Segmented Hash: an Efficient Hash
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Podcast Ch23a
Podcast Ch23a • Title: Bit Arrays • Description: Overview; bit operations in Java; BitArray class • Participants: Barry Kurtz (instructor); John Helfert and Tobie Williams (students) • Textbook: Data Structures for Java; William H. Ford and William R. Topp Bit Arrays • Applications such as compiler code generation and compression algorithms create data that includes specific sequences of bits. – Many applications, such as compilers, generate specific sequences of bits. Bit Arrays (continued) • Java binary bit handling operators |, &, and ^ act on pairs of bits and return the new value. The unary operator ~ inverts the bits of its operand. BitBit OperationsOperations x y ~x x | y x & y x ^ y 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 Bit Arrays (continued) Bit Arrays (continued) • Operator << shifts integer or char values to the left. Operators >> and >>> shift values to the right using signed or unsigned arithmetic, respectively. Assume x and y are 32-bit integers. x = 0...10110110 x << 2 = 0...1011011000 x = 101...11011100 x >> 3 = 111101...11011 x = 101...11011100 x >>> 3 = 000101...11011 Bit Arrays (continued) Before performing the bitwise operator |, &, or ^, Java performs binary numeric promotion on the operands. The type of the bitwise operator expression is the promoted type of the operands. The rules of promotion are as follows: • If either operand is of type long, the other is converted to long. • Otherwise, both operands are converted to type int. In the case of the unary operator ~, Java converts a byte, char, short to int before applying the operator, and the resulting value is an int. -
Binary Index Trees a Cumulative Frequency Array Allows Us To
Binary Index Trees A cumulative frequency array allows us to calculate the sum of the range of values in O(1), as long as there are no changes to the data once the queries start. But consider situations where we might change the value at an index in an array, then query for the sum of a range of values in the array, followed by some more changes and more queries. In this sort of situation, a cumulative frequency array would still give us O(1) query times, but it would take O(n) time to update after each change!!! (Basically, if we change one index in a cumulative frequency array, all other indexes above it would have to have this value added to it as well.) Here is a quick illustration: Current cumulative frequency array: Index 0 1 2 3 4 5 6 7 8 Value 2 4 4 4 6 8 11 13 17 Now, consider adding the value 2 to the data, recalling that index i stores the number of values less than or equal to i. The adjusted array is: Index 0 1 2 3 4 5 6 7 8 Value 2 4 5 5 7 9 12 14 18 We had to edit each index 2 or greater, which would take O(n) for a cumulative frequency array of size n. Thus, we want a new arrangement where both the query AND the update are relatively fast. The creative insight here is that perhaps if we store data in a different way, perhaps we can reduce the update time by quite a bit while incurring only a modest increase in query time. -
B-Bit Sketch Trie: Scalable Similarity Search on Integer Sketches
b-Bit Sketch Trie: Scalable Similarity Search on Integer Sketches Shunsuke Kanda Yasuo Tabei RIKEN Center for Advanced Intelligence Project RIKEN Center for Advanced Intelligence Project Tokyo, Japan Tokyo, Japan [email protected] [email protected] Abstract—Recently, randomly mapping vectorial data to algorithms intending to build sketches of non-negative inte- strings of discrete symbols (i.e., sketches) for fast and space- gers (i.e., b-bit sketches) have been proposed for efficiently efficient similarity searches has become popular. Such random approximating various similarity measures. Examples are b-bit mapping is called similarity-preserving hashing and approximates a similarity metric by using the Hamming distance. Although minwise hashing (minhash) [12]–[14] for Jaccard similarity, many efficient similarity searches have been proposed, most of 0-bit consistent weighted sampling (CWS) for min-max ker- them are designed for binary sketches. Similarity searches on nel [15], and 0-bit CWS for generalized min-max kernel [16]. integer sketches are in their infancy. In this paper, we present Thus, developing scalable similarity search methods for b-bit a novel space-efficient trie named b-bit sketch trie on integer sketches is a key issue in large-scale applications of similarity sketches for scalable similarity searches by leveraging the idea behind succinct data structures (i.e., space-efficient data structures search. while supporting various data operations in the compressed Similarity searches on binary sketches are classified -
Compact Fenwick Trees for Dynamic Ranking and Selection
