Hash Tables1 Midterm will be given in Week 11’s lecture (2 hurs)

Hash Tables 1

Midterm will be given in Week 11’s lecture (2 hurs)

Part E

Hash Tables

Hash Tables 2

01234 451-229-0004

981-101-0002

025-612-0001

• Questions: how to design a data structure, such that the following operations are all O(1)?– Deletion– Search– Insertion– Replace

Hash Tables 3

Motivations of Hash Tables

• We have n items, each contains a key and value (k, value).– The key uniquely determines the item.

• 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 4

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, N1], 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 5

Hash Functions

• A hash function h maps keys of a given type to integers in a fixed interval [0, N1]

• 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)

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Example

• 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 N10,000 and the hash functionh(x)last four digits of x

• Need a method to handle collision.

Hash Tables 7

01234

999799989999

…451-229-0004

981-101-0002

200-751-9998

025-612-0001

Example

• We design a hash table storing entries as (studentID, Name), where studentID is a eight-digit positive integer

• Our hash table uses an array of size N100 for our class of n= 49 students and the hash functionh(x)last two digits of x

• Need a method to handle collision.

Hash Tables 8

01234

97 9899

…

86xxxx04

50xxxx02

51xxxx98

50xxxx0151xxxx

02

Task: 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.

As long as the chance for collision is low, we can achieve this goal.

Setting N=1000 and looking at the last three digits will reduce the chance of collision.

How to design a Hash Function

• A hash function is usually specified as the composition of two functions:Hash code: h1: keys integers– key could be anything, e.g., your

name, an object, etc.

Compression function: h2: integers [0, N1]– The size of the array N cannot be

too large in order to save space.– Trade-off between space and

time.

• The hash code is applied first, and the compression function is applied next on the result, i.e.,

h(x) = h2(h1(x))• The goal of the hash

function is to “disperse” the keys in an apparently random way so that in most cases (1) holds.

Hash Tables 9

Integer cast:– We reinterpret the bits of the key as an integerExample: 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 10

Component sum:– We partition the bits of the key into components of fixed length

(e.g., 16 or 32 bits) a0 a1 … an1 and we sum the components (ignoring overflows) a0 a1 a2 … +an1

– 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 11

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 12

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.

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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

• Separate chaining is simple, but requires additional memory outside the table

Hash Tables 14

01234 451-229-0004 981-101-0004

025-612-0001

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 15

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

Hash Tables 16

0 1 2 3 4 5 6 7 8 9 10 11 12

41 18445932223173 0 1 2 3 4 5 6 7 8 9 10 11 12

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.

Hash Tables 17

Algorithm get(k)i h(k)p 0repeat

c A[i]if c

return null else if c.key () k

return c.element()else

i (i 1) mod Np p 1

until p Nreturn null

Linear Probing

• Search for key=20.– 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.

• Example:– h(x) x mod 13

– Insert keys 18, 41, 22, 44, 59, 32, 31, 73, 12, 20 in this order

Hash Tables 18

0 1 2 3 4 5 6 7 8 9 10 11 12

20 41 18445932223173 120 1 2 3 4 5 6 7 8 9 10 11 12

Updates with Linear Probing

• To handle insertions and deletions, we introduce a special object, called AVAILABLE, which replaces deleted elements

• 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 AVAILABLE and we return element o

– Else, we return null– Have to modify other methods

to skip available cells.

• put(k, o)– 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

Hash Tables 19

Updates with Linear Probing

• Example:– h(x) = x mod 13– Insert keys 18, 41, 22, 44,

59, 32, 31, 73, 20, 12 in this order

– Ti insert 12, we look at rank 12 and then rank 0.

Hash Tables 20

Algorithm put(k,o)

i h(k); p 0; av = -1repeat

c A[i]if av == -1 and c

A[i].key()=k A[i].element=oreturn

else if A[i].key()=k A[i].element=o return else if av > -1 and c A[av].key()=k A[av].element=o return else if av == -1 then av =i

i (i 1) mod Np p 1

until p N if av >-1 then A[av].key()=k; A[av].element=o

0 1 2 3 4 5 6 7 8 9 10 11 12

12 41 18445932223173 200 1 2 3 4 5 6 7 8 9 10 11 12

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.

Hash Tables 21

0 1 2 3 4 5 6 7 8 9 10 11 12

12 41 18445932223173 200 1 2 3 4 5 6 7 8 9 10 11 12

AA

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 nN 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 )

• 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:– small databases– compilers– browser caches

Hash Tables 22