions. Such functional sets are especially useful if we expect many insert/retrieve operations and less traversals in our code.

A characteristic function of a set $ A \subseteq U$ is a function $ f: U \rightarrow \{0,1\}$ which assigns $ f(x) = 1$ for each element $ x \in A$ and $ f(x) = 0$ for each element $ x \not\in A$ .

In our implementation, $ U$ will be the set of integers, hence we shall encode only sets of integers. Hence, the type of a set will be:

type Set = Int => Boolean

For instance, the set $ \{1,2,3\}$ will be encoded by the anonymous function:

(x: Int) => (x == 1 || x == 2 || x == 3)

Also, the empty set can be encoded as:

(x: Int) => false

while the entire set of integers may be encoded as:

(x: Int) => true

1. Write a function singleton which takes an integer and returns the set containing only that integer:

def singleton(x: Int): Set = ???

Note that singleton could have been equivalently defined as: def singleton(x: Int)(e: Int): Boolean = ???, however, the previous variant is more legible, in the sense that it highlights the idea that we are returning set objects, namely characteristic functions.

2. Write a function member which takes a set and an integer and checks if the integer is a member of the set. Note that member should be defined and called as a curry function:

def member(e: Int)(set: Set): Boolean = ???

3. Write a function ins which inserts a new element in a set. More precisely, given $ x$ and $ set$ , ins returns a new set $ \{x\} \cup set$ .

def ins(x: Int)(set: Set): Set = ???

4. Write a function fromBounds which takes two integer bounds start and stop and returns the set $ \{start, start+1, \ldots, stop\}$ . It is guaranteed that $ start \leq stop$ (you do not need to check this condition in your implementation).

def fromBounds(start: Int, stop: Int): Set = ???

5. Write the function which performs the union of two sets:

def union(set1: Set, set2: Set): Set = ???

6. Write a function which computes the complement of a set with respect to the set of integers:

def complement(s1: Set): Set = ???

7. Write a function which computes the sum of value b to all elements from a set, for given bounds. Use a tail-end recursive function:

  def sumSet(b: Int)(start: Int, stop: Int)(set: Set): Int = {
    def auxSum(crt: Int, acc: Int): Int = ???
    ???
  }

8. Generalise the previous function such that we can fold a set using any binary commutative operation over integers. Make sure this is a left fold: Folding the set: {x,y,z} with b should produce: ( (b op x) op y) op z

  def foldLeftSet
     (b:Int)                  // initial value 
     (op: (Int,Int) => Int)   // folding operation
     (start: Int, stop: Int)  // bounds (inclusive)
     (set: Set): Int = ???    // the set to be folded

9. Implement an alternative to the previous function, namely foldRight. Applying foldRight on the set {x,y,z} with b should produce: a op (b op (c op b)). Use direct recursion instead of tail recursion.

  def foldRightSet
     (b:Int)                  // initial value 
     (op: (Int,Int) => Int)   // folding operation
     (start: Int, stop: Int)  // bounds (inclusive)
     (set: Set): Int = ???    // the set to be folded

10. Implement operation filter which takes a set and returns another one containing only those elements that satisfy the predicate:

def filter(p: Int => Boolean)(set: Set): Set = ???

11. Implement a function which partitions a set into two sets. The left-most contains those elements that satisfy the predicate, while the right-most contains those elements that do not satisfy the predicate. Use pairs. A pair is constructed with simple parentheses. E.g. (1,2) is a pair of two integers. Suppose val p: (Int,Int) is another pair of two integers. Then p._1 is the left-most part of the pair while p._2 is the right-most part of the pair.

  def partition(p: Int => Boolean)(set: Set): (Set,Set) = ???

12. Implement a function forall which checks if all elements in a given range of a set satisfy a predicate (condition). (Such a condition may be that all elements from given bounds are even numbers).

  def forall(cond: Int => Boolean) // condition to be checked
            (start: Int, stop: Int) // start,stop values (inclusive)
            (set: Set): Boolean  // set to be checked
            = ???

13. Implement a function exists which checks if a predicate holds for some element from the range of a set. Hint: it is easier to implement exists using the logical relation: $ \exists x. P(X) \iff \lnot \forall x.\lnot P(X)$ .

14. Implement the function setOfDivByK which returns the set of integers divisible by a value k. Use the appropriate functions you have defined.

def setOfDivByK(k: Int): Set = ??

15. Implement the function moreDivs which verifies if set1 contains more divisors of k than set2, over the range [start,stop]. Use any combination of the previous functions you have defined for your implementation.

def moreDivs(k: Int)(start: Int, stop:Int)(set1: Set, set2: Set): Boolean = ???

• Please follow the Submission guidelines which are the same for all homework.
• To solve your homework, download the Project template, import it in IntellIJ, and you are all set. Do not rename the project manually, as this may cause problems with IntellIJ.