Compact Fenwick trees for dynamic ranking and selection Stefano Marchini Sebastiano Vigna Dipartimento di Informatica, Universit`adegli Studi di Milano, Italy October 15, 2019 Abstract The Fenwick tree [3] is a classical implicit data structure that stores an array in such a way that modifying an element, accessing an element, computing a prefix sum and performing a predecessor search on prefix sums all take logarithmic time. We introduce a number of variants which improve the classical implementation of the tree: in particular, we can reduce its size when an upper bound on the array element is known, and we can perform much faster predecessor searches. Our aim is to use our variants to implement an efficient dynamic bit vector: our structure is able to perform updates, ranking and selection in logarithmic time, with a space overhead in the order of a few percents, outperforming existing data structures with the same purpose. Along the way, we highlight the pernicious interplay between the arithmetic behind the Fenwick tree and the structure of current CPU caches, suggesting simple solutions that improve performance significantly. 1 Introduction The problem of building static data structures which perform rank and select operations on vectors of n bits in constant time using additional o(n) bits has received a great deal of attention in the last two decades starting form Jacobson's seminal work on succinct data structures. [7] The rank operator takes a position in the bit vector and returns the number of preceding ones. The select operation returns the position of the k-th one in the vector, given k. -
Hash Tables & Searching Algorithms
Search Algorithms and Tables Chapter 11 Tables • A table, or dictionary, is an abstract data type whose data items are stored and retrieved according to a key value. • The items are called records. • Each record can have a number of data fields. • The data is ordered based on one of the fields, named the key field. • The record we are searching for has a key value that is called the target. • The table may be implemented using a variety of data structures: array, tree, heap, etc. Sequential Search public static int search(int[] a, int target) { int i = 0; boolean found = false; while ((i < a.length) && ! found) { if (a[i] == target) found = true; else i++; } if (found) return i; else return –1; } Sequential Search on Tables public static int search(someClass[] a, int target) { int i = 0; boolean found = false; while ((i < a.length) && !found){ if (a[i].getKey() == target) found = true; else i++; } if (found) return i; else return –1; } Sequential Search on N elements • Best Case Number of comparisons: 1 = O(1) • Average Case Number of comparisons: (1 + 2 + ... + N)/N = (N+1)/2 = O(N) • Worst Case Number of comparisons: N = O(N) Binary Search • Can be applied to any random-access data structure where the data elements are sorted. • Additional parameters: first – index of the first element to examine size – number of elements to search starting from the first element above Binary Search • Precondition: If size > 0, then the data structure must have size elements starting with the element denoted as the first element. In addition, these elements are sorted. -
Chapter 5 Hashing
Introduction hashing performs basic operations, such as insertion, Chapter 5 deletion, and finds in average time Hashing 2 Hashing Hashing Functions a hash table is merely an of some fixed size let be the set of search keys hashing converts into locations in a hash hash functions map into the set of in the table hash table searching on the key becomes something like array lookup ideally, distributes over the slots of the hash table, to minimize collisions hashing is typically a many-to-one map: multiple keys are if we are hashing items, we want the number of items mapped to the same array index hashed to each location to be close to mapping multiple keys to the same position results in a example: Library of Congress Classification System ____________ that must be resolved hash function if we look at the first part of the call numbers two parts to hashing: (e.g., E470, PN1995) a hash function, which transforms keys into array indices collision resolution involves going to the stacks and a collision resolution procedure looking through the books almost all of CS is hashed to QA75 and QA76 (BAD) 3 4 Hashing Functions Hashing Functions suppose we are storing a set of nonnegative integers we can also use the hash function below for floating point given , we can obtain hash values between 0 and 1 with the numbers if we interpret the bits as an ______________ hash function two ways to do this in C, assuming long int and double when is divided by types have the same length fast operation, but we need to be careful when choosing first method uses -
Efficient Data Structures for High Speed Packet Processing
