Plsm: a Highly Efficient LSM-Tree Index Supporting Real-Time Big Data Analysis
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Skip Lists: a Probabilistic Alternative to Balanced Trees
Skip Lists: A Probabilistic Alternative to Balanced Trees Skip lists are a data structure that can be used in place of balanced trees. Skip lists use probabilistic balancing rather than strictly enforced balancing and as a result the algorithms for insertion and deletion in skip lists are much simpler and significantly faster than equivalent algorithms for balanced trees. William Pugh Binary trees can be used for representing abstract data types Also giving every fourth node a pointer four ahead (Figure such as dictionaries and ordered lists. They work well when 1c) requires that no more than n/4 + 2 nodes be examined. the elements are inserted in a random order. Some sequences If every (2i)th node has a pointer 2i nodes ahead (Figure 1d), of operations, such as inserting the elements in order, produce the number of nodes that must be examined can be reduced to degenerate data structures that give very poor performance. If log2 n while only doubling the number of pointers. This it were possible to randomly permute the list of items to be in- data structure could be used for fast searching, but insertion serted, trees would work well with high probability for any in- and deletion would be impractical. put sequence. In most cases queries must be answered on-line, A node that has k forward pointers is called a level k node. so randomly permuting the input is impractical. Balanced tree If every (2i)th node has a pointer 2i nodes ahead, then levels algorithms re-arrange the tree as operations are performed to of nodes are distributed in a simple pattern: 50% are level 1, maintain certain balance conditions and assure good perfor- 25% are level 2, 12.5% are level 3 and so on. -
Skip B-Trees
Skip B-Trees Ittai Abraham1, James Aspnes2?, and Jian Yuan3 1 The Institute of Computer Science, The Hebrew University of Jerusalem, [email protected] 2 Department of Computer Science, Yale University, [email protected] 3 Google, [email protected] Abstract. We describe a new data structure, the Skip B-Tree that combines the advantages of skip graphs with features of traditional B-trees. A skip B-Tree provides efficient search, insertion and deletion operations. The data structure is highly fault tolerant even to adversarial failures, and allows for particularly simple repair mechanisms. Related resource keys are kept in blocks near each other enabling efficient range queries. Using this data structure, we describe a new distributed peer-to-peer network, the Distributed Skip B-Tree. Given m data items stored in a system with n nodes, the network allows to perform a range search operation for r consecutive keys that costs only O(logb m + r/b) where b = Θ(m/n). In addition, our distributed Skip B-tree search network has provable polylogarithmic costs for all its other basic operations like insert, delete, and node join. To the best of our knowledge, all previous distributed search networks either provide a range search operation whose cost is worse than ours or may require a linear cost for some basic operation like insert, delete, and node join. ? Supported in part by NSF grants CCR-0098078, CNS-0305258, and CNS-0435201. 1 Introduction Peer-to-peer systems provide a decentralized way to share resources among machines. An ideal peer-to-peer network should have such properties as decentralization, scalability, fault-tolerance, self-stabilization, load- balancing, dynamic addition and deletion of nodes, efficient query searching and exploiting spatial as well as temporal locality in searches. -
On the Cost of Persistence and Authentication in Skip Lists*
On the Cost of Persistence and Authentication in Skip Lists⋆ Micheal T. Goodrich1, Charalampos Papamanthou2, and Roberto Tamassia2 1 Department of Computer Science, University of California, Irvine 2 Department of Computer Science, Brown University Abstract. We present an extensive experimental study of authenticated data structures for dictionaries and maps implemented with skip lists. We consider realizations of these data structures that allow us to study the performance overhead of authentication and persistence. We explore various design decisions and analyze the impact of garbage collection and virtual memory paging, as well. Our empirical study confirms the effi- ciency of authenticated skip lists and offers guidelines for incorporating them in various applications. 1 Introduction A proven paradigm from distributed computing is that of using a large collec- tion of widely distributed computers to respond to queries from geographically dispersed users. This approach forms the foundation, for example, of the DNS system. Of course, we can abstract the main query and update tasks of such systems as simple data structures, such as distributed versions of dictionar- ies and maps, and easily characterize their asymptotic performance (with most operations running in logarithmic time). There are a number of interesting im- plementation issues concerning practical systems that use such distributed data structures, however, including the additional features that such structures should provide. For instance, a feature that can be useful in a number of real-world ap- plications is that distributed query responders provide authenticated responses, that is, answers that are provably trustworthy. An authenticated response in- cludes both an answer (for example, a yes/no answer to the query “is item x a member of the set S?”) and a proof of this answer, equivalent to a digital signature from the data source. -
