Insert(v) runs in O(h) where h is the height of the BST. If we use unsorted array/vector to implement Table ADT, it can be inefficient: If we use sorted array/vector to implement Table ADT, we can improve the Search(v) performance but weakens the Insert(v) performance: The goal for this e-Lecture is to introduce BST and then balanced BST (AVL Tree) data structure so that we can implement the basic Table ADT operations: Search(v), Insert(v), Remove(v), and a few other Table ADT operations — see the next slide — in O(log N) time — which is much smaller than N. PS: Some of the more experienced readers may notice that ∃ another data structure that can implement the three basic Table ADT operations in faster time, but read on... On top of the basic three, there are a few other possible Table ADT operations: Discussion: What are the best possible implementation for the first three additional operations if we are limited to use [sorted|unsorted] array/vector? There can be more than one leaf vertex in a BST. Other interested CS instructor should contact Steven if you want to try such 'test mode'. Detailed tutorial on Quick Sort to improve your understanding of {{ track }}. This work is done mostly by my past students. Binary search works by comparing the target value to the middle element of the array. As compared to linear, binary search is much faster with Time Complexity of O(logN) whereas linear search algorithm works in O(N) time complexity. Try Search(100) (this value should not exist as we only use random integers between [1..99] to generate this random BST and thus the Search routine should check all the way from root to the only leaf in O(N) time — not efficient. Because of the way data (distinct integers for this visualization) is organised inside a BST, we can binary search for an integer v efficiently (hence the name of Binary Search Tree). If we use unsorted array/vector to implement Table ADT, it can be inefficient: If we use sorted array/vector to implement Table ADT, we can improve the Search(v) performance but weakens the Insert(v) performance: The goal for this e-Lecture is to introduce BST and then balanced BST (AVL Tree) data structure so that we can implement the basic Table ADT operations: Search(v), Insert(v), Remove(v), and a few other Table ADT operations — see the next slide — in O(log N) time — which is much smaller than N. PS: Some of the more experienced readers may notice that ∃ another data structure that can implement the three basic Table ADT operations in faster time, but read on... On top of the basic three, there are a few other possible Table ADT operations: Discussion: What are the best possible implementation for the first three additional operations if we are limited to use [sorted|unsorted] array/vector? We provide visualization for the following common BST/AVL Tree operations: There are a few other BST (Query) operations that have not been visualized in VisuAlgo: The details of these two operations are currently hidden for pedagogical purpose in a certain NUS module. For more complete implementation, we should consider duplicate integers too and we must consistently place integers that are equal to X to one subtree only (not arbitrary). Each vertex has at least 4 attributes: parent, left, right, key/value/data (there are potential other attributes). His contact is the concatenation of his name and add gmail dot com. Nodes have values. This case 3 warrants further discussions: Remove(v) runs in O(h) where h is the height of the BST. On the example BST above, try clicking Search(15) (found after just 1 comparison), Search(7) (found after 3 comparisons), Search(21) (not found after 3 comparisons). Data structure that is only efficient if there is no (or rare) update, especially the insert and/or remove operation(s) is called static data structure. Other balanced BST implementations (more or less as good or slightly better in terms of constant-factor performance) are: Red-Black Tree, B-trees/2-3-4 Tree (Bayer & McCreight, 1972), Splay Tree (Sleator and Tarjan, 1985), Skip Lists (Pugh, 1989), Treaps (Seidel and Aragon, 1996), etc. Acknowledgements Control the animation with the player controls! This part requires O(h) due to the need to find the successor vertex — on top of the earlier O(h) search-like effort. To make life easier in 'Exploration Mode', you can create a new BST using these options: We are midway through the explanation of this BST module. Vertices that are not leaf are called the internal vertices. PS: Some people call insertion of N unordered integers into a BST in O(N log N) and then performing the O(N) Inorder Traversal as 'BST sort'. At this point, stop and ponder these three Successor(v)/Predecessor(v) cases to ensure that you understand these concepts. If v is not found in the BST, we simply do nothing. If we call Remove(FindMax()), i.e. The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. Such BST is called AVL Tree, like the example shown above. We focus on AVL Tree (Adelson-Velskii & Landis, 1962) that is named after its inventor: Adelson-Velskii and Landis. Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor. the root vertex will have its parent attribute = NULL. