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Part E Hash Tables

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Part EHash Tables

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025-612-0001

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981-101-0002

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451-229-0004

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

A hash functionh maps keys of a given typewith 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

A hash functionh maps keys of a given type to integers in a fixed interval [0, N- 1]

Example:h(x) =x mod Nis 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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025-612-0001

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981-101-0002

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451-229-0004

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9997

9998

200-751-9998

9999

- 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 sizeN= 10,000 and the hash functionh(x) = last four digits of x
- Need a method to handle collision.

Hash Tables

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50xxxx01

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50xxxx02

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86xxxx04

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97

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51xxxx98

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51xxxx02

- 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 sizeN= 100 for our class of n= 49 students and the hash functionh(x) = last two digits of x
- Need a method to handle collision.

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.

Hash Tables

A hash function is usually specified as the composition of two functions:

Hash code:h1:keysintegers

key could be anything, e.g., your name, an object, etc.

Compression function:h2: integers [0, N- 1]

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

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

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

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

Multiply, Add and Divide (MAD):

h2 (y) =(ay + b)mod N

a >0 and b>0 are nonnegative integers such thata 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

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

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025-612-0001

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451-229-0004

981-101-0004

Hash Tables

the colliding item is placed in a different cell of the table

Load factor: n/N, wheren 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

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 mod13

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

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Hash Tables

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.

Algorithmget(k)

i h(k)

p0

repeat

c A[i]

if c=

returnnull

else if c.key () =k

returnc.element()

else

i(i+1)mod N

p p+1

untilp=N

returnnull

Hash Tables

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 mod13

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

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Hash Tables

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

Example:

h(x) = x mod13

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.

Algorithmput(k,o)

i h(k)

p0

repeat

c A[i]

if c=

returnA[i].key()=k and A[i].element=0

else if c.key () =k

A[i].element=o

else

i(i+1)mod N

p p+1

untilp=N

if p=N+1 the array is full.

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Hash Tables

Example:

h(x) = x mod13

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.

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Hash Tables

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/Naffects 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 is1/ (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:

small databases

compilers

browser caches

Hash Tables