Tree-Combined Trie: a Compressed Data Structure for Fast IP Address Lookup
(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 6, No. 12, 2015 Tree-Combined Trie: A Compressed Data Structure for Fast IP Address Lookup Muhammad Tahir Shakil Ahmed Department of Computer Engineering, Department of Computer Engineering, Sir Syed University of Engineering and Technology, Sir Syed University of Engineering and Technology, Karachi Karachi Abstract—For meeting the requirements of the high-speed impact their forwarding capacity. In order to resolve two main Internet and satisfying the Internet users, building fast routers issues there are two possible solutions one is IPv6 IP with high-speed IP address lookup engine is inevitable. addressing scheme and second is Classless Inter-domain Regarding the unpredictable variations occurred in the Routing or CIDR. forwarding information during the time and space, the IP lookup algorithm should be able to customize itself with temporal and Finding a high-speed, memory-efficient and scalable IP spatial conditions. This paper proposes a new dynamic data address lookup method has been a great challenge especially structure for fast IP address lookup. This novel data structure is in the last decade (i.e. after introducing Classless Inter- a dynamic mixture of trees and tries which is called Tree- Domain Routing, CIDR, in 1994). In this paper, we will Combined Trie or simply TC-Trie. Binary sorted trees are more discuss only CIDR. In addition to these desirable features, advantageous than tries for representing a sparse population reconfigurability is also of great importance; true because while multibit tries have better performance than trees when a different points of this huge heterogeneous structure of population is dense. -
Exploring the Duality Between Skip Lists and Binary Search Trees
Exploring the Duality Between Skip Lists and Binary Search Trees Brian C. Dean Zachary H. Jones School of Computing School of Computing Clemson University Clemson University Clemson, SC Clemson, SC [email protected] [email protected] ABSTRACT statically optimal [5] skip lists have already been indepen- Although skip lists were introduced as an alternative to bal- dently derived in the literature). anced binary search trees (BSTs), we show that the skip That the skip list can be interpreted as a type of randomly- list can be interpreted as a type of randomly-balanced BST balanced tree is not particularly surprising, and this has whose simplicity and elegance is arguably on par with that certainly not escaped the attention of other authors [2, 10, of today’s most popular BST balancing mechanisms. In this 9, 8]. However, essentially every tree interpretation of the paper, we provide a clear, concise description and analysis skip list in the literature seems to focus entirely on casting of the “BST” interpretation of the skip list, and compare the skip list as a randomly-balanced multiway branching it to similar randomized BST balancing mechanisms. In tree (e.g., a randomized B-tree [4]). Messeguer [8] calls this addition, we show that any rotation-based BST balancing structure the skip tree. Since there are several well-known mechanism can be implemented in a simple fashion using a ways to represent a multiway branching search tree as a BST skip list. (e.g., replace each multiway branching node with a minia- ture balanced BST, or replace (first child, next sibling) with (left child, right child) pointers), it is clear that the skip 1. -
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() { . -
Investigation of a Dynamic Kd Search Skip List Requiring [Theta](Kn) Space
Hcqulslmns ana nqulstwns et Bibliographie Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON KIA ON4 Ottawa ON KIA ON4 Canada Canada Your file Votre réference Our file Notre reldrence The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Libraq of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microforni, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othexwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. A new dynamic k-d data structure (supporting insertion and deletion) labeled the k-d Search Skip List is developed and analyzed. The time for k-d semi-infinite range search on n k-d data points without space restriction is shown to be SZ(k1og n) and O(kn + kt), and there is a space-time trade-off represented as (log T' + log log n) = Sà(n(1og n)"e), where 0 = 1 for k =2, 0 = 2 for k 23, and T' is the scaled query time. The dynamic update tirne for the k-d Search Skip List is O(k1og a). -
