Chap11. Hashing

1
Chap11. Hashing
File Strutures by Folk, Zoellick and Riccardi

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2
Chapter Objectives

• Introduce the concept of hashing
• Examine the problem of choosing a good hashing
  algorithm, presents a reasonable one in detail,
  and describe some others
• Explore several approaches for reducing
  collisions and storage of several records per
  address
• Develop and use mathematical tools for analyzing
  performance differences resulting from the use of
  different hashing techniques
• Examine problems associated with file
  deterioration (record deletions) and discuss some
  solutions
• Discuss collision resolution techniques
• Examine effects of patterns of record access on
  performance

3
Contents

• 11.1 Introduction
• 11.2 A Simple Hashing Algorithm
• 11.3 Hashing Functions and Record Distribution
• 11.4 How Much Extra Memory Should Be Used?
• 11.5 Collision Resolution by Progressive Overflow
• 11.6 Storing More Than One Record per Address
  Buckets
• 11.7 Making Deletions
• 11.8 Other Collision Resolution Techniques
• 11.9 Patterns of Record Access

4
Overview

• O(1) access to files
• Record number is obtained by a hashing function H
  applied to the primary key, H(key)
• Record numbers generated should be uniformly and
  randomly distributed such that 0 lt H(key) lt N
• A hash function is like a black box that produces
  an address every time you drop in a key
• All parts of the key should be used by the
  hashing function H so that a lot of records with
  similar keys do not all hash to the same location
• Given two random keys X, Y and N slots, the
  probability H(X)H(Y) is 1/N in this case, X and
  Y are called synonyms and a collision occurs

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Introduction
11.1 Introduction

• Hash function h(k)
• Transforms a key K into an address
• Hash vs other index
• Sequential search O(N)
• Binary search O(log2N)
• B(B) Tree index O(logkN)
• where k records in an index node
• Hash O(1)

6
A Simple Hashing Scheme (1/2)
11.1 Introduction
Record
Address
key
LOWELL
Address
4
LOWELLs home address
7
A Simple Hashing Scheme (2/2)
11.1 Introduction
ASCII Code for First Two Letters
Home Address
Name
Product
66 X 65 4,290
290
BALL
66 65
76 X 96 6,004
76 96
LOWELL
004
84 X 82 6,888
84 82
TREE
888
8
Hashing differs from indexing

• With hashing, the addresses generated appear to
  be random
• No obvious connection between the key and the
  location of the corresponding record
• So, hashing is sometimes referred to as
  randomizing
• With hashing, two different keys may be
  translated to the same address
• Two records may be sent to the same place in the
  file

9
Idea behind Hash-based Files
11.1 Introduction

• Record with hash key i is stored in node i
• All record with hash key h are stored in node h
• Primary blocks of data level nodes are stored
  sequentially
• Contents of the root node can be expressed by a
  simple function Address of data level node for
  record with primary key k
• address of node 0H(k)
• In literature on hash-based files, primary blocks
  of data level nodes are called buckets

10
e.g. Hash-based File
11.1 Introduction
11
Collision (1/2)
11.1 Introduction

• Collision
• Situation in which a record is hashed to an
  address that does not have sufficient room to
  store the record
• Perfect hashing algorithm impossible!
• Different key, same hash value
• (Different record, same address)

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Collision (2/2)
11.1 Introduction

• Solutions
• Spread out the records
• Find a hashing algorithm that distributes records
  more randomly
• Use extra memory
• Easier to find a hash algorithm that avoids
  collisions if we have a few records to distribute
  among many address
• Put more than one record at a single address

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A Simple Hashing Algorithm (1/3)
11.2 A Simple Hashing Algorithm

• Step 1. Represent the key in numerical form
• If the key is a string take the ASCII code
• e.g. LOWELL
• 76 79 87 69 76 76 32 32 32 32 32 32
• L O W E L L ( 6 blanks
  )
• If the key is a number nothing to be done

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A Simple Hashing Algorithm (2/3)
11.2 A Simple Hashing Algorithm

• Step 2. Fold and Add
• Fold
• 76 79 87 69 76 76 32 32 32 32 32
  32
• Add parts into one integer
• Suppose we use 15 bit integer expression, 32767
  is limit
• 767987697676323232323232 33820 gt 32767
  (overflow!)
• Largest addend 9090 ( ZZ )
• Largest allowable result 32767-9090 23677 -gt
  19937(??)
• Ensure no intermediate sum exceeds using mod
• 7679 8769 16448 mod 19937 16448
• 16448 7676 24124 mod 19937 4187
• 4187 3232 7419 mod 19937 7419
• 7419 3232 10651 mod 19937 10651
• 10651 3232 13883 mod 19937 13883

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A Simple Hashing Algorithm (3/3)
11.2 A Simple Hashing Algorithm

• Step 3. Divide by size of the address space
• a s mod n
• a home address
• s the sum produced in step 2
• n the number of addresses in the file
• e.g.. a 13883 mod 100 83
• A prime number is usually used for the divisor
  because primes tend to distribute remainders much
  more uniformly than do nonprimes
• So, we chose a prime number as close as possible
  to the desired size of the address space

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Hashing Functions and Record Distributions
11.3 Hashing Functions and Record Distributions

• Distributing records among address

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Uniform distribution
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Some other hashing methods
11.3 Hashing Functions and Record Distributions

• Better-than-random
• Examine keys for a pattern
• Fold parts of the key
• Divide the key by a number
• When the better-than-random methods do not work -
  randomize!
• Square the key and take the middle
• ?? ??? ??? ???? ??? ??? ??
• Radix transformation(?? ??)
• 453(10??) -gt 382(11??)

