# Introduction. hashing performs basic operations, such as insertion, better than other ADTs we ve seen so far

Save this PDF as:

Size: px
Start display at page:

Download "Introduction. hashing performs basic operations, such as insertion, better than other ADTs we ve seen so far"

## Transcription

1 Chapter 5 Hashing

2 2 Introduction hashing performs basic operations, such as insertion, deletion, and finds in average time better than other ADTs we ve seen so far

3 3 Hashing a hash table is merely an hashing converts table of some fixed size into locations in a hash searching on the key becomes something like array lookup hashing is typically a many-to-one map: multiple keys are mapped to the same array index mapping multiple keys to the same position results in a that must be resolved two parts to hashing: a hash function, which transforms keys into array indices a collision resolution procedure

4 4 Hashing Functions let K be the set of search keys hash functions map K into the set of M in the hash table h: K {0, 1,, M 1} ideally, h distributes K over the slots of the hash table, to minimize collisions if we are hashing N items, we want the number of items hashed to each location to be close to N/M example: Library of Congress Classification System hash function if we look at the first part of the call numbers (e.g., E470, PN1995) collision resolution involves going to the stacks and looking through the books almost all of CS is hashed to QA75 and QA76 (BAD)

5 5 Hashing Functions suppose we are storing a set of nonnegative integers given M, we can obtain hash values between 0 and 1 with the hash function h k = k % M when k is divided by M fast operation, but we need to be careful when choosing M example: if M = 2 p, h(k) is just the p lowest-order bits of k are all the hash values equally likely? choosing M to be a not too close to a power of 2 works well in practice

6 6 Hashing Functions we can also use the hash function below for floating point numbers if we interpret the bits as an h k = k % M two ways to do this in C, assuming long int and double types have the same length first method uses C to accomplish this task

7 7 Hashing Functions we can also use the hash function below for floating point numbers if we interpret the bits as an integer (cont.) second uses a, which is a variable that can hold objects of different types and sizes

8 8 Hashing Functions we can hash strings by combining a hash of each R is an additional parameter we get to choose if R is larger than any character value, then this approach is what you would obtain if you treated the string as a base-r

9 9 Hashing Functions K&R suggest a slightly simpler hash function, corresponding to R = 31 Weiss suggests R = 37

10 10 Hashing Functions we can use the idea for strings if our search key has parts, say, street, city, state: hash = ((street * R + city) % M) * R + state) % M; same ideas apply to hashing vectors

11 11 Hash Functions the choice of parameters can have a effect on the results of hashing compare the text's string hashing algorithm for different pairs of R and M plot of the number of words hashed to each hash table location; we use the American dictionary from the aspell program as data (305,089 words)

12 12 Hash Functions example: R = 31, M = 1024 good: words are

13 13 Hash Functions example: R = 32, M = 1024 very bad

14 14 Hash Functions example: R = 31, M = 1000 better

15 15 Hash Functions example: R = 32, M = 1000 bad

16 16 Collision Resolution hash table collision occurs when elements hash to the in the table various separate chaining open addressing linear probing other methods for dealing with collision

17 17 Separate Chaining separate chaining keep a list of all elements that to the same location each location in the hash table is a example: first 10 squares

18 18 Separate Chaining insert, search, delete in lists all proportional to of linked list insert new elements can be inserted at of list duplicates can increment other structures could be used instead of lists binary search tree another hash table linked lists good if table is and hash function is good

19 19 Separate Chaining how long are the linked lists in a hash table? value: N M where N is the number of keys and M is the size of the table is it reasonable to assume the hash table would exhibit this behavior? load factor λ = N M average length of a list = λ time to search: function + time to unsuccessful search: 1 + λ successful search: 1 + λ 2 time to evaluate the hash the list

20 20 Separate Chaining observations more important than table size general rule: make the table as large as the number of elements to be stored, λ 1 keep table size prime to ensure good

21 21 Separate Chaining declaration of hash structure

22 22 Separate Chaining hash member function

23 23 Separate Chaining routines for separate chaining

24 24 Separate Chaining routines for separate chaining

25 25 Open Addressing linked lists incur extra costs time to for new cells effort and complexity of defining second data structure a different collision strategy involves placing colliding keys in nearby slots if a collision occurs, try cells until an empty one is found bigger table size needed with M > N load factor should be below λ = 0.5 three common strategies linear probing quadratic probing double hashing

26 26 Linear Probing linear probing insert operation when k is hashed, if slot h k is open, place k there if there is a collision, then start looking for an empty slot starting with location h k + 1 in the hash table, and proceed through h k + 2,, m 1, 0, 1, 2,, h k 1 wrapping around the hash table, looking for an empty slot search operation is similar checking whether a table entry is vacant (or is one we seek) is called a

27 27 Linear Probing example: add 89, 18, 49, 58, 69 with h k f i = i = k % 10 and

28 28 Linear Probing as long as the table is, a vacant cell can be found but time to locate an empty cell can become large blocks of occupied cells results in primary deleting entries leaves some entries may no longer be found may require moving many other entries expected number of probes for search hits: ~ λ for insertion and search misses: ~ λ 2 for λ = 0.5, these values are 3 2 and 5 2, respectively