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly. Dr Felix Halim, Software Engineer, Google (Mountain View), Estudantes Pesquisadores de Graduação 1 (Jul 2011-Apr 2012) Thus the parent of 6 (and 23) is 15. Because of the way data (distinct integers for this visualization) is organised inside a BST, we can binary search for an integer v efficiently (hence the name of Binary Search Tree). A BST is called height-balanced according to the invariant above if every vertex in the BST is height-balanced. Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Rose Marie Tan Zhao Yun, Ivan Reinaldo, Undergraduate Student Researchers 2 (May 2014-Jul 2014) This work has been presented briefly at the CLI Workshop at the ACM ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). Binary search compares the target value to the middle element of the array. These values determine where they are placed within the BST. Also try practice problems to test & improve your skill level. c * log2 N, for a small constant factor c? First, what are the principles that define a Binary Search Tree? In the example above, the vertices on the left subtree of the root 15: {4, 5, 6, 7} are all smaller than 15 and the vertices on the right subtree of the root 15: {23, 50, 71} are all greater than 15. We also have a few programming problems that somewhat requires the usage of this balanced BST (like AVL Tree) data structure: Kattis - compoundwords and Kattis - baconeggsandspam. The main difference compared to Insert(v) in AVL tree is that we may trigger one of the four possible rebalancing cases several times, but not more than h = O(log N) times :O, try Remove(7) on the example above to see two chain reactions rotateRight(6) and then rotateRight(16)+rotateLeft(8) combo. Notice that only a few vertices along the insertion path: {41,20,29,32} increases their height by +1 and all other vertices will have their heights unchanged. Try the same three corner cases (but mirrored): Predecessor(6) (should be 5), Predecessor(50) (should be 23), Predecessor(4) (should be none). A Binary Search Tree (BST) is a binary tree in which each vertex has only up to 2 children that satisfies BST property: All vertices in the left subtree of a vertex must hold a value smaller than its own and all vertices in the right subtree of a vertex must hold a value larger than its own (we have assumption that all values are distinct integers in this visualization and small tweak is needed to cater for duplicates/non integer). For the example BST shown in the background, we have: {{15}, {6, 4, 5, 7}, {23, 71, 50}}. Quiz: So what is the point of learning this BST module if Hash Table can do the crucial Table ADT operations in unlikely-to-be-beaten expected O(1) time? For the example BST shown in the background, we have: {{15}, {6, 4, 5, 7}, {23, 71, 50}}. Adelson-Velskii and Landis claim that an AVL Tree (a height-balanced BST that satisfies AVL Tree invariant) with N vertices has height h < 2 * log2 N. The proof relies on the concept of minimum-size AVL Tree of a certain height h. Let Nh be the minimum number of vertices in a height-balanced AVL Tree of height h. The first few values of Nh are N0 = 1 (a single root vertex), N1 = 2 (a root vertex with either one left child or one right child only), N2 = 4, N3 = 7, N4 = 12, N5 = 20 (see the background picture), and so on (see the next two slides). It has very fast Search(v), Insert(v), and Remove(v) performance (all in expected O(1) time). To have efficient performance, we shall not maintain height(v) attribute via the O(N) recursive method every time there is an update (Insert(v)/Remove(v)) operation. Binary search is the search technique which works efficiently on the sorted lists. PS: If you want to study how these basic BST operations are implemented in a real program, you can download this BSTDemo.cpp. We use Tree Rotation(s) to deal with each of them. If we have N elements/items/keys in our BST, the upper bound height h < N if we insert the elements in ascending order (to get skewed right BST as shown above). Try Insert(60) on the example above. Search(v)/FindMin()/FindMax() operations run in O(h) where h is the height of the BST. You are allowed to use C++ STL map/set, Java TreeMap/TreeSet, or OCaml Map/Set if that simplifies your implementation (Note that Python doesn't have built-in bBST implementation). This is a big task and requires crowdsourcing. the root vertex will have its parent attribute = NULL. Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir. Project Leader & Advisor (Jul 2011-present) Deletion of a vertex with one child is not that hard: We connect that vertex's only child with that vertex's parent — try Remove(23) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead). Quiz: Inserting integers [1,10,2,9,3,8,4,7,5,6] one by one in that order into an initially empty BST will result in a BST of height: Pro-tip: You can use the 'Exploration mode' to verify the answer. VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace. We can perform an Inorder Traversal of this BST to obtain a list of sorted integers inside this BST (in fact, if we 'flatten' the BST into one line, we will see that the vertices are ordered from smallest/leftmost to largest/rightmost). If v is found in the BST, we do not report that the existing integer v is found, but instead, we perform one of the three possible removal cases that will be elaborated in three separate slides (we suggest that you try each of them one by one). For this algorithm to work properly, the data collection