Hashing Part One
Reaching for the Perfect Search
Most of this material stolen from
"File Structures" by Folk, Zoellick and Riccardi

Text File v. Binary File

Unordered Binary File
◦ average search takes N/2 file operations

Ordered Binary File
◦ average search takes Log2N file operations
◦ but keeping the data file sorted is costly

Indexed File
◦ average search takes 3 or 4 file operations

Perfect Search
◦ search time = 1file read
 Definition:
o a magic black box that converts a key to the file
Name
Field1
Field2
address of that record
Dannelly
Hash
Function
Dannelly
...
 Example Hashing Function:
o Key = Customer's Name
o Function = 1st letter x 2nd letter,
then use rightmost 4 letters.
Name
BALL
LOWELL
TREE
OLIVIER
ascii
66x65
76x79
84x82
79x76
=
=
=
=
product
4290
6004
6888
6004
RRN
290
004
888
004
 Definition:
◦ When two or more keys hash to the same address.
 Minimizing the Number of Collisions:
1) pick a hash function that avoids collisions,
i.e. one with a seemingly random distribution
◦ e.g. our previous function is terrible because letters like
"E" and "L" occur frequently, while no one's name starts
with "XZ".
2) spread out the records
◦ 300 records in a file with space for 1000 records will
have many fewer collisions than 300 records in a file with
capacity of 400


Our objective is to muddle the relationship
between the keys and the addresses.
Good Ideas:
 use both addition and multiplication
 avoid integer overflow

so mix in some subtraction and division too
 divide by prime numbers
1. pad the name with spaces
Why 19,937 ?
2. fold and add pairs of letters
19,937 is the largest
prime that insures the
next add will not
cause integer overflow.
3. mod by a prime after each add
4. divide sum by file size

Example:
Key="LOWELL" and file size = 1,000
L O
W E
76 79 | 87 69
7679 + 8769 =
16448 + 7676 =
4187 + 3232 =
7419 + 3232 =
10651 + 3232 =
L L
| 76 76 | 32 32
16,448 % 19,937
24,124 % 19,937
7,419 % 19,937
10,651 % 19,937
13,883 % 19,937
13833 % 1000 = 833
|
=
=
=
=
=
32 32 | 32 32
16,448
4,187
7,419
10,651
13,833


The simplest hash function for a string is "add
up all the characters, then divide by filesize"
For example,
◦ filesize = 100 records
◦ key = "pen"
◦ address = ( 16 + 5 + 14 ) % 100 = 35
1. Find another word with the same mapping
2. Give an improvement to this hash function
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

The optimal hash function for a set of keys:
1. will evenly distribute the keys across the address
space, and
2. every address has a equal chance of being used.

Uniform distribution is nearly impossible.
Good Mapping
key
A
B
C
D
E
address
1
2
3
4
5
6
7
8
9
10
Poor Mapping
key
A
B
C
D
E
address
1
2
3
4
5
6
7
8
9
10



Suppose we have a file of 10,000 records, finding a
hash function that will take our 10,000 keys and
yield 10,000 different addresses is essentially
impossible.
So, our 10,000 records are stored in a larger file.
How much larger than 10,000?
o 10,500?
o 12,000?
o 50,000?

It Depends
◦ larger datafile:
 more empty (wasted) space
 fewer collisions



Even with a very good hash function,
collisions will occur.
We must have an algorithm to locate
alternative addresses.
Example,
◦ Suppose "dog" and "cat" both hash to location 25.
◦ If we add "dog" first, then dog goes in location 25.
◦ If we later add "cat", where does it go?
◦ Same idea for searching. If cat is supposed to be at
25 but dog is there, where do we look next?
 "Linear Probing" or "Progressive Overflow"




When a key maps to address already in use,
just try the next one. If that one is in use, try
the next one. yadda yadda
Easy to implement.
Usually works well, especially with a nondense file and a good hash function.
Can lead to clumps of records.
 Assume these keys map to these addresses:
1.
2.
3.
4.
5.
adams = 20
bates = 22
cole = 20
dean = 21
evans = 23
 Where will each record be placed if inserted
in that order?
 Using linear probing, how many file
accesses for each?
 How many collisions is acceptable?
◦ Analysis: packing density v probing length
 Is there a collision resolution algorithm
better than linear probing?
◦ buckets