Hashing and Hash Tables

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Hashing and Hash Tables

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Title: Hashing and Hash Tables

1
Hashing and Hash Tables

Binhai Zhu
Computer Science Department, Montana State
University

2
Motivation

What are the dictionary operations?

3
Motivation

What are the dictionary operations?
(1) Insert
(2) Delete
(3) Search (most of the time, we will be
focusing on search)

4
Objective

Searching takes T(n) time in the worst case (when
the data is unorganized).
Even using binary search it takes T(log n) time
when the data are sorted.
Our Objective?

5
Objective

Searching takes T(n) time in the worst case (when
the data is unorganized).
Even using binary search it takes T(log n)
time when the data are sorted.
Our Objective?
O(1) time on average using hashing, under a
reasonable assumption.

6
Definitions

A hash table is a generalization of an array
(direct addressing is allowed), so lets first
talk about direct-address table.
Universe of keys U0,1,2,,m-1, no two elements
have the same key.
To represent a dynamic set, we use an array, or
direct address table T0..m-1, in which each
position (slot) corresponds to the key in the
universe.

7
Definitions

To represent a dynamic set, we use an array, or
direct address table T0..m-1, in which each
position (slot) corresponds to a key in the
universe.

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
8

With a direct address table T0..m-1, how do we
search an element x with key k?

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
9

With a direct address table T0..m-1, how do we
search an element x with key k?
Direct-Address-Search(T,k) return Tk

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
10

With a direct address table T0..m-1, how do we
search/insert/delete an element x with key k?
Direct-Address-Search(T,k) return Tk

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
11

With a direct address table T0..m-1, how do we
search/insert/delete an element x with key k?
Direct-Address-Search(T,k) return Tk
Direct-Address-Insert(T,x) Tkeyx ? x
Direct-Address-Delete(T,x) Tkeyx ? Nil

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
12

With a direct address table T0..m-1, how do we
search/insert/delete an element x with key k?
Direct-Address-Search(T,k) return Tk
Direct-Address-Insert(T,x) Tkeyx ? x
O(1) time!
Direct-Address-Delete(T,x) Tkeyx ? Nil

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
13

With a direct address table T0..m-1, how do we
search/insert/delete an element x with key k?
Direct-Address-Search(T,k) return Tk
Direct-Address-Insert(T,x) Tkeyx ? x
Problem?
Direct-Address-Delete(T,x) Tkeyx ? Nil

T
satellite data
0
/
key
1
/
2
2
U (universe of keys)
3
3

/
4

K (actual keys)

5
5

6
/

7
/
8
8
9
/
14
Hash Table

With direct addressing, an element with key k is
inserted in slot h(k). h is called a hash
function.
h maps the universe U of keys into the slots of a
hash table T0..m-1.
h U ? 0,1,,m-1

T
0
/
1

/
2
U (universe of keys)
3
/

K (actual keys)

6
/

8
/
9
/
15
Hash Table

With direct addressing, an element with key k is
inserted in slot h(k). h is called a hash
function.
h maps the universe U of keys into the slots of a
hash table T0..m-1.
h U ? 0,1,,m-1

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

X
Collision!
7

8
/
9
/
16
Collision

Two keys hash to the same slot --- collision.
While collision is hard to avoid, if we design
the hash function carefully we can at least
decrease the chance for collision (and in some
cases may avoid collision).

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

X
Collision!
7

8
/
9
/
17
Collision Resolution by Chaining

Two keys hash to the same slot --- collision.
While collision is hard to avoid, if we design
the hash function carefully we can at least
decrease the chance for collision (and in some
cases may avoid collision).

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

8
/
9
/
18
Collision Resolution by Chaining

Chained-Hash-Insert(T,x) insert x at the head
of list Th(keyx)
Chained-Hash-Search(T,k)

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

8
/
9
/
19
Collision Resolution by Chaining

Chained-Hash-Insert(T,x) insert x at the head
of list Th(keyx)
Chained-Hash-Search(T,k) search for an element
with key k in list Th(k)

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

8
/
9
/
20
Collision Resolution by Chaining

Chained-Hash-Insert(T,x) insert x at the head
of list Th(keyx)
Chained-Hash-Search(T,k) search for an element
with key k in list Th(k)
Chained-Hash-Delete(T,x)

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

8
/
9
/
21
Collision Resolution by Chaining

Chained-Hash-Insert(T,x) insert x at the head
of list Th(keyx)
Chained-Hash-Search(T,k) search for an element
with key k in list Th(k)
Chained-Hash-Delete(T,x) delete x from the list
Th(keyx)
Time?