should be in the sorted form. If you are a data structure and algorithm student/instructor, you are allowed to use this website directly for your classes. List of translators who have contributed ≥100 translations can be found at statistics page. Data structure that is efficient even if there are many update operations is called dynamic data structure. Such BST is called AVL Tree, like the example shown above. Without further ado, let's try Inorder Traversal to see it in action on the example BST above. We want to prepare a database of CS terminologies for all English text that ever appear in VisuAlgo system. A BST is called height-balanced according to the invariant above if every vertex in the BST is height-balanced. The most exciting new development is an automated question generator and verifier (the online quiz system) that allows student to test their knowledge of basic data structures and algorithms. Binary search is a fast search algorithm with run-time complexity of Ο(log n). A Binary Search is a sorting algorithm, that is used to search an element in a sorted array. bf(29) = -2 and bf(20) = -2 too. The first case is the easiest: Vertex v is currently one of the leaf vertex of the BST. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component. If you like VisuAlgo, the only payment that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook, Twitter, course webpage, blog review, email, etc. You can recursively check BST property on other vertices too. It is rarely used though as there are several easier-to-use (comparison-based) sorting algorithms than this. Try them to consolidate and improve your understanding about this data structure. The only limitation is that the array or list of elements must be sorted for the binary search algorithm to work on it. We have seen from earlier slides that most of our BST operations except Inorder traversal runs in O(h) where h is the height of the BST that can be as tall as N-1. For each vertex v, we define height(v): The number of edges on the path from vertex v down to its deepest leaf. We will end this module with a few more interesting things about BST and balanced BST (especially AVL Tree). root, members of left subtree of root, members of right subtree of root. It is rarely used though as there are several easier-to-use (comparison-based) sorting algorithms than this. Leaf vertex does not have any child. There can only be one root vertex in a BST. At this point, we encourage you to press [Esc] or click the X button on the bottom right of this e-Lecture slide to enter the 'Exploration Mode' and try various BST operations yourself to strengthen your understanding about this versatile data structure. You can recursively check BST property on other vertices too. Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS) Try clicking FindMin() and FindMax() on the example BST shown above. We use Tree Rotation(s) to deal with each of them. We can insert a new integer into BST by doing similar operation as Search(v). Because of the BST properties, we can find the Successor of an integer v (assume that we already know where integer v is located from earlier call of Search(v)) as follows: The operations for Predecessor of an integer v are defined similarly (just the mirror of Successor operations). Project Leader & Advisor (Jul 2011-present), Undergraduate Student Researchers 1 (Jul 2011-Apr 2012), Final Year Project/UROP students 1 (Jul 2012-Dec 2013), Final Year Project/UROP students 2 (Jun 2013-Apr 2014), Undergraduate Student Researchers 2 (May 2014-Jul 2014), Final Year Project/UROP students 3 (Jun 2014-Apr 2015), Final Year Project/UROP students 4 (Jun 2016-Dec 2017), Search(v) can now be implemented in O(log. For more complete implementation, we should consider duplicate integers too and we must consistently place integers that are equal to X to one subtree only (not arbitrary). Let N be the node selected by an insert, delete, or search operation, let P be the parent node, and let G be the grandparent node. Vertices {29,20} will no longer be height-balanced after this insertion (and will be rotated later — discussed in the next few slides), i.e. There can only be one root vertex in a BST. Hint: Go back to the previous 4 slides ago. Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir. If we have N elements/items/keys in our BST, the upper bound height h < N if we insert the elements in ascending order (to get skewed right BST as shown above). An Adelson-Velskii Landis (AVL) tree is a self-balancing BST that maintains it's height to be O(log N) when having N vertices in the AVL tree. As you should have fully understand by now, h can be as tall as O(N) in a normal BST as shown in the random 'skewed right' example above. e-Lecture: The content of this slide is hidden and only available for legitimate CS lecturer worldwide. We keep doing this until we either find the required vertex or we don't. Insert(v) and Remove(v) update operations may change the height h of the AVL Tree, but we will see rotation operation(s) to maintain the AVL Tree height to be low. But basically, in Preorder Traversal, we visit the current root before going to left subtree and then right subtree. Then, use the slide selector drop down list to resume from this slide 12-1. Operation X & Y - hidden for pedagogical purpose in an NUS module. The height of such BST is h = N-1, so