Write-Optimization in a Kernel File System
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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() { . -
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, -
Betrfs: a Right-Optimized Write-Optimized File System
BetrFS: A Right-Optimized Write-Optimized File System William Jannen, Jun Yuan, Yang Zhan, Amogh Akshintala, Stony Brook University; John Esmet, Tokutek Inc.; Yizheng Jiao, Ankur Mittal, Prashant Pandey, and Phaneendra Reddy, Stony Brook University; Leif Walsh, Tokutek Inc.; Michael Bender, Stony Brook University; Martin Farach-Colton, Rutgers University; Rob Johnson, Stony Brook University; Bradley C. Kuszmaul, Massachusetts Institute of Technology; Donald E. Porter, Stony Brook University https://www.usenix.org/conference/fast15/technical-sessions/presentation/jannen This paper is included in the Proceedings of the 13th USENIX Conference on File and Storage Technologies (FAST ’15). February 16–19, 2015 • Santa Clara, CA, USA ISBN 978-1-931971-201 Open access to the Proceedings of the 13th USENIX Conference on File and Storage Technologies is sponsored by USENIX BetrFS: A Right-Optimized Write-Optimized File System William Jannen, Jun Yuan, Yang Zhan, Amogh Akshintala, John Esmet∗, Yizheng Jiao, Ankur Mittal, Prashant Pandey, Phaneendra Reddy, Leif Walsh∗, Michael Bender, Martin Farach-Colton†, Rob Johnson, Bradley C. Kuszmaul‡, and Donald E. Porter Stony Brook University, ∗Tokutek Inc., †Rutgers University, and ‡Massachusetts Institute of Technology Abstract (microwrites). Examples include email delivery, creat- The Bε -tree File System, or BetrFS, (pronounced ing lock files for an editing application, making small “better eff ess”) is the first in-kernel file system to use a updates to a large file, or updating a file’s atime. The un- write-optimized index. Write optimized indexes (WOIs) derlying problem is that many standard data structures in are promising building blocks for storage systems be- the file-system designer’s toolbox optimize for one case cause of their potential to implement both microwrites at the expense of another. -
Algorithms and Complexity (AL)
Algorithms and Complexity (AL) Algorithms are fundamental to computer science and software engineering. The real-world performance of any software system depends on the algorithms chosen and the suitability of the various layers of implementation. Good algorithm design is therefore crucial for the performance of all software systems. Moreover, the study of algorithms provides insight into the intrinsic nature of the problem as well as possible solution techniques independent of programming language, programming paradigm, computer hardware, or any other implementation aspect. An important part of computing is the ability to select algorithms appropriate to particular purposes and to apply them, recognizing the possibility that no suitable algorithm may exist. This facility relies on understanding the range of algorithms that address an important set of well-defined problems, recognizing their strengths and weaknesses, and their suitability in particular contexts. Efficiency is a pervasive theme throughout this area. This knowledge area defines the central concepts and skills required to design, implement, and analyze algorithms for solving problems. Algorithms are essential in all advanced areas of computer science: artificial intelligence, databases, distributed computing, graphics, networking, operating systems, programming languages, security, and so on. Algorithms that have specific utility in each of these are listed in the relevant knowledge areas. Cryptography, for example, appears in the new Knowledge Area on Information Assurance and Security (IAS), while parallel and distributed algorithms appear in the Knowledge Area in Parallel and Distributed Computing (PD). As with all knowledge areas, the order of topics and their groupings do not necessarily correlate to a specific order of presentation. Different programs will teach the topics in different courses and should do so in the order they believe is most appropriate for their students. -
Assignment of Master's Thesis
CZECH TECHNICAL UNIVERSITY IN PRAGUE FACULTY OF INFORMATION TECHNOLOGY ASSIGNMENT OF MASTER’S THESIS Title: Approximate Pattern Matching In Sparse Multidimensional Arrays Using Machine Learning Based Methods Student: Bc. Anna Kučerová Supervisor: Ing. Luboš Krčál Study Programme: Informatics Study Branch: Knowledge Engineering Department: Department of Theoretical Computer Science Validity: Until the end of winter semester 2018/19 Instructions Sparse multidimensional arrays are a common data structure for effective storage, analysis, and visualization of scientific datasets. Approximate pattern matching and processing is essential in many scientific domains. Previous algorithms focused on deterministic filtering and aggregate matching using synopsis style indexing. However, little work has been done on application of heuristic based machine learning methods for these approximate array pattern matching tasks. Research current methods for multidimensional array pattern matching, discovery, and processing. Propose a method for array pattern matching and processing tasks utilizing machine learning methods, such as kernels, clustering, or PSO in conjunction with inverted indexing. Implement the proposed method and demonstrate its efficiency on both artificial and real world datasets. Compare the algorithm with deterministic solutions in terms of time and memory complexities and pattern occurrence miss rates. References Will be provided by the supervisor. doc. Ing. Jan Janoušek, Ph.D. prof. Ing. Pavel Tvrdík, CSc. Head of Department Dean Prague February 28, 2017 Czech Technical University in Prague Faculty of Information Technology Department of Knowledge Engineering Master’s thesis Approximate Pattern Matching In Sparse Multidimensional Arrays Using Machine Learning Based Methods Bc. Anna Kuˇcerov´a Supervisor: Ing. LuboˇsKrˇc´al 9th May 2017 Acknowledgements Main credit goes to my supervisor Ing. -
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. -
Leftist Heap: Is a Binary Tree with the Normal Heap Ordering Property, but the Tree Is Not Balanced. in Fact It Attempts to Be Very Unbalanced!
