Objectives:
Given the following implementation of the natural numbers, solve the next few exercises.
trait Nat case object Zero extends Nat case class Succ(x: Nat) extends Nat
4.1.1 Write a function which takes two natural numbers, and returns their sum.
def add(x: Nat, y: Nat): Nat = ???
4.1.2 Write a function which takes two natural numbers, and returns their product.
def multiply(x: Nat, y: Nat): Nat = ???
4.1.3 Write a function which takes an int and converts it to a Nat.
def toNat(x: Int): Nat = ???
Option = carrier (like a box or a container) for a single or no element, of a given type. (Ex. Some(_) or None)
We use Option to write robust functions, in case they return null or fail to return an accepted value.
4.2.1 Let's revisit the function realtrycatch now that we have a type that represents the possibility of error. If an error occurs (try function returns None), the catch function will be called instead.
def realrealtrycatch(t: => Option[Int], c: => Int): Int = { ??? }
(!) 4.2.2 Refactor the function toNat(), so that it takes an integer (a positive or negative number) and returns a “container” of a Nat.
def toNatOpt(x: Int): Option[Nat] = ???
(!) 4.2.3 Refactor the function add(), so that it takes two “containers” of Nats and returns a “container” of a Nat.
def addOpt(x: Option[Nat], y: Option[Nat]): Option[Nat] = ???
Given the following implementation of binary trees, solve the next few exercises.
trait BTree case object EmptyTree extends BTree case class Node(value: Int, left: BTree, right: BTree) extends BTree
4.3.1 Write a function which takes a BinaryTree and returns its depth.
def depth(tree: BTree): Int = ???
4.3.2 Write a function which takes a BinaryTree and returns the number of nodes in its subtree.
def subtree(tree: BTree): Int = ???
4.3.3 Write a function which takes a BinaryTree and returns the number of nodes with even number of children.
def evenChildCount(tree: BTree): Int = ???
4.3.4 Write a function which takes a BinaryTree and flattens it (turns it into a list containing the values of the nodes).
def flatten(tree: BTree): List[Int] = ???
4.3.5 Write a function which takes a BinaryTree and return the number of nodes whose values follow a certain rule.
def countNodes(tree: BTree, cond: Int => Boolean): Int = ???
4.3.6 Write a function which takes a BinaryTree and return mirrored BinaryTree.
def mirror(tree: BTree): BTree = ???
(!) 4.3.7 Write a function which takes two BinaryTree and tries to assign the second tree as a child of the first. It should return a “container” of a BinaryTree .
def append(tree1: BTree, tree2: BTree): Option[BTree] = ???
Given the following implementation of expressions, solve the next few exercises.
trait Expr case class Atom(a: Int) extends Expr case class Add(e1: Expr, e2: Expr) extends Expr case class Mult(e1: Expr, e2: Expr) extends Expr
4.4.1 Write a function which takes an Expression and evaluates it.
def evaluate(e: Expr): Int = ???
4.4.2 Write a function which takes an Expression and simplifies it. (Ex. a * (b + c) → remove parentheses → ab + ac)
def simplify(e: Expr): Expr = ???
4.4.3 Write a function which takes an Expression and removes 'useless' operations. (Ex. a * 1 → a, a + 0 → a)
def optimize(e: Expr): Expr = ???
We shall represent matrices as lists of lists, i.e. values of type [ [Integer ] ]. Each element in the outer list represents a line of the matrix.
Hence, the matrix
$ \displaystyle \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right)$
will be represented by the list [ [1,2,3],[4,5,6],[7,8,9] ].
To make signatures more legible, add the type alias to your code:
type Matrix = List[List[Int]]
which makes the type-name Matrix stand for [ [Integer] ].
4.5.1 Write a function that computes the scalar product with an integer:
$ \displaystyle 2 * \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \\ \end{array}\right)$
def scalarProd(m: Matrix)(v: Int): Matrix = ???
4.5.2 Write a function which adjoins two matrices by extending rows (horizontally):
$ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right) hjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 & 5 & 6 \\ 3 & 4 & 7 & 8\\\end{array}\right) $
def hJoin(m1: Matrix, m2: Matrix): Matrix = ???
4.5.3 Write a function which adjoins two matrices by adding new rows (vertically):
$ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right) vjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6\\ 7 & 8\\ \end{array}\right) $
def vJoin(m1: Matrix, m2: Matrix): Matrix = ???
4.5.4 Write a function which adds two matrices, element by element:
def matSum(m1: Matrix, m2: Matrix): Matrix = ???