IKI 10100I: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1...
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Transcript of IKI 10100I: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1...
IKI 10100I: Data Structures & Algorithms
Ruli Manurung(acknowledgments to Denny & Ade Azurat)
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Fasilkom UI
Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Binary Search Tree
2Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Outline
Concept of Binary Search Tree (BST)
BST operations Find Insert Remove
Running time analysis of BST operations
3Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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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Week 10
Binary Search Tree: Examples
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Week 10
Binary Search Tree: Examples
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6Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Basic Operations
FindMin, FindMax, Find
Insert
Remove
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Week 10
BinaryNode<Type> findMin(BinaryNode<Type> t) { if (t != null) while (t.left != null) t = t.left;
return t;}
FindMin
Find node with the smallest value
Algorithm: Keep going left until you reach a dead end!
Code:
8Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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;}
9Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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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10Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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}
11Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Insertion: Principle
When inserting a new element into a binary search tree, it will always become a leaf node.
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Week 10
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!
13Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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;}
14Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Removing An Element
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15Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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
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Week 10
Removing Leaf
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17Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Removing Node With 1 Child
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Removing Node With 1 Child
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19Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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;}
20Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Removing Node With 2 Children
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21Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
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Removing Node With 2 Children
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Removing Node With 2 Children
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23Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
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Removing Root
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24Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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;}
25Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Find k-th element
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k < SL + 1
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k == SL + 1
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k > SL + 1
26Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
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); }
27Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Analysis
Running time for: Insert? Find min? Remove? Find?
Average case: O(log n)
Worst case: O(n)
28Ruli Manurung (Fasilkom UI) IKI10100I: Data Structures & Algorithms
Week 10
Summary
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.