CMSC 341 Data Structures Midterm II Review Red-Black Tree Review 87
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CMSC 341 Data Structure Asymptotic Analysis Review
CMSC 341 Data Structure Asymptotic Analysis Review These questions will help test your understanding of the asymptotic analysis material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. 1. What is the purpose of asymptotic analysis? 2. Define “Big-Oh” using a formal, mathematical definition. 3. Let T1(x) = O(f(x)) and T2(x) = O(g(x)). Prove T1(x) + T2(x) = O (max(f(x), g(x))). 4. Let T(x) = O(cf(x)), where c is some positive constant. Prove T(x) = O(f(x)). 5. Let T1(x) = O(f(x)) and T2(x) = O(g(x)). Prove T1(x) * T2(x) = O(f(x) * g(x)) 6. Prove 2n+1 = O(2n). 7. Prove that if T(n) is a polynomial of degree x, then T(n) = O(nx). 8. Number these functions in ascending (slowest growing to fastest growing) Big-Oh order: Number Big-Oh O(n3) O(n2 lg n) O(1) O(lg0.1 n) O(n1.01) O(n2.01) O(2n) O(lg n) O(n) O(n lg n) O (n lg5 n) 1 9. Determine, for the typical algorithms that you use to perform calculations by hand, the running time to: a. Add two N-digit numbers b. Multiply two N-digit numbers 10. What is the asymptotic performance of each of the following? Select among: a. O(n) b. -
Data Structures and Algorithms Binary Heaps (S&W 2.4)
Data structures and algorithms DAT038/TDA417, LP2 2019 Lecture 12, 2019-12-02 Binary heaps (S&W 2.4) Some slides by Sedgewick & Wayne Collections A collection is a data type that stores groups of items. stack Push, Pop linked list, resizing array queue Enqueue, Dequeue linked list, resizing array symbol table Put, Get, Delete BST, hash table, trie, TST set Add, ontains, Delete BST, hash table, trie, TST A priority queue is another kind of collection. “ Show me your code and conceal your data structures, and I shall continue to be mystified. Show me your data structures, and I won't usually need your code; it'll be obvious.” — Fred Brooks 2 Priority queues Collections. Can add and remove items. Stack. Add item; remove the item most recently added. Queue. Add item; remove the item least recently added. Min priority queue. Add item; remove the smallest item. Max priority queue. Add item; remove the largest item. return contents contents operation argument value size (unordered) (ordered) insert P 1 P P insert Q 2 P Q P Q insert E 3 P Q E E P Q remove max Q 2 P E E P insert X 3 P E X E P X insert A 4 P E X A A E P X insert M 5 P E X A M A E M P X remove max X 4 P E M A A E M P insert P 5 P E M A P A E M P P insert L 6 P E M A P L A E L M P P insert E 7 P E M A P L E A E E L M P P remove max P 6 E M A P L E A E E L M P A sequence of operations on a priority queue 3 Priority queue API Requirement. -
Game Trees, Quad Trees and Heaps
CS 61B Game Trees, Quad Trees and Heaps Fall 2014 1 Heaps of fun R (a) Assume that we have a binary min-heap (smallest value on top) data structue called Heap that stores integers and has properly implemented insert and removeMin methods. Draw the heap and its corresponding array representation after each of the operations below: Heap h = new Heap(5); //Creates a min-heap with 5 as the root 5 5 h.insert(7); 5,7 5 / 7 h.insert(3); 3,7,5 3 /\ 7 5 h.insert(1); 1,3,5,7 1 /\ 3 5 / 7 h.insert(2); 1,2,5,7,3 1 /\ 2 5 /\ 7 3 h.removeMin(); 2,3,5,7 2 /\ 3 5 / 7 CS 61B, Fall 2014, Game Trees, Quad Trees and Heaps 1 h.removeMin(); 3,7,5 3 /\ 7 5 (b) Consider an array based min-heap with N elements. What is the worst case running time of each of the following operations if we ignore resizing? What is the worst case running time if we take into account resizing? What are the advantages of using an array based heap vs. using a BST-based heap? Insert O(log N) Find Min O(1) Remove Min O(log N) Accounting for resizing: Insert O(N) Find Min O(1) Remove Min O(N) Using a BST is not space-efficient. (c) Your friend Alyssa P. Hacker challenges you to quickly implement a max-heap data structure - "Hah! I’ll just use my min-heap implementation as a template", you think to yourself. -