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How Much Extra Memory Should Be Used?
11.4 How Much Extra Memory Should Be Used?

• The more records are packed, the more likely a
  collision will occur

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11.4 How Much Extra Memory Should Be Used?
Poisson Distribution
p(x) the probability that a given address will
have x records assigned to it after the
hashing function has been applied to all n
records ( x records? collision ??? ??)
N the number of available addresses r the
number of records to be stored x the number of
records assigned to a given address
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Predicting Collisions for Different Packing
Densities
11.4 How Much Extra Memory Should Be Used?

• of addresses no record assigned N X P(0)
• of addresses one record assigned N X P(1)
• of addresses more than two assigned
• N X P(2) P(3) P(4) ...
• of overflows 1 X NP(2) 2 X NP(3) ...
• Percentage of overflow records

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The larger space, the less overflows
11.4 How Much Extra Memory Should Be Used?
Packing Density r/N (N addresses, r
records)
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Collision Resolution by Progressive Overflow
11.5 Collision Resolution by Progressive Overflow

• Progressive overflow ( linear probing)
• Insert a new record
• 1. Take home address if empty
• 2. Otherwise, next several addresses are searched
  in sequence, until an empty one is found
• 3. If no more next space - wrapping around

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11.5 Collision Resolution by Progressive Overflow
Progressive Overflow (1/5)
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11.5 Collision Resolution by Progressive Overflow
Progressive Overflow (2/5)
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Progressive Overflow (3/5)
11.5 Collision Resolution by Progressive Overflow

• Search a record with a hash function value k
• from home address k, look at successive records,
  until Found,
• or An open address is encountered
• Worst case
• When the record does not exist and the file is
  full
• The reason to avoid overflow
• Extra searches have to occur when a record is not
  found in its home address

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Progressive Overflow (4/5)
11.5 Collision Resolution by Progressive Overflow

• - Search length of accesses required to
  retrieve a record (from secondary memory)

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Progressive Overflow (5/5)
11.5 Collision Resolution by Progressive Overflow

• With perfect hashing function average search
  length 1
• Average search length of no greater than 2.0 are
  generally considered acceptable

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Storing More Than One Record per Address Buckets
11.6 Storing More Than One Record per Address
Buckets

• Bucket a block of records sharing the same
  address (on block-addressing disk)

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Effects of Buckets on Performance
11.6 Storing More Than One Record per Address
Buckets

• of overflow records
• N X 1XP(b1) 2XP(b2) 3XP(b3)...
• N of addresses
• b of records fit in a bucket
• bN of available locations for records
• Packing density r/bN
• As the bucket size gets larger, performance
  continues to improve

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Bucket Implementation
11.6 Storing More Than One Record per Address
Buckets
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Bucket Implementation (Cont'd)
11.6 Storing More Than One Record per Address
Buckets

• Initializing and Loading
• Creating empty space
• Use hash values and find the bucket to store
• If the home bucket is full, continue to look at
  successive buckets
• Problems when
• No empty space exists
• Duplicate keys occur

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Making Deletions
11.7 Making Deletions

• The slot freed by the deletion hinders(disturb)
  later searches
• Use tombstones and reuse the freed slots

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Other Collision Resolution Techniques
11.8 Other Collision Resolution Techniques

• Double hashing avoid clustering with a second
  hash function for overflow records
• Chained progressive overflow each home address
  contains a pointer to the record with the same
  address
• Chaining with a separate overflow area move all
  overflow records to a separate overflow area
• Scatter tables Hash file contains only pointers
  to records (like indexing)

34
Linear Probing (1/2)
11.8 Other Collision Resolution Techniques

• When a synonym is identified, search forward from
  the address given by the hash function (the
  natural address) until an empty slot is located,
  and store this record there
• This is an example of open addressing (examining
  a predictable sequence of slots for an empty one)

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Linear Probing (2/2)
key
E
G
E
A
M
P
L
A
S
E
A
R
C
I
N
X
H
hash
1 0 5 1 18 3 8 9 14 7 5 5 1 13
16 12 5
0 1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18
A
11.8 Other Collision Resolution Techniques
S
insertion sequence
E
A
Memory Space
A
R
C
H
I
N
G
E
E
X
I
E
G
H
E
A
A
A
C
M
P
L
E
X
H
I
G
E
E
36
Rehashing (1/2)
11.8 Other Collision Resolution Techniques

• In linear probing, if synonym occurred,
  incremented r by 1 and searched next location
• In rehashing, use a second hash function for the
  displacement
• This method has the advantage of avoiding
  congestion, because each synonym under the first
  hash function likely uses a different
  displacement D, and this examines a different
  sequence of slots

37
Rehashing(where P3) (2/2)
key
E
G
E
A
M
P
L
A
S
E
A
R
C
I
N
X
H
hash
1 0 5 1 18 3 8 9 14 7 5 5 1 13
16 12 5
0 1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18
A
11.8 Other Collision Resolution Techniques
S
insertion sequence
E
A
Memory Space
A
R
C
H
I
N
G
E
H
E
E
X
N
H
E
A
G
A
A
M
P
L
E
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Chained progressive overflow
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Overflow File
11.8 Other Collision Resolution Techniques

• When building the file, if a collision occurs,
  place the new synonym into a separate area of the
  file called the overflow section
• ??
• ????? ????
• ?? ? ??? ?? ??? ?? ??
• ??
• Overflow section? ? ??? ?? ???? ???, ?? ??? 1??
  ?? ?? ??? ???

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scatter table

• an index that is searched by hashing
• the search of the index requires only one access
• a set of linked lists of synonyms