Hash Tables Is Balanced BST Efficient Enough?
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4 Hash Tables and Associative Arrays
4 FREE Hash Tables and Associative Arrays If you want to get a book from the central library of the University of Karlsruhe, you have to order the book in advance. The library personnel fetch the book from the stacks and deliver it to a room with 100 shelves. You find your book on a shelf numbered with the last two digits of your library card. Why the last digits and not the leading digits? Probably because this distributes the books more evenly among the shelves. The library cards are numbered consecutively as students sign up, and the University of Karlsruhe was founded in 1825. Therefore, the students enrolled at the same time are likely to have the same leading digits in their card number, and only a few shelves would be in use if the leadingCOPY digits were used. The subject of this chapter is the robust and efficient implementation of the above “delivery shelf data structure”. In computer science, this data structure is known as a hash1 table. Hash tables are one implementation of associative arrays, or dictio- naries. The other implementation is the tree data structures which we shall study in Chap. 7. An associative array is an array with a potentially infinite or at least very large index set, out of which only a small number of indices are actually in use. For example, the potential indices may be all strings, and the indices in use may be all identifiers used in a particular C++ program.Or the potential indices may be all ways of placing chess pieces on a chess board, and the indices in use may be the place- ments required in the analysis of a particular game. -
Cmsc 132: Object-Oriented Programming Ii
CMSC 132: OBJECT-ORIENTED PROGRAMMING II Hashing Department of Computer Science University of Maryland, College Park © Department of Computer Science UMD Announcements • Video “What most schools don’t teach” • http://www.youtube.com/watch?v=nKIu9yen5nc © Department of Computer Science UMD Introduction • If you need to find a value in a list what is the most efficient way to perform the search? • Linear search • Binary search • Can we have O(1)? © Department of Computer Science UMD Hashing • Remember that modulus allows us to map a number to a range • X % N → X mapped to value between 0 and N - 1 • Suppose you have 4 parking spaces and need to assign each resident a space. How can we do it? parkingSpace(ssn) = ssn % 4 • Problems?? • What if two residents are assigned the same spot? Collission! • What if we want to use name instead of ssn? • Generate integer out of the name • We just described hashing © Department of Computer Science UMD Hashing • Hashing • Technique for storing key-value entries into an array • In Java we will have an array of Objects where each Object has a key (e.g., student’s name) and a reference to data of interest (e.g., student’s grades) • The array is called the hash table • Ideally can result in O(1) search times • Hash Function • Takes a search key (Ki) and returns a location in the array (an integer index (hash index)) • A search key maps (hashes) to index i • Ideal Hash Function • If every search key corresponds to a unique element in the hash table © Department of Computer Science UMD Hashing • If we have a large range of possible search keys, but a subset of them are used, allocating a large table would a waste of significant space • Typical hash function (two steps) 1. -
Lecture 12: Hash Tables
CSI 3334, Data Structures Lecture 12: Hash Tables Date: 2012-10-10 Author(s): Philip Spencer, Chris Martin Lecturer: Fredrik Niemelä This lecture covers hashing, hash functions, collisions and how to deal with them, and probability of error in bloom filters. 1 Introduction to Hash Tables Hash Functions take input and randomly sort the input and evenly distribute it. An example of a hash function is randu: 31 ni+1 = 65539 ¤ nimod2 randu has flaws in that if you give it an odd number, it will only return odd num- bers. Also, all numbers generated by randu fit in 15 planes. An example of a bad hash function: return 9 2 Dealing with Collisions Definition 2.1 Separate Chaining - Adds to a linked list when collisions occur. Positives of Separate Chaining: There is no upper-bound on size of table and it’s easy to implement. Negatives of Separate Chaining: It gets slower as you have more collisions. Definition 2.2 Linear Probing - If the space is full, take the next space. Negatives of Linear Probing: Things linearly gather making it likely to cluster and slow down. This is known as primary clustering. This also increases the likelihood of new things being added to the cluster. It breaks when the hash table is full. 1 2 CSI 3334 – Fall 2012 Definition 2.3 Quadratic Probing - If the space is full, move i2spaces: Positives of Quadratic Probing: Avoids the primary clustering from linear probing. Negatives of Quadratic Probing: It breaks when the hash table is full. Secondary clustering occurs when things follow the same quadratic path slowing down hashing. -
Prefix Hash Tree an Indexing Data Structure Over Distributed Hash
Prefix Hash Tree An Indexing Data Structure over Distributed Hash Tables Sriram Ramabhadran ∗ Sylvia Ratnasamy University of California, San Diego Intel Research, Berkeley Joseph M. Hellerstein Scott Shenker University of California, Berkeley International Comp. Science Institute, Berkeley and and Intel Research, Berkeley University of California, Berkeley ABSTRACT this lookup interface has allowed a wide variety of Distributed Hash Tables are scalable, robust, and system to be built on top DHTs, including file sys- self-organizing peer-to-peer systems that support tems [9, 27], indirection services [30], event notifi- exact match lookups. This paper describes the de- cation [6], content distribution networks [10] and sign and implementation of a Prefix Hash Tree - many others. a distributed data structure that enables more so- phisticated queries over a DHT. The Prefix Hash DHTs were designed in the Internet style: scala- Tree uses the lookup interface of a DHT to con- bility and ease of deployment triumph over strict struct a trie-based structure that is both efficient semantics. In particular, DHTs are self-organizing, (updates are doubly logarithmic in the size of the requiring no centralized authority or manual con- domain being indexed), and resilient (the failure figuration. They are robust against node failures of any given node in the Prefix Hash Tree does and easily accommodate new nodes. Most impor- not affect the availability of data stored at other tantly, they are scalable in the sense that both la- nodes). tency (in terms of the number of hops per lookup) and the local state required typically grow loga- Categories and Subject Descriptors rithmically in the number of nodes; this is crucial since many of the envisioned scenarios for DHTs C.2.4 [Comp. -
