1. Hash Tables
1/28/2018
Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 2014
Hash Tables  1 2 3  4
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

2. Recall the Map ADT get(k) put(k, v) remove(k)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

3. Intuitive Notion of a Map
Intuitively, a map M supports the abstraction of using keys as indices
with a syntax such as M[k].
consider a restricted setting
keys that are known to be integers in a range from 0 to N − 1, for some N ≥ n.
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

4. More General Kinds of Keys
But what should we do if our keys are not integers in the range from 0 to N – 1?
hash function
map general keys to corresponding indices in a table.
For instance, the last four digits of a Social Security number.  1 2 3  4 …
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

5. Hash Functions and Hash Tables
A hash function h
maps keys 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
When implementing a map with a hash table, the goal is to store item (k, o) at index i = h(k)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

6. Example We design a hash table for a map 1 2 3 Our hash table uses 4 …
storing entries as (SSN, Name)
SSN (social security number) is a nine-digit positive integer
Our hash table uses
array of size N = 10,000 and
hash function h(x) = last four digits of x  1 2 3 4
9997
9998
9999 …
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

7. Hash Functions Hash code: h1: keys  integers
Compression function: h2: integers  [0, N - 1]
h(x) = h2(h1(x))
“disperse” the keys in an apparently random way
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

8. Hash Codes (Example 1) Memory address:
We reinterpret the memory address of the key object as an integer (default hash code of all Java objects)
Good in general, except for
numeric keys (same number in different memory addresses)
string keys (same string in different memory addresses)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

9. Hash Codes (Example 2) Integer cast:
We reinterpret the bits of the key as an integer
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
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

10. Hash Codes (Example 3) Component sum:
partition the bits of the key into components of fixed length (e.g., 16 or 32 bits)
sum the components (ignoring 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
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

11. Hash Codes (Example 4) Polynomial accumulation:
partition the bits of the key into a sequence of components of fixed length (e.g., 8, 16 or 32 bits) a0 a1 … an-1
evaluate the polynomial
p(z) = a0 + a1 z + a2 z2 + … … + an-1zn-1
at a fixed value z, ignoring overflows
Especially suitable for strings (e.g., the choice z = 33 gives at most 6 collisions on a set of 50,000 English words)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

12. Hash Codes (Example 4)
Polynomial p(z) can be evaluated in O(n) time using Horner’s rule:
The following polynomials are successively computed, each from the previous one in O(1) time
p0(z) = an-1
pi (z) = an-i-1 + zpi-1(z) (i = 1, 2, …, n -1)
We have p(z) = pn-1(z)
9x3 + 4x2 + 2x + 1
p0(z) = 9
p1(z) = 4 + z * 9 = 4 + 9z
p2(z) = 2 + z(4 + 9z) = 2+ 4z + 9z2
P3(z) = 1 + z(2 + 4z + 9z2)= 1 + 2z + 4z2 + 9z3
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

13. Compression Functions (Example 1)
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
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

14. Compression Functions (Example 2)
Multiply, Add and Divide (MAD):
h2 (y) = [ (ay + b) mod p ] mod N
p is a prime number > N
a and b are nonnegative integers in [0, p-1]
a > 0
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

15. Collisions
Different keys are hashed to the same cell in the hash table—same hash value
If key is SSN and hash function is last 4 digits
Same hash value for:
The main reason for hash values to be “dispersed”
Different ways to handle collisions
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

16. Worst-case Time Complexity
n = number of entries
Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
get(key)
put(key,value)
remove(key)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

17. Worst-case Time Complexity
n = number of entries
Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
get(key)
O(n)
O(1)
put(key,value)
remove(key)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

18. Worst-case Time Complexity
n = number of entries
Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
get(key)
O(n)
O(1)
put(key,value)
remove(key)
What is the disadvantage of Hash Tables?
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

19. Handling Collisions Additional space No additional space
Separate Chaining
No additional space
Linear probing
Quadratic probing
Double hashing
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

20. Separate Chaining
each cell in the table point to a linked list of entries
simple, but requires additional memory outside the table  1 2 3 4
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

21. Linear Probing Example: h(x) = x mod 13k
h(k) =
k mod 13
Probes 18 5 41 2 22 9 44
5,6 59 7 32 6
6,7,8 31
5,6,7,8,9,10 73 8
8,9,10,11
Open addressing
the colliding item is placed in a different cell of the table
Linear probing
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
“clustering”
Example:
h(x) = x mod 13
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 1 2 3 4 5 6 7 8 9 10 11 12
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