29 29 Linear Probing performance of linear probing (dashed) vs. more random collision resolution adequate up to λ = 0.5 Successful, Unsuccessful, Insertion

30 30 Quadratic Probing quadratic probing eliminates collision function is quadratic example: add 89, 18, 49, 58, 69 with h k f i = i 2 = k % 10 and

31 31 Quadratic Probing in linear probing, letting table get nearly hurts performance quadratic probing greatly no of finding an empty cell once the table gets larger than half full at most, collisions of the table can be used to resolve if table is half empty and the table size is prime, then we are always guaranteed to accommodate a new element could end up with situation where all keys map to the same table location

32 32 Quadratic Probing quadratic probing collisions will probe the same alternative cells clustering causes less than half an extra probe per search

33 33 Double Hashing double hashing f i = i hash 2 x apply a second hash function to x and probe across function must never evaluate to make sure all cells can be probed

34 34 Double Hashing double hashing example hash 2 x = R x mod R with R = 7 R is a prime smaller than table size insert 89, 18, 49, 58, 69

35 35 Double Hashing double hashing example (cont.) note here that the size of the table (10) is not prime if 23 inserted in the table, it would collide with 58 since hash 2 23 = 7 2 = 5 and the table size is 10, only one alternative location, which is taken

36 36 Rehashing table may get run time of operations may take too long insertions may for quadratic resolution too many removals may be intermixed with insertions solution: build a new table hash function) (with a new go through original hash table to compute a hash value for each (non-deleted) element insert it into the new table

37 37 Rehashing example: insert 13, 15, 24, 6 into a hash table of size 7 with h k = k % 7

38 38 Rehashing example (cont.) insert 23 table will be over 70% full; therefore, a new table is created

39 39 Rehashing example (cont.) new table is size 17 new hash function h k = k % 17 all old elements are inserted into new table

40 40 Rehashing rehashing run time O N since N elements and to rehash the entire table of size roughly 2N must have been N 2 insertions since last rehash rehashing may run OK if in if interactive session, rehashing operation could produce a slowdown rehashing can be implemented with could rehash as soon as the table is half full could rehash only when an insertion fails could rehash only when a certain is reached may be best, as performance degrades as load factor increases

41 41 Hash Tables with Worst-Case O(1) Access hash tables so far O(1) average case for insertions, searches, and deletions separate chaining: worst case ϴ log N log log N some queries will take nearly logarithmic time worst-case O(1) time would be better important for applications such as lookup tables for routers and memory caches if N is known in advance, and elements can be, worst-case O(1) time is achievable

42 42 Hash Tables with Worst-Case O(1) Access perfect hashing assume all N items known separate chaining if the number of lists continually increases, the lists will become shorter and shorter with enough lists, high probability of two problems number of lists might be unreasonably the hashing might still be unfortunate M can be made large enough to have probability 1 2 of no collisions if collision detected, clear table and try again with a different hash function (at most done 2 times)

43 43 Hash Tables with Worst-Case O(1) Access perfect hashing (cont.) how large must M be? theoretically, M should be N 2, which is solution: use N lists resolve collisions by using hash tables instead of linked lists each of these lists can have n 2 elements each secondary hash table will use a different hash function until it is can also perform similar operation for primary hash table total size of secondary hash tables is at most 2N

44 44 Hash Tables with Worst-Case O(1) Access perfect hashing (cont.) example: slots 1, 3, 5, 7 empty; slots 0, 4, 8 have 1 element each; slots 2, 6 have 2 elements each; slot 9 has 3 elements

45 45 Hash Tables with Worst-Case O(1) Access cuckoo hashing ϴ log N log log N bound known for a long time researchers surprised in 1990s to learn that if one of two tables were chosen as items were inserted, the size of the largest list would be ϴ log log N, which is significantly smaller main idea: use 2 tables neither more than full use a separate hash function for each item will be stored in one of these two locations collisions resolved by elements

46 46 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) example: 6 items; 2 tables of size 5; each table has randomly chosen hash function A can be placed at position 0 in Table 1 or position 2 in Table 2 a search therefore requires at most 2 table accesses in this example item deletion is trivial

47 47 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) insertion ensure item is not already in one of the tables use first hash function and if first table location is, insert there if location in first table is occupied element there and place current item in correct position in first table displaced element goes to its alternate hash position in the second table

48 48 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) example: insert A insert B (displace A)

49 49 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) insert C insert D (displace C) and E

50 50 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) insert F (displace E) (E displaces A) (A displaces B) (B relocated)

51 51 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) insert G displacements are G D B A E F C G can try G s second hash value in second table, but it also results in a displacement cycle

52 52 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) cycles if table s load value < 0.5, probability of a cycle is very insertions should require < O log N displacements if a certain number of displacements is reached on an insertion, tables can be with new hash functions

53 53 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) variations higher number of tables (3 or 4) place item in second hash slot immediately instead of other items allow each cell to store space utilization increased keys

54 54 Hash Tables with Worst-Case O(1) Access cuckoo hashing (cont.) benefits worst-case avoidance of potential for potential issues lookup and deletion times extremely sensitive to choice of hash functions time for insertion increases rapidly as load factor approaches 0.5

