INTRODUCTION TO HASHING Dr. Thomas Hicks Trinity University. Data Set - SSN's from UTSA Class

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1 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -1 - INTRODUCTION TO HASHING Dr. Thomas Hicks Trinity University Data Set - SSN's from UTSA Class Need Faster Search Techniques We need faster searching of arrays for key words. We need faster searching of files for social security numbers. Wouldn't it be nice to find a keyword or social security number in just one look? O(1)? Black-Box Address Calculator Hashing is a two phase process. One must select an Address Calculator Algorithm that he/she thinks will uniformly distribute the primary key... n h(x) n-1 Address... x -> ---->..... : Calculator... : one logical record each Direct Access File / Array Hash Techniques may be used to search internal memory arrays and direct access files. Searching an internal memory array (either static or dynamic) is a nanosecond process. Searching direct access files is a millisecond process. A HashValue is produced when a Key is Hashed through the AddressCalculator. The Hash function is used to map a key into the file/array. There are many potentially good hash functions for a large collection of data. Finding a good hash function may require a lot of searching and ingenuity. Two Decisions One must make two decisions in hashing. What Hash Function is to be used and what technique is used to resolve hash clashes/collisions..

2 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -2 - Perfect Hash Function A Perfect Hash Function is one in which each and every element is located in just one look. Do you think that it is possible to create a perfect hash function to map all 20 of the social security numbers into the file? Yes/No? Let us assume that we have 5,000 byte records /498,662,055 = /192,039,969 = /20 = /25 = 80% Loading Factor 1 Collisions - Hash Clashes A Hash Clash, or Collision, occurs when two or more keys generate the same HashValue. The second phase of hashing is to select a Hash Collision & Storage Algorithm that minimizes the damage caused by collisions. We shall examine several hash storage algorithms. What Constitutes a Good Hash Function? I. The hash function should be easy to compute. II. The hash function should uniformly distribute the data throughout the set of hash keys.

3 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -3 - a) how well does the hash function scatter random data b) how well does the hash function scatter non-random data III. The hash function should help to conserve disk space. Minimal Acceptable Hash Function Standards I. The Average Number of Hard Disk Accesses to Find the Sought Key <= 1.2 The average number of hard disk accesses is calculated by dividing the number of searches required to find each item once by the number of items. I =1 AccessQuotient = Σ (# Accesses To Find Item I) / N I N II. The Loading Factor <= 80% (1,000,000 records can be stored in 1,250,000 records or less) LoadingFactor = NoRecords/Elements / N Hash Alternative To Compiler Key Word Search One student developed a black-box address calculator function and a array management system that produced the following table. The AccessQuotient is admirable Records 1-6 do record in end Records 7-12 const var function Records of case array packed Records mod program until Records else if label then Records file with to div and Records type begin not Records downto set while or Records nil for repeat Records goto procedure The Loading Factor for the compiler hash above was not admirable. Loading Factor = 35 / 60 = 58.3% [Not Satisfy Minimal Standard]

4 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -4 - Statement of Problem A. Population of 20 prospective teachers. B. Desire 80% Loading Factor B. Desire an AccessQuotient of 1.2 or Better C. Data Answer the following -Questions 1] A Hash Function is also called an _?_[Hint AC] 2] The linked list finds an element in Order _?_{N, N 2,, N 3, Log 10 N, Log 2 N} 3] The AVL Tree finds an element in Order _?_ {N, N 2,, N 3, Log 10 N, Log 2 N} 4] The Objective of Hashing is to find an element in Order _?_ {N, N 2,, N 3, Log 10 N, Log 2 N} 5] If the 20 SSN s above were loaded into a linked list, the Access Quotient/Average Search would be _?_. [Compute to 1 decimal place] 6] If the 20 SSN s above were loaded into an AVL tree, the Access Quotient/Average Search would be _?_. [Compute to 1 decimal place] 7] If the 20 SSN s above were loaded into the best of your Linear Probing Models below, the Access Quotient/Average Search would be _?_. [Compute to 1 decimal place] 8] If the 20 SSN s above were loaded into the best of your Quadratic Probing Models below, the Access Quotient/Average Search would be _?_. [Compute to 1 decimal place] 9] Hashing is most successful for {small, medium, large, any} quantities of data. 10] {T/F} Satisfactory Hashing is always a possible. 11] Satisfactory Loading Factors are _?_% or greater. 12] Satisfactory Access Quotients are _?_ or less. 13] List the three things that constitute a good hash function?