22. Search with Linear Probing
Consider a hash table A that uses linear probing
get(k)
start at cell/bucket h(k)
probe consecutive locations until
key k is found, or
An empty cell is found, or
N cells have been unsuccessfully probed
Algorithm get(k)
i  h(k)
p  0 // number of probes
repeat
c  A[i]
if c = 
return null
else if c.getKey () = k
return c.getValue()
else
i  (i + 1) mod N
p  p + 1
until p = N
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

23. Search with Linear Probing
Consider a hash table A that uses linear probing
get(k)
start at cell/bucket h(k)
probe consecutive locations until
key k is found, or
An empty cell is found, or
N cells have been unsuccessfully probed
Algorithm get(k)
i  h(k)
p  0 // number of probes
repeat
c  A[i]
if c = 
return null
else if c.getKey () = k
return c.getValue()
else
i  (i + 1) mod N
p  p + 1
until p = N
Why stop when empty cell is found?
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

24. Updates with Linear Probing
To handle insertions and deletions
we introduce a special object, called DEFUNCT, which replaces deleted elements
Why do we need DEFUNCT, separate from EMPTY?
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

25. Why DEFUNCT? Example: h(x) = x mod 13k
h(k) =
k mod 13
New
location 18 5 41 2 22 9 44 6 59 7 32 8 31 10 73 11
Example:
h(x) = x mod 13
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 1 2 3 4 5 6 7 8 9 10 11 12
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

26. Why DEFUNCT? Example: Hint: remove 18 h(x) = x mod 13k
h(k) =
k mod 13
New
location 18 5 41 2 22 9 44 6 59 7 32 8 31 10 73 11
Example:
h(x) = x mod 13
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order
Hint: remove 18 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 1 2 3 4 5 6 7 8 9 10 11 12
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

27. Why DEFUNCT? Example: Hint: remove 18 Do we need to change get()?k
h(k) =
k mod 13
New
location 18 5 41 2 22 9 44 6 59 7 32 8 31 10 73 11
Example:
h(x) = x mod 13
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order
Hint: remove 18
Do we need to change get()? 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 1 2 3 4 5 6 7 8 9 10 11 12
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

28. Deletion with Linear Probing
remove(k)
We search for an entry with key k
If such an entry (k, o) is found, we replace it with the special item DEFUNCT and we return element o
Else, we return null
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

29. Insertion with Linear Probing
put(k, o)
throw an exception if the table is full
start at cell h(k)
probe consecutive cells until one of the following occurs
A cell i is found that is either empty or stores DEFUNCT, or
N cells have been unsuccessfully probed
We store (k, o) in cell i
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

30. Worst-case Time Complexity
n = number of entries
Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
(with collisions)
get(key)
O(n)
O(1)
put(key,value)
remove(key)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

31. Worst-case Time Complexity
n = number of entries
Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
(with collisions)
get(key)
O(n)
O(1)
put(key,value)
remove(key)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

32. Quadratic Probing Linear probing Quadratic probing
Probe (h(k) + i) mod N
where i is the i-th collision
Quadratic probing
Probe (h(k) + i2) mod N
Reduce clustering near the keys
But it could still form “secondary clustering”
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

33. Double Hashing (h(k) + i*d(k)) mod N
use a secondary hash function d(k)
Probe
(h(k) + i*d(k)) mod N
where i is the i-th collision
d(k) cannot have zero values
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

34. Double Hashing
Common choice of compression function for the secondary hash function:
d2(k) = q - (k mod q)
where
q < N
q is a prime
The possible values for d2(k) are 1, 2, … , q
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

35. Example of Double Hashing
Consider a hash table storing integer keys that handles collision with double hashing
N = 13
h(k) = k mod 13
d(k) = 7 - k mod 7 1 2 3 4 5 6 7 8 9 10 11 12 31 41 18 32 59 73 22 44 1 2 3 4 5 6 7 8 9 10 11 12
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

36. Load Factor a = n/N n = number of entries N = size of hash table
affects the performance of a hash table
Assuming hash values are like random numbers
it can be shown that the expected number of probes for an insertion is 1 / (1 - a)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

37. Desirable Load Factor Separate Chaining
Open addressing: linear probing
Load factor < 0.5
When the load factor is too high
Resize the hash table and
Rehash the entries
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

38. Performance of Hashing
The “expected” time complexity for get/put/remove
O(1)
In practice, hashing is very fast provided the load factor is not close to 100%
Applications of hash tables:
small databases
compilers
browser caches
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables

39. Time Complexity n = number of entries Counting # of comparisons
Operations
DoublyLinkedList
(unsorted)
Hash Table
(no collisions)
(with collisions)
(expected)
get(key)
O(n)
O(1)
put(key,value)
remove(key)
© 2014 Goodrich, Tamassia, Godlwasser
Hash Tables