Heaps a Heap Is a Complete Binary Tree. a Max-Heap Is A
Heaps Heaps 1 A heap is a complete binary tree. A max-heap is a complete binary tree in which the value in each internal node is greater than or equal to the values in the children of that node. A min-heap is defined similarly. 97 Mapping the elements of 93 84 a heap into an array is trivial: if a node is stored at 90 79 83 81 index k, then its left child is stored at index 42 55 73 21 83 2k+1 and its right child at index 2k+2 01234567891011 97 93 84 90 79 83 81 42 55 73 21 83 CS@VT Data Structures & Algorithms ©2000-2009 McQuain Building a Heap Heaps 2 The fact that a heap is a complete binary tree allows it to be efficiently represented using a simple array. Given an array of N values, a heap containing those values can be built, in situ, by simply “sifting” each internal node down to its proper location: - start with the last 73 73 internal node * - swap the current 74 81 74 * 93 internal node with its larger child, if 79 90 93 79 90 81 necessary - then follow the swapped node down 73 * 93 - continue until all * internal nodes are 90 93 90 73 done 79 74 81 79 74 81 CS@VT Data Structures & Algorithms ©2000-2009 McQuain Heap Class Interface Heaps 3 We will consider a somewhat minimal maxheap class: public class BinaryHeap<T extends Comparable<? super T>> { private static final int DEFCAP = 10; // default array size private int size; // # elems in array private T [] elems; // array of elems public BinaryHeap() { . -
Priority Queues and Binary Heaps Chapter 6.5
Priority Queues and Binary Heaps Chapter 6.5 1 Some animals are more equal than others • A queue is a FIFO data structure • the first element in is the first element out • which of course means the last one in is the last one out • But sometimes we want to sort of have a queue but we want to order items according to some characteristic the item has. 107 - Trees 2 Priorities • We call the ordering characteristic the priority. • When we pull something from this “queue” we always get the element with the best priority (sometimes best means lowest). • It is really common in Operating Systems to use priority to schedule when something happens. e.g. • the most important process should run before a process which isn’t so important • data off disk should be retrieved for more important processes first 107 - Trees 3 Priority Queue • A priority queue always produces the element with the best priority when queried. • You can do this in many ways • keep the list sorted • or search the list for the minimum value (if like the textbook - and Unix actually - you take the smallest value to be the best) • You should be able to estimate the Big O values for implementations like this. e.g. O(n) for choosing the minimum value of an unsorted list. • There is a clever data structure which allows all operations on a priority queue to be done in O(log n). 107 - Trees 4 Binary Heap Actually binary min heap • Shape property - a complete binary tree - all levels except the last full. -
L11: Quadtrees CSE373, Winter 2020