Efficient data structures for high speed packet processing Paolo Giaccone Notes for the class on \Computer aided simulations and performance evaluation " Politecnico di Torino November 2020 Outline 1 Applications 2 Theoretical background 3 Tables Direct access arrays Hash tables Multiple-choice hash tables Cuckoo hash 4 Set Membership Problem definition Application Fingerprinting Bit String Hashing Bloom filters Cuckoo filters 5 Longest prefix matching Patricia trie Giaccone (Politecnico di Torino) Hash, Cuckoo, Bloom and Patricia Nov. 2020 2 / 93 Applications Section 1 Applications Giaccone (Politecnico di Torino) Hash, Cuckoo, Bloom and Patricia Nov. 2020 3 / 93 Applications Big Data and probabilistic data structures 3 V's of Big Data Volume (amount of data) Velocity (speed at which data is arriving and is processed) Variety (types of data) Main efficiency metrics for data structures space time to write, to update, to read, to delete Probabilistic data structures based on different hashing techniques approximated answers, but reliable estimation of the error typically, low memory, constant query time, high scaling Giaccone (Politecnico di Torino) Hash, Cuckoo, Bloom and Patricia Nov. 2020 4 / 93 Applications Probabilistic data structures Membership answer approximate membership queries e.g., Bloom filter, counting Bloom filter, quotient filter, Cuckoo filter Cardinality estimate the number of unique elements in a dataset. e.g., linear counting, probabilistic counting, LogLog and HyperLogLog Frequency in streaming applications, find the frequency of some element, filter the most frequent elements in the stream, detect the trending elements, etc. e.g., majority algorithm, frequent algorithm, count sketch, count{min sketch Giaccone (Politecnico di Torino) Hash, Cuckoo, Bloom and Patricia Nov. -
K-Mer Data Structures Rayan Chikhi CNRS, Univ
k-mer data structures Rayan Chikhi CNRS, Univ. Lille, France CGSI - July 24, 2018 Baseline problem In-memory representation of a large set of short k-mers: e.g. ACTGAT GTATGC ATTAAA GAATTG ... (Indirect) applications ● Assembly ● Error-correction of reads ● Detection of similarity between sequences ● Detection of distances between datasets ● Alignment ● Pseudoalignment / quasi-mapping ● Detection of taxonomy ● Indexing large collections of sequencing datasets ● Quality control ● Detection of events (e.g. SNPs, indels, CNVs, alt. transcription) ● ... Goals of this lecture ● Broad sweep of state of the art, with applications ● Refresher of basic CS elements Au programme: ● Basic structures (Bloom Filters, CQF, Hashing, Perfect Hashing) ● k-mer data structures (SBT, BFT, dBG ds) ● Some reference-free applications k-mers Sequences of k consecutive letters, e.g. ACAG or TAGG for k=4 Problem statement: Framing the problem Representation of a set of k-mers: ACTGAT 6 11 Large set of k-mers : 10 - 10 elements GTATGC k in [11; 103] .. Problem statement: Operations to support Representation of a set of k-mers: - Construction (from a disk stream) ACTGAT - Membership (“is X in the set?”) GTATGC - Iteration (enumerate all elements in the set) - ... .. 106 - 1011 elements Extensions: k: 11 - 500 - Associate value(s) to k-mers (e.g. abundance) - - Navigate the de Bruijn graph Problem statement: Data structures Representation of a set of k-mers: ACTGAT “In computer science, a data structure is a GTATGC particular way of organizing and storing data in a -
CMU SCS 15-721 (Spring 2020) :: OLTP Indexes (Trie Data Structures)
ADVANCED DATABASE SYSTEMS OLTP Indexes (Trie Data Structures) @Andy_Pavlo // 15-721 // Spring 2020 Lecture #07 2 Latches B+Trees Judy Array ART Masstree 15-721 (Spring 2020) 3 LATCH IMPLEMENTATION GOALS Small memory footprint. Fast execution path when no contention. Deschedule thread when it has been waiting for too long to avoid burning cycles. Each latch should not have to implement their own queue to track waiting threads. Source: Filip Pizlo 15-721 (Spring 2020) 3 LATCH IMPLEMENTATION GOALS Small memory footprint. Fast execution path when no contention. Deschedule thread when it has been waiting for too long to avoid burning cycles. Each latch should not have to implement their own queue to track waiting threads. Source: Filip Pizlo 15-721 (Spring 2020) 4 LATCH IMPLEMENTATIONS Test-and-Set Spinlock Blocking OS Mutex Adaptive Spinlock Queue-based Spinlock Reader-Writer Locks 15-721 (Spring 2020) 5 LATCH IMPLEMENTATIONS Choice #1: Test-and-Set Spinlock (TaS) → Very efficient (single instruction to lock/unlock) → Non-scalable, not cache friendly, not OS friendly. → Example: std::atomic<T> std::atomic_flag latch; ⋮ while (latch.test_and_set(…)) { // Yield? Abort? Retry? } 15-721 (Spring 2020) 5 LATCH IMPLEMENTATIONS Choice #1: Test-and-Set Spinlock (TaS) → Very efficient (single instruction to lock/unlock) → Non-scalable, not cache friendly, not OS friendly. → Example: std::atomic<T> std::atomic_flag latch; ⋮ while (latch.test_and_set(…)) { // Yield? Abort? Retry? } 15-721 (Spring 2020) 6 LATCH IMPLEMENTATIONS Choice #2: Blocking OS Mutex → Simple to use → Non-scalable (about 25ns per lock/unlock invocation) → Example: std::mutex std::mutex m; ⋮ m.lock(); // Do something special... m.unlock(); 15-721 (Spring 2020) 6 LATCH IMPLEMENTATIONS Choice #2: Blocking OS Mutex → Simple to use → Non-scalable (about 25ns per lock/unlock invocation) → Example: std::mutex std::mutex m; pthread_mutex_t ⋮ m.lock(); futex // Do something special.. -
