Questions and Completion

To receive credit for completing the worksheet, you must complete the corresponding quiz (it is just a checkbox) on D2L when you have finished the worksheet.

If you have questions as you go through this worksheet, please feel free to post them on the discussion forum.

Option Type

Maximum of a List

Write a recursive function that returns the maximum of a list of integers.

Instead of throwing an exception when the list is empty, return an Option type. That is, your function should have the following form.

  def maximum (xs:List[Int]) : Option[Int] = {
    ...
  }

Solution: Maximum of a List

Find Element

Write a higher-order function that finds the first element in a list that satisfies a given predicate (function taking an element of the list and returning a Boolean value).

Instead of throwing an exception when no such element is found, return an Option type. That is, your function should have the following form.

  def find [X] (xs:List[X], p:X=>Boolean) : Option[X] = {
    ...
  }

Solution: Find Element

Handle an Option - Pattern Matching

Consider the following Scala function. It searches for the first String in a list of String references that is strictly longer than s and that contains s as a substring.

  def oneFind (xs:List[String], s:String) : Option[String] = {
    xs.find ((x:String) => (x.length > s.length) && x.contains (s))
  }

For example:

scala> oneFind (List ("the", "rain", "in", "spain"), "i")
res1: Option[String] = Some(rain)

scala> oneFind (List ("the", "rain", "in", "spain"), "in")
res2: Option[String] = Some(rain)

scala> oneFind (List ("the", "rain", "in", "spain"), "ain")
res3: Option[String] = Some(rain)

scala> oneFind (List ("the", "rain", "in", "spain"), "pain")
res16: Option[String] = Some(spain)

scala> oneFind (List ("the", "rain", "in", "spain"), "rain")
res4: Option[String] = None

Now write a function twoFind that calls oneFind and then uses the result (if there is one) in a second call to oneFind.

Use pattern-matching to do case analysis on the result of the first call to oneFind. Your definition of twoFind should have the following form.

  def twoFind (xs:List[String], s:String) : Option[String] = {
    ...
  }

Note that twoFind must not have return type Option[Option[String]]!

For example:

twoFind (List ("the", "rain", "in", "spain"), "i")
res1: Option[String] = None

scala> twoFind (List ("the", "rain", "in", "spain"), "in")
res2: Option[String] = None

scala> twoFind (List ("the", "rain", "in", "spain"), "n")
res3: Option[String] = None

scala> twoFind (List ("the", "rain", "in", "spain", "trained", "after", "a", "sprain"), "in")
res4: Option[String] = Some(trained)

scala> twoFind (List ("the", "rain", "in", "spain", "trained", "after", "a", "sprain"), "i")
res5: Option[String] = Some(trained)

scala> twoFind (List ("the", "rain", "in", "spain", "trained", "after", "a", "sprain"), "n")
res6: Option[String] = Some(trained)

scala> twoFind (List ("the", "rain", "in", "spain", "trained", "after", "a", "sprain"), "ain")
res7: Option[String] = Some(trained)

scala> twoFind (List ("the", "rain", "in", "spain", "trained", "after", "a", "sprain"), "pain")
res8: Option[String] = None

Solution: Handle an Option - Pattern Matching

Handle an Option - flatMap

Rewrite twoFind from the previous question to use flatMap for the Option type instead of explicit pattern-matching.

Note: if e has type Option[A] and you write e.flatMap (f), then it will have type Option[B] whenever f has type A=>Option[B]. You may find it helpful to look at the description for the Option type in the Scala API documentation.

Variations of flatMap are idiomatic in many languages that use Option types.

Solution: Handle an Option - flatMap

Tail Recursion

Scheme - Looping in Scheme

Write a Scheme expression that uses a let special form with a loop variable to print the following:

1 potato
2 potato
3 potato
4 potato

You should not define a function for this.

Solution: Scheme - Looping in Scheme

Scheme - Looping in Scheme with a Function

Write Scheme code that uses a tail-recursive function to print the following:

1 potato
2 potato
3 potato
4 potato

You should define a function for this.

Solution: Scheme - Looping in Scheme with a Function

Scala - Looping in Scala with a Function

Write Scala code that uses a tail-recursive function to print the following:

1 potato
2 potato
3 potato
4 potato

You should define a function for this.

Include a tailrec annotation so that the Scala system tells you if your function is not tail recursive.

Solution: Scala - Looping in Scala with a Function

Scheme - Counting Values

Here is Java code to count the number of integers in a linked list that are between low and high values.

public class CountingValues {
  static class Node {
    int item;
    Node next;

    public Node (int item, Node next) { 
      this.item = item; 
      this.next = next; 
    }
  }

  static int count (Node data, int low, int high) {
    int result = 0;
    while (data != null) {
      if (low <= data.item && data.item <= high) {
        result = result + 1;
      }
      data = data.next;
    }
    return result;
  }

  public static void main (String[] args) {
    Node data = null;
    for (int i = args.length - 1; i >= 0; i--) {
      data = new Node (Integer.parseInt (args[i]), data);
    }

    System.out.printf ("count (..., 5, 10) = %d%n", count (data, 5, 10));
  }
}

$ javac CountingValues.java 
$ java CountingValues 1 2 3 4 5 6 7 8 9 10 11 12
count (..., 5, 10) = 6

Your task is to rewrite the count function in Scheme. Your definition should be a tail-recursive function.

Solution: Scheme - Counting Values

Scala - Counting Values

Repeat the previous “Counting Values” task in Scala. Your definition should be a tail-recursive function.

Solution: Scala - Counting Values

Scala - Which Functions Are Tail Recursive?

Consider each of the following Scala functions. For each Scala function, is it tail recursive? Test your answers by adding a tailrec annotation and pasting the function definition into the Scala console.