Leftist heap: is a binary tree with the normal heap ordering property, but the tree is not balanced. In fact it attempts to be very unbalanced! Definition: the null path length npl(x) of node x is the length of the shortest path from x to a node without two children. The null path lengh of any node is 1 more than the minimum of the null path lengths of its children. (let npl(nil)=-1). Only the tree on the left is leftist. Null path lengths are shown in the nodes. Definition: the leftist heap property is that for every node x in the heap, the null path length of the left child is at least as large as that of the right child. This property biases the tree to get deep towards the left. It may generate very unbalanced trees, which facilitates merging! It also also means that the right path down a leftist heap is as short as any path in the heap. In fact, the right path in a leftist tree of N nodes contains at most lg(N+1) nodes. We perform all the work on this right path, which is guaranteed to be short. Merging on a leftist heap. (Notice that an insert can be considered as a merge of a one-node heap with a larger heap.) 1. (Magically and recursively) merge the heap with the larger root (6) with the right subheap (rooted at 8) of the heap with the smaller root, creating a leftist heap. Make this new heap the right child of the root (3) of h1. -
UNIVERSIDAD SAN FRANCISCO DE QUITO USFQ Optimizing Large
UNIVERSIDAD SAN FRANCISCO DE QUITO USFQ Colegio de Ciencias e Ingenierías Optimizing Large Databases: A Study on Index Structures Proyecto de Investigación . Ricardo Andres Leon Ruiz Ingeniería en Sistemas Trabajo de titulación presentado como requisito para la obtención del título de Ingeniero en Sistemas Quito, 22 de Diciembre de 2017 2 UNIVERSIDAD SAN FRANCISCO DE QUITO USFQ COLEGIO DE CIENCIAS E INGENIERÍAS HOJA DE CALIFICACIÓN DE TRABAJO DE TITULACIÓN Optimizing Large Databases: A Study on Index Structures Ricardo Andres Leon Ruiz Calificación: Nombre del profesor, Título académico Aldo Cassola, Ph.D. Firma del profesor Quito, Diciembre de 2017 3 Derechos de Autor Por medio del presente documento certifico que he leído todas las Políticas y Manuales de la Universidad San Francisco de Quito USFQ, incluyendo la Política de Propiedad Intelectual USFQ, y estoy de acuerdo con su contenido, por lo que los derechos de propiedad intelectual del presente trabajo quedan sujetos a lo dispuesto en esas Políticas. Asimismo, autorizo a la USFQ para que realice la digitalización y publicación de este trabajo en el repositorio virtual, de conformidad a lo dispuesto en el Art. 144 de la Ley Orgánica de Educación Superior. Firma del estudiante: Nombres y Apellidos: Ricardo Andres Leon Ruiz Código de estudiante: 110346 C. I.: 1714286265 Lugar, Fecha Quito, 22 de Diciembre de 2017 4 RESUMEN La siguiente investigación trata sobre comparar estructuras de índice para grandes bases de datos, tanto analíticamente, como experimentalmente. El estudio se encuentra dividido en dos partes principales. La primera parte se centra en índices de hash y B-trees. Ambas estructuras son estudiadas en el contexto del modelo de acceso de disco tradicional. -
Data Structures and Programming Spring 2016, Final Exam
