Chapter 10: Efficient Collections (Skip Lists, Trees)
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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. -
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. -
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). -
Yikes! Why Is My Systemverilog Still So Slooooow?
DVCon-2019 San Jose, CA Voted Best Paper 1st Place World Class SystemVerilog & UVM Training Yikes! Why is My SystemVerilog Still So Slooooow? Cliff Cummings John Rose Adam Sherer Sunburst Design, Inc. Cadence Design Systems, Inc. Cadence Design System, Inc. [email protected] [email protected] [email protected] www.sunburst-design.com www.cadence.com www.cadence.com ABSTRACT This paper describes a few notable SystemVerilog coding styles and their impact on simulation performance. Benchmarks were run using the three major SystemVerilog simulation tools and those benchmarks are reported in the paper. Some of the most important coding styles discussed in this paper include UVM string processing and SystemVerilog randomization constraints. Some coding styles showed little or no impact on performance for some tools while the same coding styles showed large simulation performance impact. This paper is an update to a paper originally presented by Adam Sherer and his co-authors at DVCon in 2012. The benchmarking described in this paper is only for coding styles and not for performance differences between vendor tools. DVCon 2019 Table of Contents I. Introduction 4 Benchmarking Different Coding Styles 4 II. UVM is Software 5 III. SystemVerilog Semantics Support Syntax Skills 10 IV. Memory and Garbage Collection – Neither are Free 12 V. It is Best to Leave Sleeping Processes to Lie 14 VI. UVM Best Practices 17 VII. Verification Best Practices 21 VIII. Acknowledgment 25 References 25 Author & Contact Information 25 Page 2 Yikes! Why is -
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. -
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 .......................... -
Lecture 2: Data Structures: a Short Background
Lecture 2: Data structures: A short background Storing data It turns out that depending on what we wish to do with data, the way we store it can have a signifcant impact on the running time. So in particular, storing data in certain "structured" way can allow for fast implementation. This is the motivation behind the feld of data structures, which is an extensive area of study. In this lecture, we will give a very short introduction, by illustrating a few common data structures, with some motivating problems. In the last lecture, we mentioned that there are two ways of representing graphs (adjacency list and adjacency matrix), each of which has its own advantages and disadvantages. The former is compact (size is proportional to the number of edges + number of vertices), and is faster for enumerating the neighbors of a vertex (which is what one needs for procedures like Breadth-First-Search). The adjacency matrix is great if one wishes to quickly tell if two vertices and have an edge between them. Let us now consider a diferent problem. Example 1: Scrabble problem Suppose we have a "dictionary" of strings whose average length is , and suppose we have a query string . How can we quickly tell if ? Brute force. Note that the naive solution is to iterate over the strings and check if for some . This takes time . If we only care about answering the query for one , this is not bad, as is the amount of time needed to "read the input". But of course, if we have a fxed dictionary and if we wish to check if for multiple , then this is extremely inefcient. -
A Set Intersection Algorithm Via X-Fast Trie
Journal of Computers A Set Intersection Algorithm Via x-Fast Trie Bangyu Ye* National Engineering Laboratory for Information Security Technologies, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100091, China. * Corresponding author. Tel.:+86-10-82546714; email: [email protected] Manuscript submitted February 9, 2015; accepted May 10, 2015. doi: 10.17706/jcp.11.2.91-98 Abstract: This paper proposes a simple intersection algorithm for two sorted integer sequences . Our algorithm is designed based on x-fast trie since it provides efficient find and successor operators. We present that our algorithm outperforms skip list based algorithm when one of the sets to be intersected is relatively ‘dense’ while the other one is (relatively) ‘sparse’. Finally, we propose some possible approaches which may optimize our algorithm further. Key words: Set intersection, algorithm, x-fast trie. 1. Introduction Fast set intersection is a key operation in the context of several fields when it comes to big data era [1], [2]. For example, modern search engines use the set intersection for inverted posting list which is a standard data structure in information retrieval to return relevant documents. So it has been studied in many domains and fields [3]-[8]. One of the most typical situations is Boolean query which is required to retrieval the documents that contains all the terms in the query. Besides, set intersection is naturally a key component of calculating the Jaccard coefficient of two sets, which is defined by the fraction of size of intersection and union. In this paper, we propose a new algorithm via x-fast trie which is a kind of advanced data structure. -
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. -
FORSCHUNGSZENTRUM JÜLICH Gmbh Programming in C++ Part II
FORSCHUNGSZENTRUM JÜLICH GmbH Jülich Supercomputing Centre D-52425 Jülich, Tel. (02461) 61-6402 Ausbildung von Mathematisch-Technischen Software-Entwicklern Programming in C++ Part II Bernd Mohr FZJ-JSC-BHB-0155 1. Auflage (letzte Änderung: 19.09.2003) Copyright-Notiz °c Copyright 2008 by Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre (JSC). Alle Rechte vorbehalten. Kein Teil dieses Werkes darf in irgendeiner Form ohne schriftliche Genehmigung des JSC reproduziert oder unter Verwendung elektronischer Systeme verarbeitet, vervielfältigt oder verbreitet werden. Publikationen des JSC stehen in druckbaren Formaten (PDF auf dem WWW-Server des Forschungszentrums unter der URL: <http://www.fz-juelich.de/jsc/files/docs/> zur Ver- fügung. Eine Übersicht über alle Publikationen des JSC erhalten Sie unter der URL: <http://www.fz-juelich.de/jsc/docs> . Beratung Tel: +49 2461 61 -nnnn Auskunft, Nutzer-Management (Dispatch) Das Dispatch befindet sich am Haupteingang des JSC, Gebäude 16.4, und ist telefonisch erreich- bar von Montag bis Donnerstag 8.00 - 17.00 Uhr Freitag 8.00 - 16.00 Uhr Tel.5642oder6400, Fax2810, E-Mail: [email protected] Supercomputer-Beratung Tel. 2828, E-Mail: [email protected] Netzwerk-Beratung, IT-Sicherheit Tel. 6440, E-Mail: [email protected] Rufbereitschaft Außerhalb der Arbeitszeiten (montags bis donnerstags: 17.00 - 24.00 Uhr, freitags: 16.00 - 24.00 Uhr, samstags: 8.00 - 17.00 Uhr) können Sie dringende Probleme der Rufbereitschaft melden: Rufbereitschaft Rechnerbetrieb: Tel. 6400 Rufbereitschaft Netzwerke: Tel. 6440 An Sonn- und Feiertagen gibt es keine Rufbereitschaft. Fachberater Tel. +49 2461 61 -nnnn Fachgebiet Berater Telefon E-Mail Auskunft, Nutzer-Management, E.