T
0
/
1
/
/
2
U (universe of keys)
3
/

K (actual keys)

5
If h(5)h(8)

6
/

8
/
9
/
22
Collision Resolution by Chaining

Example Let h(k) k mod 11, insert
5,28,19,15,20,33,12,17,39,11 into T0..10.

T
33
11
0
1
12
/
2
3
/
15
4
5
5
28
17
39
6
7
/
19
8
9
20
10
/
23
Hash function

A hash function which causes no collision is
called perfect hash function.
A good hash function is one which satisfies
simple uniform hashing --- each key is equally
likely to hash to any of the m slots. (It is
difficult to check this condition though.)
Now lets see some example for hash functions.
Assume that all the keys can be represented as
natural numbers.

24
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5
1)/20.61803...

25
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803
Example. K 123456, m10000.
h(k) 10000(123456 x
0.61803 mod 1)

26
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5
1)/20.61803...
Example. K 123456, m10000.
h(k) 10000(123456 x
0.61803 mod 1)
10000(76300.0041151 mod 1)

27
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803
Example. K 123456, m10000.
h(k) 10000(123456 x
0.61803 mod 1)
10000(76300.0041151 mod 1)
10000 x 0.0041151)

28
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Example. K 123456, m10000.
h(k) 10000(123456 x
0.61803 mod 1)
10000(76300.0041151 mod 1)
10000 x 0.0041151)
41.151

29
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Example. K 123456, m10000.
h(k) 10000(123456 x
0.61803 mod 1)
10000(76300.0041151 mod 1)
10000 x 0.0041151)
41.151
41

30
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Example 1. Shift folding 123-456-789 (SSN)
123456789 1368
1368 mod 1000 368.

31
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Example 1. Shift folding 123-456-789 (SSN)
123456789 1368
1368 mod 1000 368.
Example 2. Boundary folding 123-456-789
(SSN)
123654789 1566
1566 mod 1000
566.

32
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Example. k3121, 31212 9740641, so h(k)

33
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Example. k3121, 31212 9740641, so h(k)
406.
You can also encode the square into binary
representation and take the middle part.

34
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Extraction Only a part of the key is used to
compute the address.
Example 123456789, first 4 digits 1234,
last 4 digits 6789
first 2
digits of 1234 ? last digits of 6789
we have
1289

35
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Extraction Only a part of the key is used to
compute the address.
Radix Transformation k is transformed into
another number base
Example 34510 4239 , then 423 mod 100
23.

36
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Extraction Only a part of the key is used to
compute the address.
Radix Transformation k is transformed into
another number base
Example 34510 4239 , then 423 mod 100
23.
26410 3239, then 323 mod
100 23.

37
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Extraction Only a part of the key is used to
compute the address.
Radix Transformation k is transformed into
another number base
Example 34510 4239 , then 423 mod 100
23.
26410 3239, then 323 mod
100 23.
Collision is hard to avoid in
the worst case!

38
Famous Examples of Hash Functions

Division h(k) k mod m, m should be a prime
number, better close to a power of 2.
Multiplication h(k) m(kA mod 1),
A(v5 1)/20.61803.
Folding The key is divided into several parts.
These parts are combined or folded together and
are transformed in a certain way to create the
target address.
Mid-square function key is squared and the
middle part of the result is taken as the
address.
Extraction Only a part of the key is used to
compute the address.
Radix Transformation k is transformed into
another number base

39
Open Addressing

In some applications, it is hard to dynamically
allocate additional space for handling the
chaining.
So it is natural to come up with a different way
to handle collision in which all elements are
stored in the hash table itself. Then, instead of
following pointers, we simply compute the
sequences of slots to be examined.
Lets use insertion as an example.

40
Open Addressing

In some applications, it is hard to dynamically
allocate additional space for handling the
chaining.
So it is natural to come up with a different way
to handle collision in which all elements are
stored in the hash table itself. Then, instead of
following pointers, we simply compute the
sequences of slots to be examined.
Lets use insertion as an example.
To perform insertion using open addressing, we
successively examine or probe the hash table
until we find an empty slot to put the element.
Moreover, the sequence of positions probed
depends on the key being inserted i.e.,
h U x 0,1,,m-1 ? 0,1,,m-1

41
Open Addressing

To perform insertion using open addressing, we
successively examine or probe the hash table
until we find an empty slot to put the element.
Moreover, the sequence of positions probed
depends on the key being inserted i.e.,
h U x 0,1,,m-1 ? 0,1,,m-1
Apparently, for every key k, the probe
sequence
lth(k,0), h(k,1),,h(k,m-1)gt is a permutation
of lt0,1,,m-1gt
so that every position in the hash table is
eventually considered as a slot for a new key as
the table fills up.
Now, for simplicity, assume kx, and there
is no deletion.

