The document discusses data structures in Scala, focusing on queues and binary search trees (BST). It describes the functional queue implementation, operations, and their efficiencies, as well as tree representations, and provides algorithms for in-order, pre-order, and post-order traversals for BSTs. References are also included for further reading on algorithms and functional data structures.

1. Data Structures In Scala
Meetu Maltiar
Principal Consultant
Knoldus

2. Agenda
Queue
Binary Tree
Binary Tree Traversals

3. Functional Queue
Functional Queue is a data structure that has three
operations:
head: returns first element of the Queue
tail: returns a Queue without its Head
enqueue: returns a new Queue with given element at Head
Has therefore First In First Out (FIFO) property

4. Functional Queue Continued
scala> val q = scala.collection.immutable.Queue(1, 2, 3)
q: scala.collection.immutable.Queue[Int] = Queue(1, 2, 3)
scala> val q1 = q enqueue 4
q1: scala.collection.immutable.Queue[Int] = Queue(1, 2, 3, 4)
scala> q
res3: scala.collection.immutable.Queue[Int] = Queue(1, 2, 3)
scala> q1
res4: scala.collection.immutable.Queue[Int] = Queue(1, 2, 3, 4)

5. Simple Queue Implementation
class SlowAppendQueue[T](elems: List[T]) {
def head = elems.head
def tail = new SlowAppendQueue(elems.tail)
def enqueue(x: T) = new SlowAppendQueue(elems ::: List(x))
}
Head and tail operations are fast. Enqueue operation is slow as its performance is
directly proportional to number of elements.

6. Queue Optimizing Enqueue
class SlowHeadQueue[T](smele: List[T]) {
// smele is elems reversed
def head = smele.last // Not efficient
def tail = new SlowHeadQueue(smele.init) // Not efficient
def enqueue(x: T) = new SlowHeadQueue(x :: smele)
}
smele is elems reversed. Head operation is not efficient. Neither is tail operation. As both
last and init performance is directly proportional to number of elements in Queue

7. Functional Queue
class Queue[T](private val leading: List[T], private val trailing:
List[T]) {
private def mirror =
if (leading.isEmpty) new Queue(trailing.reverse, Nil)
else this
def head = mirror.leading.head
def tail = {
val q = mirror
new Queue(q.leading.tail, q.trailing)
}
def enqueue(x: T) = new Queue(leading, x :: trailing)
}

8. Binary Search Tree
BST is organized tree.
BST has nodes one of them is specified as Root node.
Each node in BST has not more than two Children.
Each Child is also a Sub-BST.
Child is a leaf if it just has a root.

9. Binary Search Property
The keys in Binary Search Tree is stored to satisfy
following property:
Let x be a node in BST.
If y is a node in left subtree of x
Then Key[y] less than equal key[x]
If y is a node in right subtree of x
Then key[x] less than equal key[y]

10. Binary Search Property
The Key of the root is 6
The keys 2, 5 and 5 in left subtree is no
larger than 6.
The key 5 in root left child is no smaller
than the key 2 in that node's left
subtree and no larger than key 5 in the
right sub tree

11. Tree Scala Representation
case class Tree[+T](value: T, left:
Option[Tree[T]], right: Option[Tree[T]])
This Tree representation is a recursive definition and has type
parameterization and is covariant due to is [+T] signature
This Tree class definition has following properties:
1. Tree has value of the given node
2. Tree has left sub-tree and it may have or do not contain value
3. Tree has right sub-tree and it may have or do not contain value
It is covariant to allow subtypes to be contained in the Tree

12. Tree In-order Traversal
BST property enables us to print out all
the Keys in a sorted order using simple
recursive In-order traversal.
It is called In-Order because it prints
key of the root of a sub-tree between
printing of the values in its left sub-
tree and printing those in its right sub-
tree

13. Tree In-order Algorithm
INORDER-TREE-WALK(x)
1. if x != Nil
2. INORDER-TREE-WALK(x.left)
3. println x.key
4. INORDER-TREE-WALK(x.right)
For our BST in example before the output expected will be:
255678

14. Tree In-order Scala
def inOrder[A](t: Option[Tree[A]], f: Tree[A] =>
Unit): Unit = t match {
case None =>
case Some(x) =>
if (x.left != None) inOrder(x.left, f)
f(x)
if (x.right != None) inOrder(x.right, f)
}

15. Tree Pre-order Algorithm
PREORDER-TREE-WALK(x)
1. if x != Nil
2. println x.key
3. PREORDER-TREE-WALK(x.left)
4. PREORDER-TREE-WALK(x.right)
For our BST in example before the output expected will be:
652578

16. Tree Pre-order Scala
def preOrder[A](t: Option[Tree[A]], f: Tree[A]
=> Unit): Unit = t match {
case None =>
case Some(x) =>
f(x)
if (x.left != None) inOrder(x.left, f)
if (x.right != None) inOrder(x.right, f)
}
Pre-Order traversal is good for creating a copy of the Tree

17. Tree Post-Order Algorithm
POSTORDER-TREE-WALK(x)
1. if x != Nil
2. POSTORDER-TREE-WALK(x.left)
3. POSTORDER-TREE-WALK(x.right)
4. println x.key
For our BST in example before the output expected will be:
255876
Useful in deleting a tree. In order to free up resources a
node in the tree can only be deleted if all the children (left
and right) are also deleted
Post-Order does exactly that. It processes left and right
sub-trees before processing current node

18. Tree Post-order Scala
def postOrder[A](t: Option[Tree[A]], f: Tree[A]
=> Unit): Unit = t match {
case None =>
case Some(x) =>
if (x.left != None) postOrder(x.left, f)
if (x.right != None) postOrder(x.right, f)
f(x)
}

19. References
1. Cormen Introduction to Algorithms
2. Binary Search Trees Wikipedia
3. Martin Odersky “Programming In Scala”
4. Daniel spiewak talk “Extreme Cleverness:
Functional Data Structures In Scala”

20. Thank You!!