Data Structures and Programming Spring 2016, Final Exam. June 21, 2016 1 1. (15 pts) True or False? (Mark for True; × for False. Score = maxf0, Right - 2 Wrongg. No explanations are needed. (1) Let A1;A2, and A3 be three sorted arrays of n real numbers (all distinct). In the comparison model, constructing a balanced binary search tree of the set A1 [ A2 [ A3 requires Ω(n log n) time. × False (Merge A1;A2;A3 in linear time; then pick recursively the middle as root (in linear time).) (2) Let T be a complete binary tree with n nodes. Finding a path from the root of T to a given vertex v 2 T using breadth-first search takes O(log n) time. × False (Note that the tree is NOT a search tree.) (3) Given an unsorted array A[1:::n] of n integers, building a max-heap out of the elements of A can be performed asymptotically faster than building a red-black tree out of the elements of A. True (O(n) for building a max-heap; Ω(n log n) for building a red-black tree.) (4) In the worst case, a red-black tree insertion requires O(1) rotations. True (See class notes) (5) In the worst case, a red-black tree deletion requires O(1) node recolorings. × False (See class notes) (6) Building a binomial heap from an unsorted array A[1:::n] takes O(n) time. True (See class notes) (7) Insertion into an AVL tree asymptotically beats insertion into an AA-tree. × False (See class notes) (8) The subtree of the root of a red-black tree is always itself a red-black tree. -
Conception Et Exploitation D'une Base De Modèles: Application Aux Data
Conception et exploitation d’une base de modèles : application aux data sciences Cyrille Ponchateau To cite this version: Cyrille Ponchateau. Conception et exploitation d’une base de modèles : application aux data sciences. Autre [cs.OH]. ISAE-ENSMA Ecole Nationale Supérieure de Mécanique et d’Aérotechique - Poitiers, 2018. Français. NNT : 2018ESMA0005. tel-01939430 HAL Id: tel-01939430 https://tel.archives-ouvertes.fr/tel-01939430 Submitted on 29 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE pour l’obtention du Grade de DOCTEUR DE L'ÉCOLE NATIONALE SUPÉRIEURE DE MÉCANIQUE ET D'AÉROTECHNIQUE (Diplôme National — Arrêté du 25 mai 2016) Ecole Doctorale : Sciences et Ingénierie des Systèmes, Mathématiques, Informatique (SISMI) Secteur de Recherche : INFORMATIQUE ET APPLICATIONS Présentée par : Cyrille PONCHATEAU ******************************************************** Conception et exploitation d’une base de modèles : application aux data sciences ******************************************************** Directeur de thèse : Ladjel BELLATRECHE -
Heapsort Previous Sorting Algorithms G G Heap Data Structure P Balanced
Previous sortinggg algorithms Insertion Sort 2 O(n ) time Heapsort Merge Sort Based off slides by: David Matuszek http://www.cis.upenn.edu/~matuszek/cit594-2008/ O(n) space PtdbMttBPresented by: Matt Boggus 2 Heap data structure Balanced binary trees Binary tree Recall: The dthfdepth of a no de iitditis its distance f rom th e root The depth of a tree is the depth of the deepest node Balanced A binary tree of depth n is balanced if all the nodes at depths 0 through n-2 have two children Left-justified n-2 (Max) Heap property: no node has a value greater n-1 than the value in its parent n Balanced Balanced Not balanced 3 4 Left-jjyustified binary trees Buildinggp up to heap sort How to build a heap A balanced binary tree of depth n is left- justified if: n it has 2 nodes at depth n (the tree is “full”)or ), or k How to maintain a heap it has 2 nodes at depth k, for all k < n, and all the leaves at dept h n are as fa r le ft as poss i ble How to use a heap to sort data Left-justified Not left-justified 5 6 The heappp prop erty siftUp A node has the heap property if the value in the Given a node that does not have the heap property, you can nodide is as large as or larger t han th e va lues iiin its give it the heap property by exchanging its value with the children value of the larger child 12 12 12 12 14 8 3 8 12 8 14 8 14 8 12 Blue node has Blue node has Blue node does not Blue node does not Blue node has heap property heap property have heap property have heap property heap property All leaf nodes automatically have