42
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

43
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

T
0
1
2
3
4
5
6
7
8
Example. Insert keys 10,22,31,4,15,28,17,88,59
into T. h(k,i)h(k)i mod m, h(k)k mod m.
9
10
44
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

T
0
h(10,0)(100) mod 11 10
1
2
3
4
5
6
7
8
Example. Insert keys 10,22,31,4,15,28,17,88,59
into T. h(k,i)h(k)i mod m, h(k)k mod m.
9
10
10
45
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

T
0
22
h(10,0)(100) mod 11 10 h(22,0)
0 h(31,0)9 h(4,0)4 h(15,0)(40) mod 11
4
1
2
3
4
4
5
6
7
8
Example. Insert keys 10,22,31,4,15,28,17,88,59
into T. h(k,i)h(k)i mod m, h(k)k mod m.
9
31
10
10
46
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

T
0
22
h(10,0)(100) mod 11 10 h(22,0)
0 h(31,0)9 h(4,0)4 h(15,0)(40) mod 11
4 h(15,1)(41) mod 11 5
1
2
3
4
4
5
15
6
7
8
Example. Insert keys 10,22,31,4,15,28,17,88,59
into T. h(k,i)h(k)i mod m, h(k)k mod m.
9
31
10
10
47
Open Addressing

Hash-Insert(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj Nil
then Tj ? k
return j
else i ? i 1
7. until im
8. error hash table overflow

T
0
22
1
88
2
3
4
4
5
15
28
6
17
7
8
59

Example. Insert keys 10,22,31,4,15,28,17,88,59
into T.
h(k,i)h(k)i mod m,
h(k)k mod m.

9
31
10
10
48
Open Addressing

Hash-Search(T,k)
1. i ? 0
2. repeat j ? h(k,i)
if Tj k
then return j
i ? i 1
6. until TjNil or im
7. return Nil

T
0
22
1
88
i 0 j ? h(15,0)4 Tj ! 15 i 1 j ?
h(15,1)5 Tj 15 return 5
2
3
4
4
5
15
28
6
17
7
8
59
Example. Search 15 in T. h(k,i)h(k)i mod m,
h(k)k mod m.
9
31
10
10
49
Open Addressing

How about deletion?
You can simply use Hash-Search
to find the key first. Then what?
1. i ? 0
2. repeat j ? h(k,i)
if Tj ! Nil and Tjk
then Tj ? Nil? exit
i ? i 1
6. until TjNil or im

T
0
22
1
88
2
3
4
4
5
15
28
6
17
7
8
59
Example. Delete 4,15 in T. h(k,i)h(k)i mod
m, h(k)k mod m.
9
31
10
10
50
Open Addressing

How about deletion?
You can simply use Hash-Search
to find the key first. Then what?
1. i ? 0
2. repeat j ? h(k,i)
if Tj ! Nil and Tj k
then Tj ? Nil?, exit
i ? i 1
6. until TjNil or im

T
0
22
1
88
Delete 15 i 0 j ? h(15,0)4 Tj Nil exit
2
3
4
Nil
5
15
28
6
17
7
8
59
Example. Delete 4,15 in T. h(k,i)h(k)i mod
m, h(k)k mod m.
9
31
10
10
51
Open Addressing

How about deletion?
You can simply use Hash-Search
to find the key first.
1. i ? 0
2. repeat j ? h(k,i)
if Tj ! Nil and Tj k
then Tj ? deleted, exit
i ? i 1
6. until TjNil or im

T
0
22
1
88
Delete 15 i 0 j ? h(15,0)4 Tj deleted i
1 j ? h(15,1)5 Tj15
2
3
4
deleted
5
15
28
6
17
7
8
59
Example. Delete 4,15 in T. h(k,i)h(k)i mod
m, h(k)k mod m.
9
31
10
10
52
Open Addressing

How about deletion?
You can simply use Hash-Search
to find the key first.
1. i ? 0
2. repeat j ? h(k,i)
if Tj ! Nil and Tj k
then Tj ? deleted, exit
i ? i 1
6. until TjNil or im

T
0
22
1
88
Delete 15 i 0 j ? h(15,0)4 Tj deleted i
1 j ? h(15,1)5 Tj15 15 is deleted!
2
3
4
deleted
deleted
5
28
6
17
7
8
59
Example. Delete 4,15 in T. h(k,i)h(k)i mod
m, h(k)k mod m.
9
31
10
10
53
Linear probing

That is sth we have just seen.
h is an ordinary hash function i.e.,
h U ? 0,1,2,,m-1
h(k,i) h(k) i mod m.
Initial slot probed is exactly Th(k).

