Midterm will be given in Week 11’s
lecture (2 hurs)
Hash Tables
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Part E
Hash Tables
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Hash Tables
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025-612-0001
981-101-0002
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451-229-0004
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Hash Tables
• Questions: how to design a data structure, such that the
following operations are all O(1)?
–
–
–
–
Deletion
Search
Insertion
Replace
Hash Tables
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Motivations of Hash
Tables
• We have n items, each contains a key and value (k, value).
– The key uniquely determines the item.
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Each key could be anything, e.g., a number in [0, 232], a
string of length 32, etc.
• How to store the n items such that
given the key k, we can find the position of the item
with key= k in O(1) time.
– Another constraint: space required is O(n).
• Linked list? Space O(n) and Time O(n).
• Array? Time O(1) and space: too big, e.g.,
• If the key is an integer in [0, 2 32], then the space required is 2 32.
• if the key is a string of length 30, the space required is 26 30.
• Hash Table: space O(n) and time O(1).
Hash Tables
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Basic ideas of Hash Tables
• A hash function h maps keys of a given type with a wide
range to integers in a fixed interval [0, N - 1], where N is
the size of the hash table such that
• if k≠k’ then h(k)≠h(k’)
….. (1) .
Problem:
It is hard to design a function h such that (1)
holds.
What we can do:
• We can design a function h so that with high
chance, (1) holds.
• i.e., (1) may not always holds, but (1) holds for
most of the n keys.
Hash Tables
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Hash Functions
• A hash function h maps keys of a given type to integers in a
fixed interval [0, N - 1]
• Example:
h(x) = x mod N
is a hash function for integer keys
• The integer h(x) is called the hash value of key x
• A hash table for a given key type consists of
– Hash function h
– Array (called table) of size N
• the goal is to store item (k, o) at index i = h(k)
Hash Tables
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Example
Hash Tables
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025-612-0001
981-101-0002
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451-229-0004
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• We design a hash table storing
entries as (HKID, Name),
where HKID is a nine-digit
positive integer
• Our hash table uses an array
of size N = 10,000 and the
hash function
h(x) = last four digits of x
• Need a method to handle
collision.
9997
9998
9999
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200-751-9998
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Example
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50xxxx01
50xxxx02
51xxxx02
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86xxxx04
…
• We design a hash table storing
entries as (studentID, Name), 97 
where studentID is a eight51xxxx98
98
digit positive integer
99 
Task: to store the n items such
• Our hash table uses an array
that given the key k, we can
of size N = 100 for our class of
find the position of the item
n= 49 students and the hash
with key= k in O(1) time.
As long as the chance for collision
function
is low, we can achieve this
h(x) = last two digits of x
goal.
• Need a method to handle
Setting N=1000 and looking at
collision.
the last three digits will
reduce the chance of collision.
Hash Tables
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How to design a Hash Function
• A hash function is usually
• The hash code is
specified as the composition of
applied first, and the
two functions:
compression function is
applied next on the
Hash code:
result, i.e.,
h1: keys  integers
h(x) = h2(h1(x))
– key could be anything, e.g., your
name, an object, etc.
• The goal of the hash
Compression function:
function is to “disperse”
h2: integers  [0, N - 1]
the keys in an
– The size of the array N cannot be
apparently random way
too large in order to save space.
so that in most cases (1)
– Trade-off between space and time.
holds.
Hash Tables
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Integer cast:
– We reinterpret the bits of the key as an integer
Example: characters is mapped to its ASCII code.
A=65=01000001, B=66=01000010, a=97,
b=98,
,=44,
.=46.
– Suitable for keys of length less than or equal to the number of
bits of the integer type (e.g., byte, short, int and float in Java)
Hash Tables
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Component sum:
– We partition the bits of the key into components of fixed length
(e.g., 16 or 32 bits) a0 a1 … an-1 and we sum the components
(ignoring overflows) a0 + a1 + a2 + … +an-1
– Example 1: AB=0100000101000010
h(AB)= 01000001
+ 01000010
10000011.
Example 2:
h(100000011000001001001000)=
10000001
10000010
01001000
01001011 (ignore overflows)
– Suitable for numeric keys of fixed length greater than or equal to
the number of bits of the integer type (e.g., long and double in
Java)
Hash Tables
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Compression Functions
• Division:
– h2 (y) = y mod N
– The size N of the hash table is usually chosen to be a
prime
– The reason has to do with number theory and is
beyond the scope of this course
– Example: keys: {200, 205, 210, 215, 220, 600}.
If N=100, 200 and 600 have the same code, i.e., 0.
It is better to choose N=101.
Hash Tables
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Compression Functions
• Multiply, Add and Divide (MAD):
– h2 (y) = (ay + b) mod N
– a >0 and b>0 are nonnegative integers such that
a mod N  0 and N is a prime number.
Example: Keys={200, 205, 210, 215, 220, 600}.
N=101. a=3 and b=7.
h(200)=(600+7) mod 101 = 607 mod 101=1.
H(205)=(615+7) mod 101 = 622 mod 101=16.
h(210)=(630+7) mod 101 = 637 mod 101=31.
h(215)=(645+7) mod 101 = 652 mod 101=46.
h(220)=(660+7) mod 101 = 667 mod 101=61.
