Objectives � Memahami sifat dari Binary Search Tree (BST) Struktur Data & Algoritme � Memahami operasi-operasi pada BST � Memahami kelebihan dan kekurangan dari BST ( Data Structures & Algorithms ) Binary Search Tree Denny ( denny@cs.ui.ac.id ) Suryana Setiawan ( setiawan@cs.ui.ac.id ) Fakultas I lm u Kom puter Universitas I ndonesia Sem ester Genap - 2 0 0 4 / 2 0 0 5 Version 2 .0 - I nternal Use Only SDA/ TOPI C/ V2.0/ 2 Outline Properties of Binary Search Tree � Properties of Binary Search Tree (BST) � 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 � Operation the values of all the keys in the right subtree are � Insert larger than the key in X. � find � So, the key should be comparable. � remove X <X >X SDA/ TOPI C/ V2.0/ 3 SDA/ TOPI C/ V2.0/ 4 1
Binary Search Tree Binary Search Tree 3 3 1 1 7 2 9 2 1 3 2 3 2 1 2 1 5 2 3 6 1 3 SDA/ TOPI C/ V2.0/ 5 SDA/ TOPI C/ V2.0/ 6 Basic Operations Insertion � insert � Penyisipan sebuah elemen baru dalam binary search tree, elemen tersebut pasti akan menjadi leaf � findMin and findMax � remove 10 15 2 12 1 5 14 3 6 SDA/ TOPI C/ V2.0/ 7 SDA/ TOPI C/ V2.0/ 8 2
Insertion: algorithm Insertion BinaryNode insert(int x, BinaryNode t) � Insert X into a binary search tree: { � start from the root. if (t == null) { � If the value of X is less than the value of the root, then t = new BinaryNode (x, null, null); X should be inserted on the left sub-tree. } else if (x < t.element) { � On the other hand, if the value of X is greater than the t.left = insert (x, t.left); value of the root, then X should be inserted on the } else if (x > t.element) { right sub-tree. t.right = insert (x, t.right); � Remember that, a sub tree is also a tree. So, the } else { problem to insert an element in the sub-tree is same throw new DuplicateItem(“exception”); as the problem to insert an element in the root. } � So? return t; } � We can attack this problem with recursive approach. SDA/ TOPI C/ V2.0/ 9 SDA/ TOPI C/ V2.0/ 10 FindMin FindMax � Mencari node yang memiliki nilai terkecil. � Mencari node yang memiliki nilai terbesar � Algorithm: � Algorithm? � ke kiri terus sampai buntu….:) � Code? � Code: BinaryNode findMin (BinaryNode t) { if (t == null) throw exception ; while (t.left != null) { t = t.left; } return t; } SDA/ TOPI C/ V2.0/ 11 SDA/ TOPI C/ V2.0/ 12 3
Find Remove � Diberikan sebuah nilai yang harus dicari dalam sebuah BST. Jika ada elemen tersebut, return node 8 tersebut. Jika tidak ada, return null. 12 � Algorithm? 4 4 � Code? 7 9 6 6 2 1 1 5 3 5 5 3 6 SDA/ TOPI C/ V2.0/ 13 SDA/ TOPI C/ V2.0/ 14 Remove Removing 6 � if the node is a leaf (has no child), no problemo… delete it immediately 8 � if the node has one child: its parent adjusts a child reference to bypass the node. 12 4 � if the node has two children? � replace the item in this node with the smallest item in the right subtree and then remove that node, or 6 1 � replace the item in this node with the biggest item in the left subtree and then remove that node � introduce new sub-problems: 5 3 � removeMin, removeMax SDA/ TOPI C/ V2.0/ 15 SDA/ TOPI C/ V2.0/ 16 4
After 6 removed removeMin BinaryNode removeMin(BinaryNode t) { 8 if (t == null) throw exception ; 12 4 if (t.left != null) { t.left = removeMin (t.left); } else { 6 1 t = t.right; } return t; } 3 5 SDA/ TOPI C/ V2.0/ 17 SDA/ TOPI C/ V2.0/ 18 removeMax Removing 2 � code? 7 9 2 1 5 3 4 SDA/ TOPI C/ V2.0/ 19 SDA/ TOPI C/ V2.0/ 20 5
After 2 deleted Removing Root 9 7 7 2 12 9 2 2 3 10 14 1 5 1 5 X 3 9 11 3 4 4 SDA/ TOPI C/ V2.0/ 21 SDA/ TOPI C/ V2.0/ 22 Remove Find k-th element BinaryNode remove(int x, BinaryNode t) { if (t == null) throw exception; if (x < t.element) { t.left = remove(x, t.left); X X X } else if (x > t.element) { t.right = remove(x, t.right); } else if (t.left != null && t.right != null) { t.element = findMin(t.right).element; S R S R S R t.right = removeMin(t.right); S L S L S L } else { t = (t.left != null) ? t.left : t.right; } return t; k < S L + 1 k == S L + 1 k > S L + 1 } SDA/ TOPI C/ V2.0/ 23 SDA/ TOPI C/ V2.0/ 24 6
Find k-th element Analysis BinaryNode findKth(int k, BinaryNode t) � Runnning time for: { � insert? if (t == null) throw exception; � Find min? int leftSize = (t.left != null) ? � remove? t.left.size : 0; � Find? � Worst case: O(n) if (k <= leftSize ) { return findKth (k, t.left); } else if (k == leftSize + 1) { return t; } else { return findKth ( k - leftSize - 1, t.right); } } SDA/ TOPI C/ V2.0/ 25 SDA/ TOPI C/ V2.0/ 26 Summary Further Reading � Binary Search Tree maintains the order of the tree. � http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/1998/handout/handout16.html � Each node should be comparable � Chapter 18 � All operations take O(log n) - average case, when the tree equally balanced. � All operations will take O(n) - worst case, when the all the height of the tree equals with the total of the nodes. SDA/ TOPI C/ V2.0/ 27 SDA/ TOPI C/ V2.0/ 28 7
What’s Next � AVL tree SDA/ TOPI C/ V2.0/ 29 8
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