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 -
Introduction to Hash Table and Hash Function
Introduction to Hash Table and Hash Function This is a short introduction to Hashing mechanism Introduction Is it possible to design a search of O(1)– that is, one that has a constant search time, no matter where the element is located in the list? Let’s look at an example. We have a list of employees of a fairly small company. Each of 100 employees has an ID number in the range 0 – 99. If we store the elements (employee records) in the array, then each employee’s ID number will be an index to the array element where this employee’s record will be stored. In this case once we know the ID number of the employee, we can directly access his record through the array index. There is a one-to-one correspondence between the element’s key and the array index. However, in practice, this perfect relationship is not easy to establish or maintain. For example: the same company might use employee’s five-digit ID number as the primary key. In this case, key values run from 00000 to 99999. If we want to use the same technique as above, we need to set up an array of size 100,000, of which only 100 elements will be used: Obviously it is very impractical to waste that much storage. But what if we keep the array size down to the size that we will actually be using (100 elements) and use just the last two digits of key to identify each employee? For example, the employee with the key number 54876 will be stored in the element of the array with index 76. -
Balanced Search Trees and Hashing Balanced Search Trees
Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the desired node The path length can vary from log2(n+1) to O(n) depending on how balanced or unbalanced the tree is The shape of the tree is determined by the values of the items and the order in which they were inserted CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 2 Examples 10 40 20 20 60 30 40 10 30 50 70 50 Can you get the same tree with 60 different insertion orders? 70 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 3 2-3 Trees Each internal node has two or three children All leaves are at the same level 2 children = 2-node, 3 children = 3-node CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 4 2-3 Trees (continued) 2-3 trees are not binary trees (duh) A 2-3 tree of height h always has at least 2h-1 nodes i.e. greater than or equal to a binary tree of height h A 2-3 tree with n nodes has height less than or equal to log2(n+1) i.e less than or equal to the height of a full binary tree with n nodes CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 5 Definition of 2-3 Trees T is a 2-3 tree of height h if T is empty (height 0), OR T is of the form r TL TR Where r is a node that contains one data item and TL and TR are 2-3 trees, each of height h-1, and the search key of r is greater than any in TL and less than any in TR, OR CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 6 Definition of 2-3 Trees (continued) T is of the form r TL TM TR Where r is a node that contains two data items and TL, TM, and TR are 2-3 trees, each of height h-1, and the smaller search key of r is greater than any in TL and less than any in TM and the larger search key in r is greater than any in TM and smaller than any in TR. -
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. -
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 -
Lecture 26 — Hash Tables
Parallel and Sequential Data Structures and Algorithms — Lecture 26 15-210 (Spring 2012) Lecture 26 — Hash Tables Parallel and Sequential Data Structures and Algorithms, 15-210 (Spring 2012) Lectured by Margaret Reid-Miller — 24 April 2012 Today: - Hashing - Hash Tables 1 Hashing hash: transitive verb1 1. (a) to chop (as meat and potatoes) into small pieces (b) confuse, muddle 2. ... This is the definition of hash from which the computer term was derived. The idea of hashing as originally conceived was to take values and to chop and mix them to the point that the original values are muddled. The term as used in computer science dates back to the early 1950’s. More formally the idea of hashing is to approximate a random function h : α β from a source (or universe) set U of type α to a destination set of type β. Most often the source! set is significantly larger than the destination set, so the function not only chops and mixes but also reduces. In fact the source set might have infinite size, such as all character strings, while the destination set always has finite size. Also the source set might consist of complicated elements, such as the set of all directed graphs, while the destination are typically the integers in some fixed range. Hash functions are therefore many to one functions. Using an actual randomly selected function from the set of all functions from α to β is typically not practical due to the number of such functions and hence the size (number of bits) needed to represent such a function. -
Hashing (Part 2)
Hashing (part 2) CSE 2011 Winter 2011 14 March 2011 1 Collision Handling Separate chaining Pbi(Probing (open a ddress ing ) Linear probing Quadratic probing Double hashing 2 1 Quadratic Probing Linear probing: Quadratic probing Insert item (k, e) A[i] is occupied i = h(k) Try A[(i+1) mod N]: used A[i] is occupied Try A[(i+22) mod N]: used Try A[(i+1) mod N]: used Try A[(i+32) mod N] Try A[(i+2) mod N] and so on and so on until an empty cell is found May not be able to find an empty cell if N is not prime, or the hash table is at least half full 3 Double Hashing Double hashing uses a Insert item (k, e) secondary hash function i = h(()k) d(k) and handles collisions A[i] is occupied by placing an item in the first available cell of the Try A[(i+d(k))mod N]: used series Try A[(i+2d(k))mod N]: used (i j d(k)) mod N Try A[(i+3d(k))mod N] for j 0, 1, … , N 1 and so on until The secondary hash an empty cell is found function d(k) cannot have zero values The table size N must be a prime to allow probing of all the cells 4 2 Example of Double Hashing kh(k ) d (k ) Probes Consider a hash 18 5 3 5 tbltable s tor ing itinteger 41 2 1 2 22 9 6 9 keys that handles 44 5 5 5 10 collision with double 59 7 4 7 32 6 3 6 hashing 315 4 590 N 13 73 8 4 8 h(k) k mod 13 d(k) 7 k mod 7 0123456789101112 Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 31 41 18 32 59 73 22 44 0123456789101112 5 Double Hashing (2) d(k) should be chosen to minimize clustering Common choice of compression map for the secondary hash function: d(k) q k mod q where q N q is a prime The possible values for d(k) are 1, 2, … , q Note: linear probing has d(k) 1.