we have h < N. Discussion: Do you know how to get skewed left BST instead? This attribute is saved in each vertex so we can access a vertex's height in O(1) without having to recompute it every time. But this time, instead of reporting that the new integer is not found, we create a new vertex in the insertion point and put the new integer there. Discussion: Is there other tree rotation cases for Insert(v) operation of AVL Tree? BST (and especially balanced BST like AVL Tree) is an efficient data structure to implement a certain kind of Table (or Map) Abstract Data Type (ADT). Root vertex does not have a parent. They allow fast lookup, insertion, and deletion. On the example BST above, height(11) = height(32) = height(50) = height(72) = height(99) = 0 (all are leaves). We will end this module with a few more interesting things about BST and balanced BST (especially AVL Tree). We have seen from earlier slides that most of our BST operations except Inorder traversal runs in O(h) where h is the height of the BST that can be as tall as N-1. Try them to consolidate and improve your understanding about this data structure. Deletion of a leaf vertex is very easy: We just remove that leaf vertex — try Remove(5) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead). Control the animation with the player controls! We recommend using Google Chrome to access VisuAlgo. Deletion of a vertex with two children is as follow: We replace that vertex with its successor, and then delete its duplicated successor in its right subtree — try Remove(6) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead). Removing v without doing anything else will disconnect the BST. This is of order O(log k). Some other implementation separates key (for ordering of vertices in the BST) with the actual satellite data associated with the keys. But basically, in Preorder Traversal, we visit the current root before going to left subtree and then right subtree. So can we have BST that has height closer to log2 N, i.e. This key holds the value to be searched. Other balanced BST implementations (more or less as good or slightly better in terms of constant-factor performance) are: Red-Black Tree, B-trees/2-3-4 Tree (Bayer & McCreight, 1972), Splay Tree (Sleator and Tarjan, 1985), Skip Lists (Pugh, 1989), Treaps (Seidel and Aragon, 1996), etc. See that all vertices are height-balanced, an AVL Tree. Inorder Traversal is a recursive method whereby we visit the left subtree first, exhausts all items in the left subtree, visit the current root, before exploring the right subtree and all items in the right subtree. c * log2 N, for a small constant factor c? VB.NET Searching Arrays & Binary Search Algorithm in Visual Basic 2008 Arrays can be searched in two ways: with the BinarySearch method, which works on sorted arrays and is extremely fast, and with the IndexOf (and LastIndexOf) methods, which work regardless of the order of the elements. This part is also clearly O(1) — on top of the earlier O(h) search-like effort. Sometimes root vertex is not included as part of the definition of internal vertex as the root of a BST with only one vertex can actually fit into the definition of a leaf too. We know that for any other AVL Tree of N vertices (not necessarily the minimum-size one), we have N ≥ Nh. Query operations (the BST structure remains unchanged): Predecessor(v) (and similarly Successor(v)), and. The second case is also not that hard: Vertex v is an (internal/root) vertex of the BST and it has exactly one child. The height of such BST is h = N-1, so we have h < N. Discussion: Do you know how to get skewed left BST instead? The binary search algorithm applies to direct access of contiguous memory, so an array is used to store the data for a binary search algorithm. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. Data structure that is only efficient if there is no (or rare) update, especially the insert and/or remove operation(s) is called static data structure. Drop an email to visualgo.info at gmail dot com if you want to activate this CS lecturer-only feature and you are really a CS lecturer (show your University staff profile). There are several known implementations of balanced BST, too many to be visualized and explained one by one in VisuAlgo. Let’s start with basic terminology so we may share the same language and investigate related concepts. Please login if you are a repeated visitor or register for an (optional) free account first. zh, id, kr, vn, th. root, members of left subtree of root, members of right subtree of root. Discuss the answer above! When you are ready to continue with the explanation of balanced BST (we use AVL Tree as our example), press [Esc] again or switch the mode back to 'e-Lecture Mode' from the top-right corner drop down menu. We then go to the right subtree/stop/go the left subtree, respectively. Removing v without doing anything else will disconnect the BST. Visit VisuAlgo to help visualize binary search trees. It's one of the most important algorithms of the modern era and quite easy to understand. Once the system is ready, we will invite VisuAlgo visitors to contribute, especially if you are not a native English speaker. The (integer) key of each vertex is drawn inside the circle that represent that vertex. There are several known implementations of balanced BST, too many to be visualized and explained one by one in VisuAlgo. Keyboard shortcuts are: Return to 'Exploration Mode' to start exploring! We also have a few programming problems that somewhat requires the usage of this balanced BST (like AVL Tree) data structure: Kattis - compoundwords and Kattis - baconeggsandspam. First, what are the principles that define a Binary Search Tree? Another data structure that can be used to implement Table ADT is Hash Table. Try the same three corner cases (but mirrored): Predecessor(6) (should be 5), Predecessor(50) (should be 23), Predecessor(4) (should be none). Not all attributes will be used for all vertices, e.g. Go to full screen mode (F11) to enjoy this setup. Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for a real examination in NUS. BST and especially balanced BST (e.g. On the example BST above, height(11) = height(32) = height(50) = height(72) = height(99) = 0 (all are leaves). To have efficient performance, we shall not maintain height(v) attribute via the O(N) recursive method every time there is an update (Insert(v)/Remove(v)) operation. Inorder Traversal runs in O(N), regardless of the height of the BST. The answers should be 4 and 71 (both after 4 comparisons). For a few more interesting questions about this data structure, please practice on BST/AVL training module (no login is required). In AVL Tree, we will later see that its height h < 2 * log N (tighter analysis exist, but we will use easier analysis in VisuAlgo where c = 2). Deletion of a vertex with two children is as follow: We replace that vertex with its successor, and then delete its duplicated successor in its right subtree — try Remove(6) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead). However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this. This special requirement of Table ADT will be made clearer in the next few slides. Just like a standard binary tree, each node can have at most two children. Please login if you are a repeated visitor or register for an (optional) free account first. By now you should be aware that this h can be as tall as O(N) in a normal BST as shown in the random 'skewed right' example above. VisuAlgo is free of charge for Computer Science community on earth. See the example shown above for N = 15 (a perfect BST which is rarely achievable in real life — try inserting any other integer and it will not be perfect anymore). For a few more interesting questions about this data structure, please practice on BST/AVL training module (no login is required). If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim. VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond. Erin Teo Yi Ling, Wang Zi, Final Year Project/UROP students 4 (Jun 2016-Dec 2017) The answers should be 4 and 71 (both after 4 comparisons). List of translators who have contributed ≥100 translations can be found at statistics page. Você pode clicar neste link para ler nosso paper de 2012 sobre este sistema (ele ainda não era chamado VisuAlgo em 2012).Este trabalho foi feito em sua maioria por meus estudantes anteriores. However, you are NOT allowed to download VisuAlgo (client-side) files and host it on your own website as it is plagiarism. A binary search tree is a special implementation of a binary tree and is commonly used for tree interview questions. When you are ready to continue with the explanation of balanced BST (we use AVL Tree as our example), press [Esc] again or switch the mode back to 'e-Lecture Mode' from the top-right corner drop down menu. Go to full screen mode (F11) to enjoy this setup. It is a searching technique that is better then the liner search technique as the number of iterations decreases in the binary search. Binary search works on a sorted array. Not all attributes will be used for all vertices, e.g. Calling rotateRight(Q) on the left picture will produce the right picture. First, we set the current vertex = root and then check if the current vertex is smaller/equal/larger than integer v that we are searching for. In the example above, (key) 15 has 6 as its left child and 23 as its right child. This method involves a binary search for two vertices in a k long path so that an additional vertex can be accommodated in the current path. See the example shown above for N = 15 (a perfect BST which is rarely achievable in real life — try inserting any other integer and it will not be perfect anymore). We will now introduce BST data structure. So, is there a way to make our BSTs 'not that tall'? Currently, the general public can only use the 'training mode' to access these online quiz system. Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com. Simple Solution is to traverse nodes in Inorder and store result in an NUS module Singapore. 4 comparisons ) we either find the Successor ( v ) — 'previous smaller element. In VisuAlgo system have also written public notes about VisuAlgo in various languages: zh id! Submission to our grading server your skill level not designed to work on it Tarjan in 1985 are... Start with basic terminology so we may do this N times giving a summation of ( client-side ) files host! ) inventors: Georgy Adelson-Velskii and Evgenii Landis, back in 1962 this BSTDemo.cpp contains questions 12! Prepare a database of CS terminologies for all vertices, e.g keyboard shortcuts are: Return to 'Exploration '... 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