/* The Scala library includes many other functions on lists that are common.
 * Below we define our own versions of these functions for pedagogical purposes,
 * but normally the library versions would be used instead.
 */

def append [X] (xs:List[X], ys:List[X]) : List[X] =
  xs match
    case Nil   => ys
    case z::zs => z :: append (zs, ys)

def flatten [X] (xss:List[List[X]]) : List[X] =
  xss match
    case Nil     => Nil
    case ys::yss => ys ::: flatten (yss)

/* The "take" function takes the first n elements of a list. */
def take [X] (n:Int, xs:List[X]) : List[X] =
  if n <= 0 then
    Nil
  else
    xs match
      case Nil   => Nil
      case y::ys => y :: take (n - 1, ys)

/* The "drop" function drop the first n elements of a list. */
def drop [X] (n:Int, xs:List[X]) : List[X] =
  if n <= 0 then
    xs
  else 
    xs match
      case Nil   => Nil
      case y::ys => drop (n - 1, ys)

val sampleList : List[Int] = (1 to 12).toList
val takeresult : List[Int] = take (3, sampleList)
val dropresult : List[Int] = drop (3, sampleList)

/* Reverse a list.  This version is straightforward, but inefficient.  Revisited later on. */
def reverse [X] (xs:List[X]) : List[X] =
  xs match
    case Nil   => Nil
    case y::ys => reverse (ys) ::: List (y)

/* zip takes two lists and creates a single list containing pairs from the two lists
 * that occur at the same position.  The definition shows more sophisticated pattern
 * matching: the use of wildcards; overlapping patterns; and decomposing pairs and
 * lists simultaneously.  Note that the "xs" and "ys" in the third case shadow the
 * "xs" and "ys" parameters to the function.
 */
def zip [X,Y] (xs:List[X], ys:List[Y]) : List[(X,Y)] = (xs, ys) match
  case (Nil, _)       => Nil
  case (_, Nil)       => Nil
  case (x::xs, y::ys) => (x, y) :: zip (xs, ys)

val ziplist = zip (List (1, 2, 3), List ("the", "rain", "in", "spain"))

/* unzip decomposes a list of pairs into a pair of lists.
 * The recursive case illustrates pattern-matching the result of the
 * recursive call in order to apply an operation to the elements.
 */
def unzip [X,Y] (ps:List[(X,Y)]) : (List[X], List[Y]) =
  ps match
    case Nil        => (Nil, Nil)
    case (x, y)::qs => 
      val (xs, ys) = unzip (qs)
      (x :: xs, y :: ys)

val unziplist = unzip (ziplist)

def reverse2 [X] (xs:List[X]) : List[X] =
  def aux (xs:List[X], result:List[X]) : List[X] = xs match
    case Nil   => result
    case y::ys => aux (ys, y :: result)

  aux (xs, Nil)

Lists

flatMap for Lists

What is the result of evaluating the following Scala expressions?

  def repeat [X] (x:X, n:Int) : List[X] = {
    if n == 0 then 
      Nil
    else 
      x :: repeat (x, n - 1)
  }

  val xs:List[(Char,Int)] = List (('a', 2), ('b', 4), ('c', 8))

  val ys = xs.map ((p:(Char,Int)) => repeat (p._1, p._2))

  val zs = xs.flatMap ((p:(Char,Int)) => repeat (p._1, p._2))

What types do ys and zs have, and how does flatMap differ from map?
Try to express flatMap using map and flatten.

Solution: flatMap for Lists

Classes in Scala

Java Translation

Translate the following Java class definition into a corresponding Scala class definition. You will need a companion object definition to model the static components.

In order to definition the Scala class and object at the same time in the Scala REPL, you must enter :paste, paste your definitions together, and then press control-D.

class Counter {
  private static int numCounters = 0;

  final int id;
  int count;

  Counter (int initial) {
    this.id = numCounters;
    this.count = initial;
    numCounters++;
  }

  static int getNumCounters () {
    return numCounters;
  }

  int getId () {
    return this.id;
  }

  int getNextCount () {
    return this.count++;
  }
}

Solution: Classes - Java Translation

Algebraic Data Types via Case classes in Scala

MyList Case Class

Define your own MyList class in Scala (as in the slides).

Now implement the following functions:

enum MyList[+X] {
  ...
}
import MyList._

  def length [X] (xs:MyList[X]) : Int = {
    ...
  }

  def toList [X] (xs:MyList[X]) : List[X] = {
    ...
  }

  def fromList [X] (xs:List[X]) : MyList[X] = {
    ...
  }

  def append [X] (xs:MyList[X], ys:MyList[X]) : MyList[X] = {
    ...
  }

  def map [X,Y] (xs:MyList[X], f:X=>Y) : MyList[Y] = {
    ...
  }

Solution: MyList Case Class

Binary Tree Case Class

Define your own Tree class for binary trees in Scala (as in the slides).

Now implement the following functions:

enum Tree[+X] {
  case Leaf
  case Node(l:Tree[X], c:X, r:Tree[X])
}
import Tree.{Leaf,Node}

val tree1:Tree[Int] = Leaf

val tree2:Tree[Int] = Node (Leaf, 5, Leaf)

val tree3:Tree[Int] = Node (Node (Leaf, 2, Leaf), 3, Node (Leaf, 4, Leaf))

// Find the size of a binary tree.  Leaves have size zero here.
def size [X] (t:Tree[X]) : Int = {
  ...
}

// Insert a number into a sorted binary tree.
def insert [X] (x:X, t:Tree[X], lt:(X,X)=>Boolean) : Tree[X] = {
  ...
}

// Put the elements of the tree into a list using an in-order traversal.
def inorder [X] (t:Tree[X]) : List[X] = {
  ...
}

Solution: Binary Tree Case Class

Solutions