Cuckoo Hashing
Cuckoo Hashing Rasmus Pagh* BRICSy, Department of Computer Science, Aarhus University Ny Munkegade Bldg. 540, DK{8000 A˚rhus C, Denmark. E-mail: [email protected] and Flemming Friche Rodlerz ON-AIR A/S, Digtervejen 9, 9200 Aalborg SV, Denmark. E-mail: ff[email protected] We present a simple dictionary with worst case constant lookup time, equal- ing the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. (Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738{761, 1994). The space usage is similar to that of binary search trees, i.e., three words per key on average. Besides being conceptually much simpler than previous dynamic dictionaries with worst case constant lookup time, our data structure is interesting in that it does not use perfect hashing, but rather a variant of open addressing where keys can be moved back in their probe sequences. An implementation inspired by our algorithm, but using weaker hash func- tions, is found to be quite practical. It is competitive with the best known dictionaries having an average case (but no nontrivial worst case) guarantee. Key Words: data structures, dictionaries, information retrieval, searching, hash- ing, experiments * Partially supported by the Future and Emerging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT). This work was initiated while visiting Stanford University. y Basic Research in Computer Science (www.brics.dk), funded by the Danish National Research Foundation. z This work was done while at Aarhus University. 1 2 PAGH AND RODLER 1. INTRODUCTION The dictionary data structure is ubiquitous in computer science. -
Space- and Time-Efficient String Dictionaries
Tokushima University Ph.D. Thesis Space- and Time-Efficient String Dictionaries z間¹率hB間¹率noD文W列辞ø Author: Supervisor: Shunsuke Kanda Prof. Masao Fuketa A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Advanced Technology and Science Department of Information Science and Intelligent Systems March 2018 iii Abstract In modern computer science, the management of massive data is a fundamental problem because the amount of data is growing faster than we can easily handle them. Such data are often represented as strings such as documents, Web contents and genomics data; therefore, data structures and algorithms for space-efficient string processing have been developed by many researchers. In this thesis, we focus on a string dictionary that is an in-memory data structure for storing a set of strings. It has been traditionally used to manage vocabulary in natural language processing and information retrieval. The size of the dictionaries is not problematic because of Heaps’ Law. However, string dictionaries in recent applications, such as Web search engines, RDF stores, geographic information systems and bioinformatics, need to handle very large datasets. As the space usage of string dictionaries is a significant issue in those applications, it is necessary to develop space-efficient data structures. If limited to static applications, existing data structures have already achieved very high space efficiency by exploiting succinct data structures and text compression techniques. For example, state-of-the-art string dictionaries can be implemented in space up to 5% of the original dataset size. However, there remain trade-off problems among space efficiency, lookup-time performance and construction costs. -
Double Hashing
Double Hashing Intro & Coding Hashing Hashing - provides O(1) time on average for insert, search and delete Hash function - maps a big number or string to a small integer that can be used as index in hash table. Collision - Two keys resulting in same index. Hashing – a Simple Example ● Arbitrary Size à Fix Size 0 1 2 3 4 Hashing – a Simple Example ● Arbitrary Size à Fix Size Insert(0) 0 % 5 = 0 0 1 2 3 4 0 Hashing – a Simple Example ● Arbitrary Size à Fix Size Insert(0) 0 % 5 = 0 Insert(5321) 5321 % 5 = 1 0 1 2 3 4 0 5321 Hashing – a Simple Example ● Arbitrary Size à Fix Size Insert(0) 0 % 5 = 0 Insert(5321) 5321 % 5 = 1 Insert(-8002) -8002 % 5 = 3 0 1 2 3 4 0 5321 -8002 Hashing – a Simple Example ● Arbitrary Size à Fix Size Insert(0) 0 % 5 = 0 Insert(5321) 5321 % 5 = 1 Insert(-8002) -8002 % 5 = 3 Insert(20000) 20000 % 5 = 0 0 1 2 3 4 0 5321 -8002 Modular Hashing ● Overall a good simple, general approach to implement a hash map ● Basic formula: ○ h(x) = c(x) mod m ■ Where c(x) converts x into a (possibly) large integer ● Generally want m to be a prime number ○ Consider m = 100 ○ Only the least significant digits matter ■ h(1) = h(401) = h(4372901) Collision Resolution ● A strategy for handling the case when two or more keys to be inserted hash to the same index. ● Closed Addressing § Separate Chaining ● Open Addressing § Linear Probing § Quadratic Probing § Double Hashing Separate Chaining ● Make each cell of hash table point to a linked list of records that have same hash function value Key Hash Value S 2 0 E 0 1 A 0 2 R 4 3 C 4 4 H 4