55 55 Hash Tables with Worst-Case O(1) Access hopscotch hashing improves on linear probing algorithm linear probing tries cells in sequential order, starting from hash location, which can be long due to primary and secondary clustering instead, hopscotch hashing places a bound on of the probe sequence results in worst-case constant-time lookup can be parallelized

56 56 Hash Tables with Worst-Case O(1) Access hopscotch hashing (cont.) if insertion would place an element too far from its hash location, go backward and other elements evicted elements cannot be placed farther than the maximal length each position in the table contains information about the current element inhabiting it, plus others that to it

57 57 Hash Tables with Worst-Case O(1) Access hopscotch hashing (cont.) example: MAX_DIST = 4 each bit string provides 1 bit of information about the current position and the next 3 that follow 1: item hashes to current location; 0: no

58 58 Hash Tables with Worst-Case O(1) Access hopscotch hashing (cont.) example: insert H in 9 try in position 13, but too far, so try candidates for eviction (10, 11, 12) evict G in 11

59 59 Hash Tables with Worst-Case O(1) Access hopscotch hashing (cont.) example: insert I in 6 position 14 too far, so try positions 11, 12, 13 G can move down one position 13 still too far; F can move down one

60 60 Hash Tables with Worst-Case O(1) Access hopscotch hashing (cont.) example: insert I in 6 position 12 still too far, so try positions 9, 10, 11 B can move down three now slot is open for I, fourth from 6

61 61 Hash Tables with Worst-Case O(1) Access universal hashing in principle, we can end up with a situation where all of our keys are hashed to the in the hash table (bad) more realistically, we could choose a hash function that does not distribute the keys to avoid this, we can choose the hash function so that it is independent of the keys being stored yields provably good performance on average

62 62 Hash Tables with Worst-Case O(1) Access universal hashing (cont.) let H be a finite collection of functions mapping our set of keys K to the range {0,1,, M 1} H is a collection if for each pair of distinct keys k, l K, the number of hash functions h H for which h k = h(l) is at most H /M that is, with a randomly selected hash function h H, the chance of a between distinct k and l is not more than the probability 1 M of a collision if h k and h l were chosen randomly and independently from {0,1,, M 1}

63 63 Hash Tables with Worst-Case O(1) Access universal hashing (cont.) example: choose a prime p sufficiently large that every key k is in the range 0 to p 1 (inclusive) let A = 0, 1,, p 1 and B = 1,, p 1 then the family h a,b k = ak + b mod p mod M a A, b B is a universal class of hash functions

64 64 Hash Tables with Worst-Case O(1) Access extendible hashing amount of data too large to fit in main consideration is then the number of disk accesses assume we need to store N records and M = 4 records fit in one disk block current problems if probing or separate chaining is used, collisions could cause to be examined during a search rehashing would be expensive in this case

65 65 Hash Tables with Worst-Case O(1) Access extendible hashing (cont.) allows search to be performed in insertions require a bit more use B-tree disk accesses as M increases, height of B-tree could make height = 1, but multi-way branching would be extremely high

66 66 Hash Tables with Worst-Case O(1) Access extendible hashing (cont.) example: 6-bit integers root contains 4 pointers determined by first 2 bits each leaf has up to 4 elements

67 67 Hash Tables with Worst-Case O(1) Access extendible hashing (cont.) example: insert place in third leaf, but full split leaf into 2 leaves, determined by 3 bits

68 68 Hash Tables with Worst-Case O(1) Access extendible hashing (cont.) example: insert first leaf split

69 69 Hash Tables with Worst-Case O(1) Access extendible hashing (cont.) considerations several directory may be required if the elements in a leaf agree in more than D+1 leading bits number of bits to distinguish bit strings does not work well with duplicates: does not work at all) ( > M

70 70 Hash Tables with Worst-Case O(1) Access final points choose hash function carefully watch separate chaining: close to 1 probing hashing: 0.5 hash tables have some not possible to find min/max not possible to exact string is known for a string unless the binary search trees can do this, and O(log N) is only slightly worse than O(1)

71 71 Hash Tables with Worst-Case O(1) Access final points (cont.) hash tables good for symbol table gaming remembering locations to avoid recomputing through transposition table spell checkers

### Chapter 5 Hashing. Introduction. Hashing. Hashing Functions. hashing performs basic operations, such as insertion,

Introduction Chapter 5 Hashing hashing performs basic operations, such as insertion, deletion, and finds in average time 2 Hashing a hash table is merely an of some fixed size hashing converts into locations

### Lecture 4. Hashing Methods

Lecture 4 Hashing Methods 1 Lecture Content 1. Basics 2. Collision Resolution Methods 2.1 Linear Probing Method 2.2 Quadratic Probing Method 2.3 Double Hashing Method 2.4 Coalesced Chaining Method 2.5

### Open Addressing: Linear Probing (cont.)

Open Addressing: Linear Probing (cont.) Cons of Linear Probing () more complex insert, find, remove methods () primary clustering phenomenon items tend to cluster together in the bucket array, as clustering