5 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -5-14] Suppose we are going to try to use Linear Probing to resolve the collisions for 1,000 records. How many elements must there be in the data set? 15] Suppose we are going to try to use Quadratic Probing to resolve the collisions for 1,500 records. How many elements must there be in the data set? 16] Suppose we are going to try to use Linear Probing to resolve the collisions for 1,000 records. We have also decided to use Modularization to distribute our data. Write the Hash Function. 17] Suppose we are going to try to use Quadratic Probing to resolve the collisions for 1,500 records. We have also decided to use Modularization to distribute our data. Write the Hash Function. 18] What is Primary Clustering? 19] Why might it be better to select a prime number when using modularization in the hash function? 20] Why might the Quadratic Probing be better than the Linear Probing? 21] In a Linear Probing, the first attempt to place the collision moves 1 cell up; if that slot is full, the second attempt to place the collision _?_ cells up; if that slot is full, the third attempt to place the collision _?_ cells up; if that slot is full, the fourth attempt to place the collision _?_ cells up. 22] In a Quadratic Probing, the first attempt to place the collision moves 1 cell up; if that slot is full, the second attempt to place the collision _?_ cells up; if that slot is full, the third attempt to place the collision _?_ cells up; if that slot is full, the fourth attempt to place the collision _?_ cells up. 23] Almost all applications involve Insertion, Deletion, and Searching. In only the space below, discuss Deletion from a Linear Probing Model.

6 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -6-24] There are two major decisions that must be made when hashing. List them. 25] Be able to solve any of the following:

7 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -7 - Linear Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Hash Function is Modularization [Thus Must Mod By? + 1 To Fill Slots 1-? ] #1 Linear Probing Collision Resolution 29 Put In Next Available Cell Upward Wrap Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = 7 [ 2] h( ) = 6 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

8 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -8 - Linear Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Hash Function Folding & Mod (First 3 Digits + Second Three Digits + Third Three Digits) Mod By? + 1 #2 Linear Probing Collision Resolution 29 Put In Next Available Cell Upward Wrap Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = 7 [ 2] h( ) = _16 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

9 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing -9 - Linear Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Hash Function is Truncate & Mod [Truncate last six digits - Mod By + 1 To Fill Slots 1- ] #3 Linear Probing Collision Resolution 29 Put In Next Available Cell Upward Wrap Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _18_ [ 2] h( ) = _25_ [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

10 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing Linear Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Hash Function is Truncate & Mod [Truncate first two digits; truncate last two digits - Mod By + 1 To Fill Slots 1- ] #4 Linear Probing Collision Resolution 29 Put In Next Available Cell Upward Wrap Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _14_ [ 2] h( ) = _21_ 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

11 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing Quadratic Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Hash Function is Modularization [Thus Must Mod By + 1 To Fill Slots 1- ] #5 Quadratic Probing 29 Collision Resolution Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = 7 [ 2] h( ) = 6 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

12 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing Quadratic Probing Is Always Pre-Designed For 80% Loading Factor 20 Students Requires 25 Slots] Modularization To Nearest Prime > 80% Loading [Thus Must Mod By To Fill Slots 1-29] #6 Quadratic Probing 29 Collision Resolution Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _15_ [ 2] h( ) = 9_ 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = Total # Searches = Loading Factor = % Average Search =

13 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing Linked List Overflow Is Always Pre-Designed For?% Loading Factor 20 Students Requires 20+ Slots] Hash Function is Modularization [Thus Must Mod By? + 1 To Fill Slots 1-? ] #7 Linked List Overflow 29 Collision Resolution Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _2 [ 2] h( ) = _16_ 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = / 1 15 Total # Searches = Loading Factor = % Average Search = / 1 1

14 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing Binary Tree Overflow Is Always Pre-Designed For?% Loading Factor 20 Students Requires 20+ Slots] Hash Function is Modularization [Thus Must Mod By? + 1 To Fill Slots 1-? ] #8 Binary Tree Overflow 29 Collision Resolution Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _2 [ 2] h( ) = _16_ 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = / 1 15 Total # Searches = Loading Factor = % Average Search = / 3881 / 1 1

15 Dr. Thomas E. Hicks Data Abstractions Homework - Hashing AVL Tree Overflow Is Always Pre-Designed For?% Loading Factor 20 Students Requires 20+ Slots] Hash Function is Modularization [Thus Must Mod By? + 1 To Fill Slots 1-? ] #9 AVL Tree Overflow 29 Collision Resolution Place Last Four Digits Of SSN In Slot 28 First Two Done For You! [ 1] h( ) = _2 [ 2] h( ) = _16_ 25 [ 3] h( ) = [ 4] h( ) = 24 [ 5] h( ) = [ 6] h( ) = 23 [ 7] h( ) = [ 8] h( ) = 22 [ 9] h( ) = [10] h( ) = 21 [12] h( ) = 20 [14] h( ) = 19 [16] h( ) = 18 [18] h( ) = 17 [20] h( ) = / 1 15 Total # Searches = Loading Factor = % Average Search = / 3881 / 1 1