54
Quadratic probing

h is an ordinary hash function i.e.,
h U ? 0,1,2,,m-1
h(k,i) h(k) C1i C2i2 mod m, C1, C2 are
two non-zero constants.
Initial slot probed is also Th(k), but when
igt0 it is intuitively better than linear probing.

55
Quadratic probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h(k)C1i C2i2 mod m,
h(k)k mod m,
C11, C23.

T
0
1
2
3
4
5
6
7
8
9
10
56
Quadratic probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h(k)C1i C2i2 mod m,
h(k)k mod m,
C11, C23.

T
0
22
1
h(10,0)10 h(22,0)0 h(31,0)9 h(4,0)4 h(15,0)4
2
3
4
4
5
6
7
8
9
31
10
10
57
Quadratic probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h(k)C1i C2i2 mod m,
h(k)k mod m,
C11, C23.

T
0
22
1
h(10,0)10 h(22,0)0 h(31,0)9 h(4,0)4 h(15,0)4
h(15,1)413 mod 11 8 h(28,0)6
2
3
4
4
5
6
28
7
8
15
9
31
10
10
58
Quadratic probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h(k)C1i C2i2 mod m,
h(k)k mod m,
C11, C23.

T
0
22
1
h(17,0)6 h(17,1)10 h(17,2)623x22 mod 11
9 h(17,3)633x32 mod 11 3
2
3
17
4
4
5
6
28
7
8
15
9
31
10
10
59
Quadratic probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h(k)C1i C2i2 mod m,
h(k)k mod m,
C11, C23.

T
0
22
1
88
2
3
17
4
4
5
6
28
59
7
8
15
9
31
10
10
60
Double probing

h(k,i) h1(k) ih2(k) mod m,
h1, h2 are two auxiliary hash functions.
Initial slot probed is also Th1(k).
If m 2j, then h2 should better be an odd
function.
Example. h1(k)k mod m, h2(k)1k mod m,
mm-2.

61
Double probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h1(k)ih2(k) mod m,
h1(k)k mod m,
h2(k)1 k mod (m-1).

T
0
1
2
3
4
5
6
7
8
9
10
62
Double probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h1(k)ih2(k) mod m,
h1(k)k mod m,
h2(k)1 k mod (m-1).

T
0
22
h(10,0)10 h(22,0)0 h(31,0)9 h(4,0)4 h(15,0)4
h(15,1)41x6 mod 11 10 h(15,2)42x6 mod
11 5
1
2
3
4
4
5
6
7
8
9
31
10
10
63
Double probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h1(k)ih2(k) mod m,
h1(k)k mod m,
h2(k)1 k mod (m-1).

T
0
22
h(28,0)6 h(17,0)6 h(17,1)61x8 mod 11
3 h(88,2)02x9 mod 11 7 h(59,2)42x10
mod 11 2
1
2
3
17
4
4
5
15
6
28
7
8
9
31
10
10
64
Double probing

Example.
Insert keys 10,22,31,4,15,28,17,88,59 into T.
h(k,i)h1(k)ih2(k) mod m,
h1(k)k mod m,
h2(k)1 k mod (m-1).

T
0
22
h(28,0)6 h(17,0)6 h(17,1)61x8 mod 11
3 h(88,2)02x9 mod 11 7 h(59,2)42x10
mod 11 2
1
59
2
3
17
4
4
5
15
6
28
88
7
8
9
31
10
10
65
Analysis of hashing (in general tough)

In a hash table with size m, we want to store n
elements with collision resolved by chaining. Let
a n/m.
Theorem. An unsuccessful search takes expected
time O(1a), under the assumption of simple
uniform hashing.

66
Analysis of hashing (in general tough)

Theorem 11.1. An unsuccessful search takes
expected time O(1a), under the assumption of
simple uniform hashing.
Sketch of proof.
Under the assumption of simple uniform hashing,
any new key
k is equally likely to hash to any of the m
slots. The expected time to search unsuccessfully
for the key k is the expected time to search to
the end of list Th(k), which has expected size
a. Therefore, including the cost for computing
h(k), the cost for this unsuccessful search is
O(1a). ?