### Cpt S 223. School of EECS, WSU

Hashing & Hash Tables 1 Overview Hash Table Data Structure : Purpose To support insertion, deletion and search in average-case constant t time Assumption: Order of elements irrelevant ==> data structure

### Hashing. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

Hashing CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Overview Hashing Technique supporting insertion, deletion and

### Hashing. Dr. Ronaldo Menezes Hugo Serrano. Ronaldo Menezes, Florida Tech

Hashing Dr. Ronaldo Menezes Hugo Serrano Agenda Motivation Prehash Hashing Hash Functions Collisions Separate Chaining Open Addressing Motivation Hash Table Its one of the most important data structures

### Hashing. 1. Introduction. 2. Direct-address tables. CmSc 250 Introduction to Algorithms

Hashing CmSc 250 Introduction to Algorithms 1. Introduction Hashing is a method of storing elements in a table in a way that reduces the time for search. Elements are assumed to be records with several

### Hash Tables. Hashing Probing Separate Chaining Hash Function

Hash Tables Hashing Probing Separate Chaining Hash Function Introduction In Chapter 4 we saw: linear search O( n ) binary search O( log n ) Can we improve the search operation to achieve better than O(

### CMSC 341 Hashing (Continued) Based on slides from previous iterations of this course

CMSC 341 Hashing (Continued) Based on slides from previous iterations of this course Today s Topics Review Uses and motivations of hash tables Major concerns with hash tables Properties Hash function Hash

### HASH TABLES. Goal is to store elements k,v at index i = h k

CH 9.2 : HASH TABLES 1 ACKNOWLEDGEMENT: THESE SLIDES ARE ADAPTED FROM SLIDES PROVIDED WITH DATA STRUCTURES AND ALGORITHMS IN C++, GOODRICH, TAMASSIA AND MOUNT (WILEY 2004) AND SLIDES FROM JORY DENNY AND

### CSE 214 Computer Science II Searching

CSE 214 Computer Science II Searching Fall 2017 Stony Brook University Instructor: Shebuti Rayana shebuti.rayana@stonybrook.edu http://www3.cs.stonybrook.edu/~cse214/sec02/ Introduction Searching in a

### Lecture 16. Reading: Weiss Ch. 5 CSE 100, UCSD: LEC 16. Page 1 of 40

Lecture 16 Hashing Hash table and hash function design Hash functions for integers and strings Collision resolution strategies: linear probing, double hashing, random hashing, separate chaining Hash table

### Hashing. October 19, CMPE 250 Hashing October 19, / 25

Hashing October 19, 2016 CMPE 250 Hashing October 19, 2016 1 / 25 Dictionary ADT Data structure with just three basic operations: finditem (i): find item with key (identifier) i insert (i): insert i into

### HASH TABLES. Hash Tables Page 1

HASH TABLES TABLE OF CONTENTS 1. Introduction to Hashing 2. Java Implementation of Linear Probing 3. Maurer s Quadratic Probing 4. Double Hashing 5. Separate Chaining 6. Hash Functions 7. Alphanumeric

### Chapter 11: Indexing and Hashing

Chapter 11: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree Index Files Static Hashing Dynamic Hashing Comparison of Ordered Indexing and Hashing Index Definition in SQL

### Chapter 27 Hashing. Objectives

Chapter 27 Hashing 1 Objectives To know what hashing is for ( 27.3). To obtain the hash code for an object and design the hash function to map a key to an index ( 27.4). To handle collisions using open

### Fast Lookup: Hash tables

CSE 100: HASHING Operations: Find (key based look up) Insert Delete Fast Lookup: Hash tables Consider the 2-sum problem: Given an unsorted array of N integers, find all pairs of elements that sum to a

### The dictionary problem

6 Hashing The dictionary problem Different approaches to the dictionary problem: previously: Structuring the set of currently stored keys: lists, trees, graphs,... structuring the complete universe of

### Lecture 10 March 4. Goals: hashing. hash functions. closed hashing. application of hashing

Lecture 10 March 4 Goals: hashing hash functions chaining closed hashing application of hashing Computing hash function for a string Horner s rule: (( (a 0 x + a 1 ) x + a 2 ) x + + a n-2 )x + a n-1 )

### Why do we need hashing?

jez oar Hashing Hashing Ananda Gunawardena Many applications deal with lots of data Search engines and web pages There are myriad look ups. The look ups are time critical. Typical data structures like

### Hash Tables and Hash Functions

Hash Tables and Hash Functions We have seen that with a balanced binary tree we can guarantee worst-case time for insert, search and delete operations. Our challenge now is to try to improve on that...