L11: Quadtrees CSE373, Winter 2020 Quadtrees CSE 373 Winter 2020 Instructor: Hannah C. Tang Teaching Assistants: Aaron Johnston Ethan Knutson Nathan Lipiarski Amanda Park Farrell Fileas Sam Long Anish Velagapudi Howard Xiao Yifan Bai Brian Chan Jade Watkins Yuma Tou Elena Spasova Lea Quan L11: Quadtrees CSE373, Winter 2020 Announcements ❖ Homework 4: Heap is released and due Wednesday ▪ Hint: you will need an additional data structure to improve the runtime for changePriority(). It does not affect the correctness of your PQ at all. Please use a built-in Java collection instead of implementing your own. ▪ Hint: If you implemented a unittest that tested the exact thing the autograder described, you could run the autograder’s test in the debugger (and also not have to use your tokens). ❖ Please look at posted QuickCheck; we had a few corrections! 2 L11: Quadtrees CSE373, Winter 2020 Lecture Outline ❖ Heaps, cont.: Floyd’s buildHeap ❖ Review: Set/Map data structures and logarithmic runtimes ❖ Multi-dimensional Data ❖ Uniform and Recursive Partitioning ❖ Quadtrees 3 L11: Quadtrees CSE373, Winter 2020 Other Priority Queue Operations ❖ The two “primary” PQ operations are: ▪ removeMax() ▪ add() ❖ However, because PQs are used in so many algorithms there are three common-but-nonstandard operations: ▪ merge(): merge two PQs into a single PQ ▪ buildHeap(): reorder the elements of an array so that its contents can be interpreted as a valid binary heap ▪ changePriority(): change the priority of an item already in the heap 4 L11: Quadtrees CSE373, -
Search Trees
Lecture III Page 1 “Trees are the earth’s endless effort to speak to the listening heaven.” – Rabindranath Tagore, Fireflies, 1928 Alice was walking beside the White Knight in Looking Glass Land. ”You are sad.” the Knight said in an anxious tone: ”let me sing you a song to comfort you.” ”Is it very long?” Alice asked, for she had heard a good deal of poetry that day. ”It’s long.” said the Knight, ”but it’s very, very beautiful. Everybody that hears me sing it - either it brings tears to their eyes, or else -” ”Or else what?” said Alice, for the Knight had made a sudden pause. ”Or else it doesn’t, you know. The name of the song is called ’Haddocks’ Eyes.’” ”Oh, that’s the name of the song, is it?” Alice said, trying to feel interested. ”No, you don’t understand,” the Knight said, looking a little vexed. ”That’s what the name is called. The name really is ’The Aged, Aged Man.’” ”Then I ought to have said ’That’s what the song is called’?” Alice corrected herself. ”No you oughtn’t: that’s another thing. The song is called ’Ways and Means’ but that’s only what it’s called, you know!” ”Well, what is the song then?” said Alice, who was by this time completely bewildered. ”I was coming to that,” the Knight said. ”The song really is ’A-sitting On a Gate’: and the tune’s my own invention.” So saying, he stopped his horse and let the reins fall on its neck: then slowly beating time with one hand, and with a faint smile lighting up his gentle, foolish face, he began.. -
CS210-Data Structures-Module-29-Binary-Heap-II
Data Structures and Algorithms (CS210A) Lecture 29: • Building a Binary heap on 풏 elements in O(풏) time. • Applications of Binary heap : sorting • Binary trees: beyond searching and sorting 1 Recap from the last lecture 2 A complete binary tree How many leaves are there in a Complete Binary tree of size 풏 ? 풏/ퟐ 3 Building a Binary heap Problem: Given 풏 elements {푥0, …, 푥푛−1}, build a binary heap H storing them. Trivial solution: (Building the Binary heap incrementally) CreateHeap(H); For( 풊 = 0 to 풏 − ퟏ ) What is the time Insert(푥,H); complexity of this algorithm? 4 Building a Binary heap incrementally What useful inference can you draw from Top-down this Theorem ? approach The time complexity for inserting a leaf node = ?O (log 풏 ) # leaf nodes = 풏/ퟐ , Theorem: Time complexity of building a binary heap incrementally is O(풏 log 풏). 5 Building a Binary heap incrementally What useful inference can you draw from Top-down this Theorem ? approach The O(풏) time algorithm must take O(1) time for each of the 풏/ퟐ leaves. 6 Building a Binary heap incrementally Top-down approach 7 Think of alternate approach for building a binary heap In any complete 98 binaryDoes treeit suggest, how a manynew nodes approach satisfy to heapbuild propertybinary heap ? ? 14 33 all leaf 37 11 52 32 nodes 41 21 76 85 17 25 88 29 Bottom-up approach 47 75 9 57 23 heap property: “Every ? node stores value smaller than its children” We just need to ensure this property at each node. -
