NCUE CSIE Wireless Communications and Networking Laboratory
CHAPTER 8
Hashing
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Hashing
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Definition:
In hashing the dictionary pairs are stored in a table, ht, called the
hash table. The hash table is partitioned into b buckets,
ht[0],…,ht[b-1]. The address or location of a pair is determined by
a hash function, h, which maps keys into buckets. Thus, for any
key k, h(k) is an integer in the range 0 through b-1.
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Hash Table
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index
Bucket
x
H(x)
Slot0
Slot1
0
1
2
b-1
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Terminologies
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  utilization  Collision
The identifier density of a hash table is the ratio n/T, where n is the
number of identifiers in the table. The loading density or loading
factor of a hash table is   n s  b (T: distinct possible value of
identifiers. s: number of slots per bucket. b: bucket number)
Identifier density:
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Terminologies
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Overflow:
Since many keys typically have the same home bucket, it is possible
that the home bucket for a new dictionary pair is full at the time we
wish to insert this pair into the dictionary.
not necessarily
collision
overflow
necessarily
Collision:
A collision occurs when the home bucket for the new pair is not empty
at the time of insertion.
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Hashing Function
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Hashing Function Design :
(1) Easy to Compute
(2) Minimize the Number of Collisions
(3) Uniform Hash function
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Hashing Function
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Hashing Function :
 Mid-Square
 Division (Modulus)
 Folding
 Digit Analysis
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Mid-square
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The mid-square hash function determines the home bucket for a key
by squaring the key and then using an appropriate number of bits from
the middle of the square to obtain the bucket address.
ex)
We assume the key = 8125，and hashing table has 1000
buckets.
(8125)2  66015625
so address is “156” or “015”
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Divisions
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The home bucket is obtained by using the modulo (%) operator. The
key x is divided by some number M, and the remainder is used as the
home bucket for x.
f(x) = x mod M
ex)
prime
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Folding
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In this method the key k is partitioned into several parts, all but
possibly the last being of the same length. These partitons are then
added together to obtain the hash address for k.
There are two ways of carrying out this additon. (1) Shift (2) Boundary
ex1)
We assume the key = 12320324111220，and hashing table
has 1000 buckets.
123|203|241|112|20
(1) 123+203+241+112+020=699
(2) 123+302+241+211+020=897
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Folding
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ex2)
(1) Shift
(2) Boundary
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Digit Analysis
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All the keys in the table are known in advance. Each key is interpreted
as a number using some radix r. The same radix is used for all the
keys in the table. Using this radix, the digits of each key are examined.
ex)
phone number
address
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Overflow Handling
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 Linear Open Addressing (Linear Probing)
 Quadratic Probing
 Rehashing
 Chaining
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Linear Probing
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When the overflow occurs, we search the hash table buckets in the
order (H(x)+1, H(x)+2…), until the hash table is full or reaching the
first unfilled bucket.
ex)
0
10
0
1
2
10
1
Insert 55
75
3
3
55
4
4
6
75
43
5
6
10
1
2
5
0
43
Insert 25
2
75
3
55
4
25
5
43
6
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Linear Probing
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Advantages:
Simple 、Easy to Implement。
Disadvantages:
When the clustering occurs, the search time will increase
rapidly。
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Quadratic Probing
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When the overflow occurs, we search the hash table buckets by using
ex)
 22
 12
1
2
 22
H(x), overflow
Key k, hash function H
1st search： H(k)
2nd search： (H(k)+12)%b
3th search： (H(k)-12)%b
4th search： (H(k)+22)%b
5th search： (H(k)-22)%b
Nth search：(H(k)±((B-1)/2)2)%b
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Rehashing
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The rehashing method is to use a series of hash functions h1,h2,…,hm.
Buckets hi(k), 1≦i≦m are examined in the order.
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Chaining
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Many of the comparisons can be saved if we maintain lists of keys,
one list per bucket, each list containing all the synonyms for that
bucket.
0
10
0
1
2
1
75
3
6
Insert 55
2
Insert 25
75
55
3
4
5
10
25
4
43
5
6
43
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Question:
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Assume that a hash function has the following
characteristics:
keys 257 and 567 hash to 3
keys 987 and 313 hash to 6
keys 734, 189 and 575 hash to 5
keys 122 and 391 hash to 8
Assume that insertions are done in order 257, 987, 122,
575, 189, 734, 567, 313, 391
(1) Indicate the position of the data if open probe addressingis used to
resolve collision
(2) Indicate the position of the data if chining with separate
lists is used to resolve collision
Hash Table
0
1
2
3
4
5
6
7
8
9
10
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Question:
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If H(x) = x mod 7 and separate chaining resolves
collisions, What does the hash table look like after the
following insertions occur: 8, 10, 24, 15, 32, 17?
Assume that each table item contains only a search key.
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Question:
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Suppose the hashing function f(x) = x mod 11 is used to
hash a list of input value (in the given order) into a hash
table implemented by the array
bucked[0],bucket[1],…bucket[10]. The inputs are
10,100,32,45,126,3,24,200,and 53. Each bucket can hold
only one number. Overflow is resolved by quadratic
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probing, which examines buckets f(x), (f(x)+ i ) mod
2
11,and (f(x)- i ) mod 11, i=1 to 5.Show the final contents
in bucket[0] to bucket[10].
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Ans:
0
32
1
100
2
45
3
3
4
5
126
6
24
7
8
53
9
200
10
10
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Question:
NCUE CSIE Wireless Communications and Networking Laboratory
For each hash table below, show the result of inserting
the following sequence of key values, in the given order ,
into an initially empty hash table of that type:
26,17,20,9,34,32,15,21.
In both cases , assume a hash table size of 11 and a hash
function h(x) = x mod 11.
(1)Static hash table that uses chaining
(2)Hash table that uses linear probing
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(1)
(2)
0
0
32
1
34
2
2
21
3
3
1
→34
4
→26→15
4
26
5
→17
5
15
6
6
17
7
7
8
8
9
→20→9
9
20
10
→32→21
10
9
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Reference
NCUE CSIE Wireless Communications and Networking Laboratory
 Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Freed
〝Fundamentals of Data Structures in C〞, W. H. Freeman & Co
Ltd, 1992.
 Ellis Horowitz, Sartaj Sahni, and Dinesh Mehta
〝Fundamentals of Data Structures in C++〞 Silicon Pr, 2006
 Richard F.Gilberg, Behrouz A. Forouzan,
〝Data Structures: A Pseudocode Approach with C〞, S
Baker & Taylor Books, 2004
 Fred Buckley, and Marty Lewinter 〝A Friendly Introduction to Graph
Theory〞 Prentice Hall, 2002
 〝資料結構-使用C語言〞蘇維雅譯，松崗，2004
 〝資料結構-使用C語言〞 蔡明志編著，全華，2004
 〝資料結構(含精選試題)〞洪逸編著，鼎茂，2005
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