### Hash Tables. Gunnar Gotshalks. Maps 1

Hash Tables Maps 1 Definition A hash table has the following components» An array called a table of size N» A mathematical function called a hash function that maps keys to valid array indices hash_function:

### Algorithms with numbers (2) CISC4080, Computer Algorithms CIS, Fordham Univ.! Instructor: X. Zhang Spring 2017

Algorithms with numbers (2) CISC4080, Computer Algorithms CIS, Fordham Univ.! Instructor: X. Zhang Spring 2017 Acknowledgement The set of slides have used materials from the following resources Slides

### Algorithms with numbers (2) CISC4080, Computer Algorithms CIS, Fordham Univ. Acknowledgement. Support for Dictionary

Algorithms with numbers (2) CISC4080, Computer Algorithms CIS, Fordham Univ. Instructor: X. Zhang Spring 2017 Acknowledgement The set of slides have used materials from the following resources Slides for

### Chapter 12: Indexing and Hashing

Chapter 12: Indexing and Hashing Basic Concepts Ordered Indices B+-Tree Index Files B-Tree Index Files Static Hashing Dynamic Hashing Comparison of Ordered Indexing and Hashing Index Definition in SQL

### BINARY HEAP cs2420 Introduction to Algorithms and Data Structures Spring 2015

BINARY HEAP cs2420 Introduction to Algorithms and Data Structures Spring 2015 1 administrivia 2 -assignment 10 is due on Thursday -midterm grades out tomorrow 3 last time 4 -a hash table is a general storage

### lecture23: Hash Tables

lecture23: Largely based on slides by Cinda Heeren CS 225 UIUC 18th July, 2013 Announcements mp6.1 extra credit due tomorrow tonight (7/19) lab avl due tonight mp6 due Monday (7/22) Hash tables A hash

### CS 241 Analysis of Algorithms

CS 241 Analysis of Algorithms Professor Eric Aaron Lecture T Th 9:00am Lecture Meeting Location: OLB 205 Business HW5 extended, due November 19 HW6 to be out Nov. 14, due November 26 Make-up lecture: Wed,

### Hashing 1. Searching Lists

Hashing 1 Searching Lists There are many instances when one is interested in storing and searching a list: A phone company wants to provide caller ID: Given a phone number a name is returned. Somebody

### Chapter 12: Indexing and Hashing

Chapter 12: Indexing and Hashing Database System Concepts, 5th Ed. See www.db-book.com for conditions on re-use Chapter 12: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree

### Table ADT and Sorting. Algorithm topics continuing (or reviewing?) CS 24 curriculum

Table ADT and Sorting Algorithm topics continuing (or reviewing?) CS 24 curriculum A table ADT (a.k.a. Dictionary, Map) Table public interface: // Put information in the table, and a unique key to identify

### Data Structures. Topic #6

Data Structures Topic #6 Today s Agenda Table Abstract Data Types Work by value rather than position May be implemented using a variety of data structures such as arrays (statically, dynamically allocated)

### HASH TABLES.

1 HASH TABLES http://en.wikipedia.org/wiki/hash_table 2 Hash Table A hash table (or hash map) is a data structure that maps keys (identifiers) into a certain location (bucket) A hash function changes the

### CS301 - Data Structures Glossary By

CS301 - Data Structures Glossary By Abstract Data Type : A set of data values and associated operations that are precisely specified independent of any particular implementation. Also known as ADT Algorithm

### CS 310 Hash Tables, Page 1. Hash Tables. CS 310 Hash Tables, Page 2

CS 310 Hash Tables, Page 1 Hash Tables key value key value Hashing Function CS 310 Hash Tables, Page 2 The hash-table data structure achieves (near) constant time searching by wasting memory space. the

### Hash Tables. Hash functions Open addressing. March 07, 2018 Cinda Heeren / Geoffrey Tien 1

Hash Tables Hash functions Open addressing Cinda Heeren / Geoffrey Tien 1 Hash functions A hash function is a function that map key values to array indexes Hash functions are performed in two steps Map

### Chapter 11: Indexing and Hashing

Chapter 11: Indexing and Hashing Database System Concepts, 6 th Ed. See www.db-book.com for conditions on re-use Chapter 11: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree

### CSE 373 Autumn 2012: Midterm #2 (closed book, closed notes, NO calculators allowed)

Name: Sample Solution Email address: CSE 373 Autumn 0: Midterm # (closed book, closed notes, NO calculators allowed) Instructions: Read the directions for each question carefully before answering. We may

### Chapter 20 Hash Tables

Chapter 20 Hash Tables Dictionary All elements have a unique key. Operations: o Insert element with a specified key. o Search for element by key. o Delete element by key. Random vs. sequential access.

### 6.7 b. Show that a heap of eight elements can be constructed in eight comparisons between heap elements. Tournament of pairwise comparisons

Homework 4 and 5 6.7 b. Show that a heap of eight elements can be constructed in eight comparisons between heap elements. Tournament of pairwise comparisons 6.8 Show the following regarding the maximum

### CS 251, LE 2 Fall MIDTERM 2 Tuesday, November 1, 2016 Version 00 - KEY

CS 251, LE 2 Fall 2016 MIDTERM 2 Tuesday, November 1, 2016 Version 00 - KEY W1.) (i) Show one possible valid 2-3 tree containing the nine elements: 1 3 4 5 6 8 9 10 12. (ii) Draw the final binary search

### CMSC 341 Hashing. Based on slides from previous iterations of this course

CMSC 341 Hashing Based on slides from previous iterations of this course Hashing Searching n Consider the problem of searching an array for a given value q If the array is not sorted, the search requires