Functional Data Structures with Isabelle/HOL
Functional Data Structures with Isabelle/HOL Tobias Nipkow Fakultät für Informatik Technische Universität München 2021-6-14 1 Part II Functional Data Structures 2 Chapter 6 Sorting 3 1 Correctness 2 Insertion Sort 3 Time 4 Merge Sort 4 1 Correctness 2 Insertion Sort 3 Time 4 Merge Sort 5 sorted :: ( 0a::linorder) list ) bool sorted [] = True sorted (x # ys) = ((8 y2set ys: x ≤ y) ^ sorted ys) 6 Correctness of sorting Specification of sort :: ( 0a::linorder) list ) 0a list: sorted (sort xs) Is that it? How about set (sort xs) = set xs Better: every x occurs as often in sort xs as in xs. More succinctly: mset (sort xs) = mset xs where mset :: 0a list ) 0a multiset 7 What are multisets? Sets with (possibly) repeated elements Some operations: f#g :: 0a multiset add mset :: 0a ) 0a multiset ) 0a multiset + :: 0a multiset ) 0a multiset ) 0a multiset mset :: 0a list ) 0a multiset set mset :: 0a multiset ) 0a set Import HOL−Library:Multiset 8 1 Correctness 2 Insertion Sort 3 Time 4 Merge Sort 9 HOL/Data_Structures/Sorting.thy Insertion Sort Correctness 10 1 Correctness 2 Insertion Sort 3 Time 4 Merge Sort 11 Principle: Count function calls For every function f :: τ 1 ) ::: ) τ n ) τ define a timing function Tf :: τ 1 ) ::: ) τ n ) nat: Translation of defining equations: E[[f p1 ::: pn = e]] = (Tf p1 ::: pn = T [[e]] + 1) Translation of expressions: T [[g e1 ::: ek]] = T [[e1]] + ::: + T [[ek]] + Tg e1 ::: ek All other operations (variable access, constants, constructors, primitive operations on bool and numbers) cost 1 time unit 12 Example: @ E[[ [] @ ys = ys ]] = (T@ [] ys = T [[ys]] + 1) = (T@ [] ys = 2) E[[ (x # xs)@ ys = x #(xs @ ys) ]] = (T@ (x # xs) ys = T [[x #(xs @ ys)]] + 1) = (T@ (x # xs) ys = T@ xs ys + 5) T [[x #(xs @ ys)]] = T [[x]] + T [[xs @ ys]] + T# x (xs @ ys) = 1 + (T [[xs]] + T [[ys]] + T@ xs ys) + 1 = 1 + (1 + 1 + T@ xs ys) + 1 13 if and case So far we model a call-by-value semantics Conditionals and case expressions are evaluated lazily. -
Leftist Trees
DEMO : Purchase from www.A-PDF.com to remove the watermark5 Leftist Trees 5.1 Introduction......................................... 5-1 5.2 Height-Biased Leftist Trees ....................... 5-2 Definition • InsertionintoaMaxHBLT• Deletion of Max Element from a Max HBLT • Melding Two Max HBLTs • Initialization • Deletion of Arbitrary Element from a Max HBLT Sartaj Sahni 5.3 Weight-Biased Leftist Trees....................... 5-8 University of Florida Definition • Max WBLT Operations 5.1 Introduction A single-ended priority queue (or simply, a priority queue) is a collection of elements in which each element has a priority. There are two varieties of priority queues—max and min. The primary operations supported by a max (min) priority queue are (a) find the element with maximum (minimum) priority, (b) insert an element, and (c) delete the element whose priority is maximum (minimum). However, many authors consider additional operations such as (d) delete an arbitrary element (assuming we have a pointer to the element), (e) change the priority of an arbitrary element (again assuming we have a pointer to this element), (f) meld two max (min) priority queues (i.e., combine two max (min) priority queues into one), and (g) initialize a priority queue with a nonzero number of elements. Several data structures: e.g., heaps (Chapter 3), leftist trees [2, 5], Fibonacci heaps [7] (Chapter 7), binomial heaps [1] (Chapter 7), skew heaps [11] (Chapter 6), and pairing heaps [6] (Chapter 7) have been proposed for the representation of a priority queue. The different data structures that have been proposed for the representation of a priority queue differ in terms of the performance guarantees they provide. -
