IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture20.

IKI 10100: Data Structures & Algorithms

Ruli Manurung(acknowledgments to Denny & Ade Azurat)

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Fasilkom UI

Ruli Manurung (Fasilkom UI) IKI10100: Lecture 20th Mar 2007

Binary Search Tree

2Ruli Manurung (Fasilkom UI) IKI10100: Lecture 20th Mar 2007

Outline

Concept of Binary Search Tree (BST)

BST operations Find Insert Remove

Running time analysis of BST operations


Binary Search Tree: Properties

Elements have keys (no duplicates allowed).

For every node X in the tree, the values of all the keys in the left subtree are smaller than the key in X and the values of all the keys in the right subtree are larger than the key in X.

The keys must be comparable.

X

<X >X


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Binary Search Tree: Examples


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Binary Search Tree: Examples


Basic Operations

FindMin, FindMax, Find

Insert

Remove


FindMin

Find node with the smallest value

Algorithm: Keep going left until you reach a dead end!

Code:BinaryNode<Type> findMin(BinaryNode<Type> t) { if (t != null) while (t.left != null) t = t.left;

return t;}


FindMax

Find node with the largest value

Algorithm: Keep going right until you reach a dead end!

Code:BinaryNode<Type> findMax(BinaryNode<Type> t) { if (t != null) while (t.right != null) t = t.right;

return t;}


Find

You are given an element to find in a BST. If it exists, return the node. If not, return null.

Algorithm?

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Find: Implementation

BinaryNode<Type> find(Type x, BinaryNode<T> t)

{

while(t!=null)

{

if(x.compareTo(t.element)<0)

t = t.left;

else if(x.compareTo(t.element)>0)

t = t.right;

else

return t; // Match

}

return null; // Not found

}


Insertion: Principle

When inserting a new element into a binary search tree, it will always become a leaf node.

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Insertion: Algorithm

To insert X into a binary search tree: Start from the root If the value of X < the value of the root:

X should be inserted in the left sub-tree. If the value of X > the value of the root:

X should be inserted in the right sub-tree.

Remember that a sub-tree is also a tree.

We can implement this recursively!


Insertion: Implementation

BinaryNode<Type> insert(Type x, BinaryNode<Type> t) { if (t == null) t = new BinaryNode<Type>(x); else if(x.compareTo(t.element)<0) t.left = insert (x, t.left); else if(x.compareTo(t.element)>0) t.right = insert (x, t.right); else throw new DuplicateItemException(x);

return t;}


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Removing An Element

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Removing An Element: Algorithm

If the node is a leaf, simply delete it.

If the node has one child, adjust parent’s child reference to bypass the node.

If the node has two children: Replace the node’s element with the smallest element

in the right subtree and then remove that node, or Replace the node’s element with the largest element in

the left subtree and then remove that node

Introduces new sub-problems: removeMin: Alternatively, removeMax


Removing Leaf

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Removing Node With 1 Child

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Removing Node With 1 Child

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removeMin

BinaryNode<Type> removeMin(BinaryNode<Type> t)

{

if (t == null)

throw new ItemNotFoundException();

else if (t.left != null)

{

t.left = removeMin(t.left);

return t;

}

else

return t.right;

}


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Removing Node With 2 Children


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Removing Root

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Remove BinaryNode<Type> remove(Type x, BinaryNode<Type> t){ if (t == null) throw new ItemNotFoundException(); if (x.compareTo(t.element)<0) t.left = remove(x, t.left); else if(x.compareTo(t.element)>0) t.right = remove(x, t.right); else if (t.left!=null && t.right != null) { t.element = findMin(t.right).element; t.right = removeMin(t.right); } else { if(t.left!=null) t=t.left; else t=t.right; } return t;}


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k < SL + 1

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k == SL + 1

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k > SL + 1

Find k-th element


Find k-th element

BinaryNode<Type> findKth(int k, BinaryNode<Type> t)

{

if (t == null) throw exception;

int leftSize = (t.left != null) ?

t.left.size : 0;

if (k <= leftSize )

return findKth (k, t.left);

else if (k == leftSize + 1)

return t;

else

return findKth ( k - leftSize - 1, t.right);

}


Analysis

Running time for: Insert? Find min? Remove? Find?

Average case: O(log n)

Worst case: O(n)


Binary Search Tree maintains the order of the tree.

Each node should be comparable

All operations take O(log n) - average case, when the tree is equally balanced.

All operations will take O(n) - worst case, when the height of the tree equals the number of nodes.

Summary

IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture20.

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