### Introduction. for large input, even access time may be prohibitive we need data structures that exhibit times closer to O(log N) binary search tree

Chapter 4 Trees 2 Introduction for large input, even access time may be prohibitive we need data structures that exhibit running times closer to O(log N) binary search tree 3 Terminology recursive definition

Cuckoo Hashing for Undergraduates Rasmus Pagh IT University of Copenhagen March 27, 2006 Abstract This lecture note presents and analyses two simple hashing algorithms: Hashing with Chaining, and Cuckoo

### Outline for Today. How can we speed up operations that work on integer data? A simple data structure for ordered dictionaries.

van Emde Boas Trees Outline for Today Data Structures on Integers How can we speed up operations that work on integer data? Tiered Bitvectors A simple data structure for ordered dictionaries. van Emde

### STANDARD ADTS Lecture 17 CS2110 Spring 2013

STANDARD ADTS Lecture 17 CS2110 Spring 2013 Abstract Data Types (ADTs) 2 A method for achieving abstraction for data structures and algorithms ADT = model + operations In Java, an interface corresponds

### We assume uniform hashing (UH):

We assume uniform hashing (UH): the probe sequence of each key is equally likely to be any of the! permutations of 0,1,, 1 UH generalizes the notion of SUH that produces not just a single number, but a

### Hash Tables. Johns Hopkins Department of Computer Science Course : Data Structures, Professor: Greg Hager

Hash Tables What is a Dictionary? Container class Stores key-element pairs Allows look-up (find) operation Allows insertion/removal of elements May be unordered or ordered Dictionary Keys Must support

### Database System Concepts, 6 th Ed. Silberschatz, Korth and Sudarshan See for conditions on re-use

Chapter 11: Indexing and Hashing Database System Concepts, 6 th Ed. See www.db-book.com for conditions on re-use Chapter 12: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files Static

### CS1020 Data Structures and Algorithms I Lecture Note #15. Hashing. For efficient look-up in a table

CS1020 Data Structures and Algorithms I Lecture Note #15 Hashing For efficient look-up in a table Objectives 1 To understand how hashing is used to accelerate table lookup 2 To study the issue of collision

### Hashing Algorithms. Hash functions Separate Chaining Linear Probing Double Hashing

Hashing Algorithms Hash functions Separate Chaining Linear Probing Double Hashing Symbol-Table ADT Records with keys (priorities) basic operations insert search create test if empty destroy copy generic

### Dictionaries and Hash Tables

Dictionaries and Hash Tables 0 1 2 3 025-612-0001 981-101-0002 4 451-229-0004 Dictionaries and Hash Tables 1 Dictionary ADT The dictionary ADT models a searchable collection of keyelement items The main

### ECE 242 Data Structures and Algorithms. Hash Tables II. Lecture 25. Prof.

ECE 242 Data Structures and Algorithms http://www.ecs.umass.edu/~polizzi/teaching/ece242/ Hash Tables II Lecture 25 Prof. Eric Polizzi Summary previous lecture Hash Tables Motivation: optimal insertion

### Data Structures and Algorithm Analysis (CSC317) Hash tables (part2)

Data Structures and Algorithm Analysis (CSC317) Hash tables (part2) Hash table We have elements with key and satellite data Operations performed: Insert, Delete, Search/lookup We don t maintain order information

### Hashing. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong

Department of Computer Science and Engineering Chinese University of Hong Kong In this lecture, we will revisit the dictionary search problem, where we want to locate an integer v in a set of size n or

### CS/COE 1501

CS/COE 1501 www.cs.pitt.edu/~lipschultz/cs1501/ Hashing Wouldn t it be wonderful if... Search through a collection could be accomplished in Θ(1) with relatively small memory needs? Lets try this: Assume

### Balanced Search Trees

Balanced Search Trees Computer Science E-22 Harvard Extension School David G. Sullivan, Ph.D. Review: Balanced Trees A tree is balanced if, for each node, the node s subtrees have the same height or have

### This lecture. Iterators ( 5.4) Maps. Maps. The Map ADT ( 8.1) Comparison to java.util.map

This lecture Iterators Hash tables Formal coursework Iterators ( 5.4) An iterator abstracts the process of scanning through a collection of elements Methods of the ObjectIterator ADT: object object() boolean

### STRUKTUR DATA. By : Sri Rezeki Candra Nursari 2 SKS

STRUKTUR DATA By : Sri Rezeki Candra Nursari 2 SKS Literatur Sjukani Moh., (2007), Struktur Data (Algoritma & Struktur Data 2) dengan C, C++, Mitra Wacana Media Utami Ema. dkk, (2007), Struktur Data (Konsep

### Summer Final Exam Review Session August 5, 2009

15-111 Summer 2 2009 Final Exam Review Session August 5, 2009 Exam Notes The exam is from 10:30 to 1:30 PM in Wean Hall 5419A. The exam will be primarily conceptual. The major emphasis is on understanding

### Hash Tables. Hash functions Open addressing. November 24, 2017 Hassan Khosravi / Geoffrey Tien 1

Hash Tables Hash functions Open addressing November 24, 2017 Hassan Khosravi / Geoffrey Tien 1 Review: hash table purpose We want to have rapid access to a dictionary entry based on a search key The key