H(600)=(3600 +7) mod 101=3607 mod 101=72.
Hash Tables
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Collision Handling
• Collisions occur when
different elements are
mapped to the same cell
• Separate Chaining: let
each cell in the table
point to a linked list of
entries that map there
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025-612-0001
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451-229-0004
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• Separate chaining is
simple, but requires
additional memory
outside the table
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Open Addressing
• The colliding item is placed in a
different cell of the table
• Load factor: n/N, where n is the number of
items to store and N the size of the hash table.
• n/N≤1.
• To get a reasonable performance, n/N<0.5.
Hash Tables
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Linear Probing
• Linear probing handles
collisions by placing the
colliding item in the next
(circularly) available table
cell
• Each table cell inspected is
referred to as a “probe”
• Colliding items lump
together, causing future
collisions to cause a longer
sequence of probes
• Example:
– h(x) = x mod 13
– Insert keys 18, 41, 22, 44,
59, 32, 31, 73, in this order
0 1 2 3 4 5 6 7 8 9 10 11 12
41
18 44 59 32 22 31 73
0 1 2 3 4 5 6 7 8 9 10 11 12
Hash Tables
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Search with Linear Probing
• Consider a hash table A that
uses linear probing
• get(k)
– We start at cell h(k)
– We probe consecutive
locations until one of the
following occurs
• An item with key k is found,
or
• An empty cell is found, or
• N cells have been
unsuccessfully probed
– To ensure the efficiency, if k is
not in the table, we want to
find an empty cell as soon as
possible. The load factor can
NOT be close to 1.
Algorithm get(k)
i  h(k)
p0
repeat
c  A[i]
if c = 
return null
else if c.key () = k
return c.element()
else
i  (i + 1) mod N
pp+1
until p = N
return null
Hash Tables
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Linear Probing
• Search for key=20.
• Example:
– h(x) = x mod 13
– Insert keys 18, 41, 22, 44,
59, 32, 31, 73, 12, 20 in this
order
– h(20)=20 mod 13 =7.
– Go through rank 8, 9, …, 12, 0.
• Search for key=15
– h(15)=15 mod 13=2.
– Go through rank 2, 3 and
return null.
0 1 2 3 4 5 6 7 8 9 10 11 12
20 41
18 44 59 32 22 31 73 12
0 1 2 3 4 5 6 7 8 9 10 11 12
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Updates with Linear Probing
• To handle insertions and
deletions, we introduce a
special object, called
AVAILABLE, which replaces
deleted elements
• remove(k)
• put(k, o)
– We search for an entry with key
k
– If such an entry (k, o) is found,
we replace it with the special
item AVAILABLE and we
return element o
– Else, we return null
– Have to modify other methods
to skip available cells.
Hash Tables
– We throw an exception if the
table is full
– We start at cell h(k)
– We probe consecutive cells until
one of the following occurs
• A cell i is found that is either
empty or stores AVAILABLE,
or
• N cells have been unsuccessfully
probed
– We store entry (k, o) in cell i
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Updates with Linear Probing
Algorithm put(k,o)
i  h(k); p  0;
av = -1
repeat
• Example:
c  A[i]
– h(x) = x mod 13
if av == -1 and c= 
– Insert keys 18, 41, 22, 44,
A[i].key()=k
59, 32, 31, 73, 20, 12 in this
A[i].element=o
order
return
– Ti insert 12, we look at rank
else if A[i].key()=k
12 and then rank 0.
A[i].element=o
return
else if av > -1 and c= 
A[av].key()=k
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A[av].element=o
return
else
if av == -1 then av =i
12 41
18 44 59 32 22 31 73 20
i  (i + 1) mod N
0 1 2 3 4 5 6 7 8 9 10 11 12
pp+1
until p = N
Hash Tables
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if av >-1 then A[av].key()=k; A[av].element=o
A complete example
• Example:
– h(x) = x mod 13
– Insert keys 18, 41, 22, 44,
59, 32, 31, 73, 20, 12 in this
order
– Remove(): 20, 12
– Get(11): check the cell after
AVAILABLE cells.
– Insert keys 10, 11. 10 is at
rank 12 and 11 is at rank 0.
The Available cells are hard to
deal with.
Separate Chaining approach
is simpler.
0 1 2 3 4 5 6 7 8 9 10 11 12
A
A
12 41
18 44 59 32 22 31 73 20
0 1 2 3 4 5 6 7 8 9 10 11 12
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Performance of Hashing
• In the worst case, searches,
insertions and removals on a
hash table take O(n) time
• The worst case occurs when all
the keys inserted into the map
collide
• The load factor a = n/N affects
the performance of a hash table
• Assuming that the hash values
are like random numbers, it can
be shown that the expected
number of probes for an
insertion with open addressing
is
1 / (1 - a)
• The expected running
time of all the operations
in a hash table is O(1)
• In practice, hashing is very
fast provided the load
factor is not close to 100%
• Applications of hash
tables:
Hash Tables
– small databases
– compilers
– browser caches
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