Worksheet Algebraic Data Types and Pattern Matching

1. Option Type
2. Classes in Scala
- 2.1. Java Translation
3. Algebraic Data Types via Case classes in Scala
- 3.1. MyList Case Class
- 3.2. Binary Tree Case Class
4. Solutions

1. Option Type

1.1. 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

1.2. 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

1.3. 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

1.4. 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 Scala API documentation for the Option type at https://www.scala-lang.org/api/current/scala/Option.html.

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

Solution: Handle an Option - flatMap

2. Classes in Scala

2.1. 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

3. Algebraic Data Types via Case classes in Scala

3.1. 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

3.2. 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] = {
  ...
}

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

Solution: Binary Tree Case Class

4. Solutions

4.1. Solution: Maximum of a List

def maximum (xs:List[Int]) : Option[Int] = {
  xs match {
    case Nil   => None
    case y::ys => 
      maximum (ys) match {
        case None => Some (y)
        case Some (z) => Some (y max z)
      }
  }
}

Suppose we start with maximum (List(11,31,21)). Computation goes as follows:

 maximum (List(11,31,21))
=> maximum (List(31,21)) match [with y=11] { ... }
=> (maximum (List(21)) match [with y=31] {...}) match [with y=11] { ... }
=> ((maximum (List()) match [with y=21] {...}) match [with y=31] {...}) match [with y=11] { ... }
=> ((None match [with y=21] {...}) match [with y=31] {...}) match [with y=11] { ... }
=> (Some(21) match [with y=31] {...}) match [with y=11] { ... }
=> Some(31 max 21) match [with y=11] { ... }
=> Some(31) match [with y=11] { ... }
=> Some(11 max 31)
=> Some(31)

It’s actually easier to understand if you starting at Nil by running on progressively longer lists:

 maximum(List())
=> None

maximum(List(21))
=> maximum(List()) match {...}
=> None match {...}
=> Some(21)

maximum(List(31,21))
=> maximum(List(21)) match {...}
=> Some(21) match {...}
=> Some(31 max 21)
=> Some(31)

maximum(List(11,31,21))
=> maximum(List(31,21)) match {...}
=> Some(31) match {...}
=> Some(11 max 31)
=> Some(31)

Recall that match is simply short-hand for if-else and is evaluated left to right. So the definition above is equivalent to:

def maximum (xs:List[Int]) : Option[Int] = {
  if (xs == Nil) {            //case Nil
    None
  } else {                    // case y::ys
    val y = xs.head
    val ys = xs.tail
    val zoption = maximum (ys)
    if (zoption == None) {    // case None
      Some (y)
    } else {                  // case Some(z)
      val z = zoption.get
      Some (y max z)
    }
  }
}

4.2. Solution: Find Element

def find [X] (xs:List[X], p:X=>Boolean) : Option[X] = {
  xs match {
    case Nil            => None
    case y::ys if p (y) => Some (y)
    case _::ys          => find (ys, p)
  }
}

This one is easier since tail recursive:

def p = (x:Int) => (x==5)

find(List(11,5,21), p)
=> find(List(5,21), p)
=> Some(5)

find(List(11,21), p)
=> find(List(21), p)
=> None

4.3. Solution: Handle an Option - Pattern Matching

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

def twoFind (xs:List[String], s:String) : Option[String] = {
  oneFind (xs, s) match {
    case None     => None
    case Some (t) => oneFind (xs, t)
  }
}

4.4. Solution: Handle an Option - flatMap

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

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

4.5. Solution: Classes - Java Translation

object Counter:
  private var numCounters = 0

  def getNumCounters : Int = numCounters
  def incNumCounters : Unit = numCounters = numCounters + 1

class Counter (initial:Int):
  private val id:Int = Counter.getNumCounters
  private var count:Int = initial
  Counter.incNumCounters

  def getId () : Int = id
  def getNextCount () : Int = { val tmp = count; count = count + 1; tmp }

4.6. Solution: MyList Case Class

enum MyList[+X]:
  case MyNil
  case MyCons(head:X, tail:MyList[X])
import MyList.*

def length [X] (xs:MyList[X]) : Int =
  xs match
    case MyNil => 0
    case MyCons (_, ys) => 1 + length (ys)

def toList [X] (xs:MyList[X]) : List[X] =
  xs match
    case MyNil => Nil
    case MyCons (y, ys) => y::toList(ys)

def fromList [X] (xs:List[X]) : MyList[X] =
  xs match
    case Nil => MyNil
    case y::ys => MyCons (y, fromList (ys)) 

def append [X] (xs:MyList[X], ys:MyList[X]) : MyList[X] =
  xs match
    case MyNil => ys
    case MyCons (z, zs) => MyCons (z, append (zs, ys)) 

def map [X,Y] (xs:MyList[X], f:X=>Y) : MyList[Y] =
  xs match
    case MyNil => MyNil
    case MyCons (y, ys) => MyCons (f (y), map (ys, f)) 

val xs = MyCons(11, MyCons(21, MyCons(31, MyNil)))
val len = length (xs)

4.7. Solution: Binary Tree Case Class

import javax.swing.text.AbstractDocument.LeafElement
enum Tree[+X]:
  case Leaf
  case Node(l:Tree[X], c:X, r:Tree[X])
import Tree.*

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 =
  t match
    case Leaf => 0
    case Node (t1, _, t2) => size (t1) + 1 + size (t2)

// Insert a number into a sorted binary tree.
def insert [X] (x:X, t:Tree[X], lt:(X,X)=>Boolean) : Tree[X] =
  t match
    case Leaf => Node (Leaf, x, Leaf)
    case Node (t1, c, t2) if (lt (x, c)) => Node (insert (x, t1, lt), c, t2)
    case Node (t1, c, t2) if (lt (c, x)) => Node (t1, c, insert (x, t2, lt))
    case Node (t1, c, t2)                => Node (t1, c, t2)

// Put the elements of the tree into a list using an in-order traversal.
def inorder [X] (t:Tree[X]) : List[X] =
  t match
    case Leaf => Nil
    case Node (t1, c, t2) => inorder (t1) ::: List (c) ::: inorder (t2)

def map [X,Y] (t:Tree[X], f:X=>Y) : Tree[Y] =
  t match
    case Leaf => Leaf
    case Node (t1, c, t2) => Node (inorder (t1), f (c), inorder (t2))