### Lecture 12 March 21, 2007

6.85: Advanced Data Structures Spring 27 Oren Weimann Lecture 2 March 2, 27 Scribe: Tural Badirkhanli Overview Last lecture we saw how to perform membership quueries in O() time using hashing. Today we

### CS 206 Introduction to Computer Science II

CS 206 Introduction to Computer Science II 04 / 16 / 2018 Instructor: Michael Eckmann Today s Topics Questions? Comments? Graphs Shortest weight path (Dijkstra's algorithm) Besides the final shortest weights

### CSE 5311 Notes 9: Hashing

CSE 53 Notes 9: Hashing (Last updated 7/5/5 :07 PM) CLRS, Chapter Review: 2: Chaining - related to perfect hashing method 3: Hash functions, skim universal hashing 4: Open addressing COLLISION HANDLING

CPSC 9 admin notes! TAs Office hours next week! Monday during LA 9 - Pearl! Monday during LB Andrew! Monday during LF Marika! Monday during LE Angad! Tuesday during LH 9 Giorgio! Tuesday during LG - Pearl!

### Hash Tables Hash Tables Goodrich, Tamassia

Hash Tables 0 1 2 3 4 025-612-0001 981-101-0002 451-229-0004 Hash Tables 1 Hash Functions and Hash Tables A hash function h maps keys of a given type to integers in a fixed interval [0, N 1] Example: h(x)

### Chapter 13: Query Processing

Chapter 13: Query Processing! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions 13.1 Basic Steps in Query Processing 1. Parsing

### SILT: A Memory-Efficient, High- Performance Key-Value Store

SILT: A Memory-Efficient, High- Performance Key-Value Store SOSP 11 Presented by Fan Ni March, 2016 SILT is Small Index Large Tables which is a memory efficient high performance key value store system

### Chapter 11: Indexing and Hashing" Chapter 11: Indexing and Hashing"

Chapter 11: Indexing and Hashing" Database System Concepts, 6 th Ed.! Silberschatz, Korth and Sudarshan See www.db-book.com for conditions on re-use " Chapter 11: Indexing and Hashing" Basic Concepts!

### Priority Queue Sorting

Priority Queue Sorting We can use a priority queue to sort a list of comparable elements 1. Insert the elements one by one with a series of insert operations 2. Remove the elements in sorted order with

### Indexing and Hashing

C H A P T E R 1 Indexing and Hashing This chapter covers indexing techniques ranging from the most basic one to highly specialized ones. Due to the extensive use of indices in database systems, this chapter

### Practical Session 8- Hash Tables

Practical Session 8- Hash Tables Hash Function Uniform Hash Hash Table Direct Addressing A hash function h maps keys of a given type into integers in a fixed interval [0,m-1] 1 Pr h( key) i, where m is

### ! A relational algebra expression may have many equivalent. ! Cost is generally measured as total elapsed time for

Chapter 13: Query Processing Basic Steps in Query Processing! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions 1. Parsing and

### ECE 242 Data Structures and Algorithms. Hash Tables I. Lecture 24. Prof.

ECE 242 Data Structures and Algorithms http//www.ecs.umass.edu/~polizzi/teaching/ece242/ Hash Tables I Lecture 24 Prof. Eric Polizzi Motivations We have to store some records and perform the following

### INF2220: algorithms and data structures Series 3

Universitetet i Oslo Institutt for Informatikk A. Maus, R.K. Runde, I. Yu INF2220: algorithms and data structures Series 3 Topic Map and Hashing (Exercises with hints for solution) Issued: 7. 09. 2016

### 4. Trees. 4.1 Preliminaries. 4.2 Binary trees. 4.3 Binary search trees. 4.4 AVL trees. 4.5 Splay trees. 4.6 B-trees. 4. Trees

4. Trees 4.1 Preliminaries 4.2 Binary trees 4.3 Binary search trees 4.4 AVL trees 4.5 Splay trees 4.6 B-trees Malek Mouhoub, CS340 Fall 2002 1 4.1 Preliminaries A Root B C D E F G Height=3 Leaves H I J

### CSE 373 Autumn 2010: Midterm #2 (closed book, closed notes, NO calculators allowed)

Name: Email address: CSE 373 Autumn 2010: Midterm #2 (closed book, closed notes, NO calculators allowed) Instructions: Read the directions for each question carefully before answering. We may give partial

### CS2210 Data Structures and Algorithms

CS2210 Data Structures and Algorithms Lecture 5: Hash Tables Instructor: Olga Veksler 0 1 2 3 025-612-0001 981-101-0002 4 451-229-0004 2004 Goodrich, Tamassia Outline Hash Tables Motivation Hash functions

### Radix Searching. The insert procedure for digital search trees also derives directly from the corresponding procedure for binary search trees:

Radix Searching The most simple radix search method is digital tree searching - the binary search tree with the branch in the tree according to the bits of keys: at the first level the leading bit is used,