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. -
Comparison of Dictionary Data Structures
A Comparison of Dictionary Implementations Mark P Neyer April 10, 2009 1 Introduction A common problem in computer science is the representation of a mapping between two sets. A mapping f : A ! B is a function taking as input a member a 2 A, and returning b, an element of B. A mapping is also sometimes referred to as a dictionary, because dictionaries map words to their definitions. Knuth [?] explores the map / dictionary problem in Volume 3, Chapter 6 of his book The Art of Computer Programming. He calls it the problem of 'searching,' and presents several solutions. This paper explores implementations of several different solutions to the map / dictionary problem: hash tables, Red-Black Trees, AVL Trees, and Skip Lists. This paper is inspired by the author's experience in industry, where a dictionary structure was often needed, but the natural C# hash table-implemented dictionary was taking up too much space in memory. The goal of this paper is to determine what data structure gives the best performance, in terms of both memory and processing time. AVL and Red-Black Trees were chosen because Pfaff [?] has shown that they are the ideal balanced trees to use. Pfaff did not compare hash tables, however. Also considered for this project were Splay Trees [?]. 2 Background 2.1 The Dictionary Problem A dictionary is a mapping between two sets of items, K, and V . It must support the following operations: 1. Insert an item v for a given key k. If key k already exists in the dictionary, its item is updated to be v. -
Binary Search Tree
ADT Binary Search Tree! Ellen Walker! CPSC 201 Data Structures! Hiram College! Binary Search Tree! •" Value-based storage of information! –" Data is stored in order! –" Data can be retrieved by value efficiently! •" Is a binary tree! –" Everything in left subtree is < root! –" Everything in right subtree is >root! –" Both left and right subtrees are also BST#s! Operations on BST! •" Some can be inherited from binary tree! –" Constructor (for empty tree)! –" Inorder, Preorder, and Postorder traversal! •" Some must be defined ! –" Insert item! –" Delete item! –" Retrieve item! The Node<E> Class! •" Just as for a linked list, a node consists of a data part and links to successor nodes! •" The data part is a reference to type E! •" A binary tree node must have links to both its left and right subtrees! The BinaryTree<E> Class! The BinaryTree<E> Class (continued)! Overview of a Binary Search Tree! •" Binary search tree definition! –" A set of nodes T is a binary search tree if either of the following is true! •" T is empty! •" Its root has two subtrees such that each is a binary search tree and the value in the root is greater than all values of the left subtree but less than all values in the right subtree! Overview of a Binary Search Tree (continued)! Searching a Binary Tree! Class TreeSet and Interface Search Tree! BinarySearchTree Class! BST Algorithms! •" Search! •" Insert! •" Delete! •" Print values in order! –" We already know this, it#s inorder traversal! –" That#s why it#s called “in order”! Searching the Binary Tree! •" If the tree is -
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. -
An Experimental Study of Dynamic Biased Skip Lists
AN EXPERIMENTAL STUDY OF DYNAMIC BIASED SKIP LISTS by Yiqun Zhao Submitted in partial fulfillment of the requirements for the degree of Master of Computer Science at Dalhousie University Halifax, Nova Scotia August 2017 ⃝c Copyright by Yiqun Zhao, 2017 Table of Contents List of Tables ................................... iv List of Figures .................................. v Abstract ...................................... vii List of Abbreviations and Symbols Used .................. viii Acknowledgements ............................... ix Chapter 1 Introduction .......................... 1 1.1 Dynamic Data Structures . .2 1.1.1 Move-to-front Lists . .3 1.1.2 Red-black Trees . .3 1.1.3 Splay Trees . .3 1.1.4 Skip Lists . .4 1.1.5 Other Biased Data Structures . .7 1.2 Our Work . .8 1.3 Organization of The Thesis . .8 Chapter 2 A Review of Different Search Structures ......... 10 2.1 Move-to-front Lists . 10 2.2 Red-black Trees . 11 2.3 Splay Trees . 14 2.3.1 Original Splay Tree . 14 2.3.2 Randomized Splay Tree . 18 2.3.3 W-splay Tree . 19 2.4 Skip Lists . 20 Chapter 3 Dynamic Biased Skip Lists ................. 24 3.1 Biased Skip List . 24 3.2 Modifications . 29 3.2.1 Improved Construction Scheme . 29 3.2.2 Lazy Update Scheme . 32 ii 3.3 Hybrid Search Algorithm . 34 Chapter 4 Experiments .......................... 36 4.1 Experimental Setup . 38 4.2 Analysis of Different C1 Sizes . 40 4.3 Analysis of Varying Value of t in H-DBSL . 46 4.4 Comparison of Data Structures using Real-World Data . 49 4.5 Comparison with Data Structures using Generated Data . 55 Chapter 5 Conclusions ..........................