CS 270 Algorithms
CS 270 Algorithms Week 10 Oliver Kullmann Binary heaps Sorting Heapification Building a heap 1 Binary heaps HEAP- SORT Priority 2 Heapification queues QUICK- 3 Building a heap SORT Analysing 4 QUICK- HEAP-SORT SORT 5 Priority queues Tutorial 6 QUICK-SORT 7 Analysing QUICK-SORT 8 Tutorial CS 270 General remarks Algorithms Oliver Kullmann Binary heaps Heapification Building a heap We return to sorting, considering HEAP-SORT and HEAP- QUICK-SORT. SORT Priority queues CLRS Reading from for week 7 QUICK- SORT 1 Chapter 6, Sections 6.1 - 6.5. Analysing 2 QUICK- Chapter 7, Sections 7.1, 7.2. SORT Tutorial CS 270 Discover the properties of binary heaps Algorithms Oliver Running example Kullmann Binary heaps Heapification Building a heap HEAP- SORT Priority queues QUICK- SORT Analysing QUICK- SORT Tutorial CS 270 First property: level-completeness Algorithms Oliver Kullmann Binary heaps In week 7 we have seen binary trees: Heapification Building a 1 We said they should be as “balanced” as possible. heap 2 Perfect are the perfect binary trees. HEAP- SORT 3 Now close to perfect come the level-complete binary Priority trees: queues QUICK- 1 We can partition the nodes of a (binary) tree T into levels, SORT according to their distance from the root. Analysing 2 We have levels 0, 1,..., ht(T ). QUICK- k SORT 3 Level k has from 1 to 2 nodes. Tutorial 4 If all levels k except possibly of level ht(T ) are full (have precisely 2k nodes in them), then we call the tree level-complete. CS 270 Examples Algorithms Oliver The binary tree Kullmann 1 ❚ Binary heaps ❥❥❥❥ ❚❚❚❚ ❥❥❥❥ ❚❚❚❚ ❥❥❥❥ ❚❚❚❚ Heapification 2 ❥ 3 ❖ ❄❄ ❖❖ Building a ⑧⑧ ❄ ⑧⑧ ❖❖ heap ⑧⑧ ❄ ⑧⑧ ❖❖❖ 4 5 6 ❄ 7 ❄ HEAP- ⑧ ❄ ⑧ ❄ SORT ⑧⑧ ❄ ⑧⑧ ❄ Priority 10 13 14 15 queues QUICK- is level-complete (level-sizes are 1, 2, 4, 4), while SORT ❥ 1 ❚❚ Analysing ❥❥❥❥ ❚❚❚❚ QUICK- ❥❥❥ ❚❚❚ SORT ❥❥❥ ❚❚❚❚ 2 ❥❥ 3 ♦♦♦ ❄❄ ⑧ Tutorial ♦♦ ❄❄ ⑧⑧ ♦♦♦ ⑧ 4 ❄ 5 ❄ 6 ❄ ⑧⑧ ❄❄ ⑧ ❄ ⑧ ❄ ⑧⑧ ❄ ⑧⑧ ❄ ⑧⑧ ❄ 8 9 10 11 12 13 is not (level-sizes are 1, 2, 3, 6). -
AVL Tree, Bayer Tree, Heap Summary of the Previous Lecture
DATA STRUCTURES AND ALGORITHMS Hierarchical data structures: AVL tree, Bayer tree, Heap Summary of the previous lecture • TREE is hierarchical (non linear) data structure • Binary trees • Definitions • Full tree, complete tree • Enumeration ( preorder, inorder, postorder ) • Binary search tree (BST) AVL tree The AVL tree (named for its inventors Adelson-Velskii and Landis published in their paper "An algorithm for the organization of information“ in 1962) should be viewed as a BST with the following additional property: - For every node, the heights of its left and right subtrees differ by at most 1. Difference of the subtrees height is named balanced factor. A node with balance factor 1, 0, or -1 is considered balanced. As long as the tree maintains this property, if the tree contains n nodes, then it has a depth of at most log2n. As a result, search for any node will cost log2n, and if the updates can be done in time proportional to the depth of the node inserted or deleted, then updates will also cost log2n, even in the worst case. AVL tree AVL tree Not AVL tree Realization of AVL tree element struct AVLnode { int data; AVLnode* left; AVLnode* right; int factor; // balance factor } Adding a new node Insert operation violates the AVL tree balance property. Prior to the insert operation, all nodes of the tree are balanced (i.e., the depths of the left and right subtrees for every node differ by at most one). After inserting the node with value 5, the nodes with values 7 and 24 are no longer balanced.