### Database Technology. Topic 7: Data Structures for Databases. Olaf Hartig.

Topic 7: Data Structures for Databases Olaf Hartig olaf.hartig@liu.se Database System 2 Storage Hierarchy Traditional Storage Hierarchy CPU Cache memory Main memory Primary storage Disk Tape Secondary

### CSCI2100B Data Structures Trees

CSCI2100B Data Structures Trees Irwin King king@cse.cuhk.edu.hk http://www.cse.cuhk.edu.hk/~king Department of Computer Science & Engineering The Chinese University of Hong Kong Introduction General Tree

### Figure 1: Chaining. The implementation consists of these private members:

Hashing Databases and keys. A database is a collection of records with various attributes. It is commonly represented as a table, where the rows are the records, and the columns are the attributes: Number

### Chapter 11: Indexing and Hashing

Chapter 11: Indexing and Hashing Database System Concepts, 6 th Ed. See www.db-book.com for conditions on re-use Chapter 12: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree

### Recitation 9. Prelim Review

Recitation 9 Prelim Review 1 Heaps 2 Review: Binary heap min heap 1 2 99 4 3 PriorityQueue Maintains max or min of collection (no duplicates) Follows heap order invariant at every level Always balanced!

### Symbol Tables. For compile-time efficiency, compilers often use a symbol table: associates lexical names (symbols) with their attributes

For compile-time efficiency, compilers often use a symbol table: associates lexical names (symbols) with their attributes What items should be entered? variable names defined constants procedure and function

### Plot SIZE. How will execution time grow with SIZE? Actual Data. int array[size]; int A = 0;

How will execution time grow with SIZE? int array[size]; int A = ; for (int i = ; i < ; i++) { for (int j = ; j < SIZE ; j++) { A += array[j]; } TIME } Plot SIZE Actual Data 45 4 5 5 Series 5 5 4 6 8 Memory

### Database Applications (15-415)

Database Applications (15-415) DBMS Internals- Part V Lecture 13, March 10, 2014 Mohammad Hammoud Today Welcome Back from Spring Break! Today Last Session: DBMS Internals- Part IV Tree-based (i.e., B+

### of characters from an alphabet, then, the hash function could be:

Module 7: Hashing Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS 39217 E-mail: natarajan.meghanathan@jsums.edu Hashing A very efficient method for implementing

### Hash Tables. Lower bounds on sorting Hash functions. March 05, 2018 Cinda Heeren / Geoffrey Tien 1

Hash Tables Lower bounds on sorting Hash functions Cinda Heeren / Geoffrey Tien 1 Detour back to sorting Lower bounds on worst case Selection sort, Insertion sort, Quicksort, Merge sort Worst case complexities

### Lecture 12. Lecture 12: The IO Model & External Sorting

Lecture 12 Lecture 12: The IO Model & External Sorting Lecture 12 Today s Lecture 1. The Buffer 2. External Merge Sort 2 Lecture 12 > Section 1 1. The Buffer 3 Lecture 12 > Section 1 Transition to Mechanisms

### Data Structure for Language Processing. Bhargavi H. Goswami Assistant Professor Sunshine Group of Institutions

Data Structure for Language Processing Bhargavi H. Goswami Assistant Professor Sunshine Group of Institutions INTRODUCTION: Which operation is frequently used by a Language Processor? Ans: Search. This

### CMSC 132: Object-Oriented Programming II. Hash Tables

CMSC 132: Object-Oriented Programming II Hash Tables CMSC 132 Summer 2017 1 Key Value Map Red Black Tree: O(Log n) BST: O(n) 2-3-4 Tree: O(log n) Can we do better? CMSC 132 Summer 2017 2 Hash Tables a

### University of Illinois at Urbana-Champaign Department of Computer Science. Final Examination

University of Illinois at Urbana-Champaign Department of Computer Science Final Examination CS 225 Data Structures and Software Principles Spring 2010 7-10p, Wednesday, May 12 Name: NetID: Lab Section

### Chapter 12: Query Processing

Chapter 12: Query Processing Overview Catalog Information for Cost Estimation \$ Measures of Query Cost Selection Operation Sorting Join Operation Other Operations Evaluation of Expressions Transformation

### Implementation of Linear Probing (continued)

Implementation of Linear Probing (continued) Helping method for locating index: private int findindex(long key) // return -1 if the item with key 'key' was not found int index = h(key); int probe = index;

### Addresses in the source program are generally symbolic. A compiler will typically bind these symbolic addresses to re-locatable addresses.

1 Memory Management Address Binding The normal procedures is to select one of the processes in the input queue and to load that process into memory. As the process executed, it accesses instructions and

### Algorithms in Systems Engineering IE172. Midterm Review. Dr. Ted Ralphs

Algorithms in Systems Engineering IE172 Midterm Review Dr. Ted Ralphs IE172 Midterm Review 1 Textbook Sections Covered on Midterm Chapters 1-5 IE172 Review: Algorithms and Programming 2 Introduction to

Lecture IV Query Processing Kyumars Sheykh Esmaili Basic Steps in Query Processing 2 Query Optimization Many equivalent execution plans Choosing the best one Based on Heuristics, Cost Will be discussed