Hashing Powerpoint

7/26/2019 Hashing Powerpoint

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Dictionaries, Tables

Hashing

TCSS 342


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The Dictionary ADT

a dictionary (table) is an abstract model of a

database

like a priority queue, a dictionary stores

key-element pairs

the main operation supported by a

dictionary is searching by key


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Examples

Telephone directory

Library catalogue

Books in print: key ISBN

FAT (File Allocation Table)


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Main Issues

Size

Operations: search, insert, delete, ??? Create

reports??? List?

What will be stored in the dictionary?

How will be items identified?


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The Dictionary ADT

simple container methods:

size()

isEmpty()

elements()

query methods:

findElement(k)

findAllElements(k)


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The Dictionary ADT

update methods:

insertItem(k, e)

removeElement(k)

removeAllElements(k)

special element

NO_SUCH_KEY, returned by an unsuccessful

search


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Implementing a Dictionary

with a Sequence unordered sequence

searching and removing takes O(n) time

inserting takes O(1) time

applications to log files (frequent insertions,

rare searches and removals) 34 14 12 22 18

34 14 12 22 18


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Implementing a Dictionary

with a Sequence array-based ordered sequence

(assumes keys can be ordered)

- searching takes O(log n) time (binary search)- inserting and removing takes O(n) time

- application to look-up tables

(frequent searches, rare insertions and removals)

12 14 18 22 34


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Binary Search

narrow down the search range in stages

high-low game

findElement(22)

2 4 5 7 8 9 12 14 17 19 22 25 27 28 33 37

low highmid

14


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Binary Search

2 4 5 7 8 9 12 14 17 19 22 25 27 28 33 37

low highmid

25

2 4 5 7 8 9 12 14 17 19 22 25 27 28 33 37

low highmid

19

2 4 5 7 8 9 12 14 17 19 22 25 27 28 33 37

low = mid = high

22


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Pseudocode for Binary Search

AlgorithmBinarySearch(S, k, low, high)

if low > high thenreturn NO_SUCH_KEY

elsemid (low+high) / 2

if k = key(mid) thenreturn key(mid)

else if k < key(mid) thenreturn BinarySearch(S, k, low, mid-1)

elsereturn BinarySearch(S, k, mid+1, high)


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Running Time of Binary

Search The range of candidate items to be searched

is halved after each comparison

Comparison Search Range

0 n

1 n/2

2i n/2i

log 2n 1


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Running Time of Binary

Search In the array-based implementation, access

by rank takes O(1) time, thusbinary search

runs in O(log n) time

Binary Search is applicable only to Random

Access structures (Arrays, Vectors)


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Implementations

Sorted? Non Sorted?

Elementary: Arrays, vectors linked lists

Orgainization: None (log file), Sorted, Hashed

Advanced: balanced trees


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Skip Lists

Simulate Binary Search on a linked list.

Linked list allows easy insertion and

deletion.

http://www.epaperpress.com/s_man.html


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A FAT Example

Directory: Key: file name. Data (time, date, size) location of first block in the FAT table.

If first block is in physical location #23 (Diskblock number) look up position #23 in the FAT.Either shows end of file or has the block numberon disk.

Example: Directory entry: block # 4 FAT: x x x F 5 6 10 x 23 25

3

The file occupies blocks 4,5,6,10, 3.


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Hashing

Place item with key k in position h(k).

Hope: h(k) is 1-1.

Requires: unique key (unless multiple items

allowed). Key must be protected from

change (use abstract class that provides only

a constructor).

Keys must be comparable.


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Hashing Problem

RT&T is a large phone company, and they

want to provide enhanced caller ID

capability: given a phone number, return the callers name

phone numbers are in the range 0 toR = 10101

n is the number of phone numbers usedwant to do this as efficiently as possible


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Generalized indexing

Hash table

Data storage associated with a key

The key need not be an integer


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Hash Tables

A data structure

The location of an item is determined

directly as a function of the item itselfNot by a sequence of trial and error comparisons

Commonly used to provide faster searching

O(n) for linear searches

O (logn) for binary search

O(1) for hash table


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Another Solution

A Hash Table is an alternative solution with

O(1) expected query time and O(n + N)

space, whereN is the size of the table Like an array, but with a function to map

the large range of keys into a smaller one

e.g., take the original key, mod the size of thetable, and use that as an index


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Example

Insert item (401-863-7639, Roberto) into a table of

size 5

4018637639 mod 5 = 4, so item (401-863-7639,Roberto) is stored in slot 4 of the table

A lookup uses the same process: map the key to an

index, then check the array cell at that index

401-

863-7639

Roberto

0 1 2 3 4


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Collision

Insert (401-863-9350, Andy)

And insert (401-863-2234, Devin). We have

a collision!


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Collision Resolution

How to deal with two keys which map to

the same cell of the array?

Use chaining

Set up lists of items with the same index


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Hash Function

Function hdefined by h(i) = i

Determines the location of an item iin the

hash table

Called a hash function.

To reduce the large size of a hash table

use h(i) = imod25;


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Popular Hash-Code Maps

The polynomial is computed withHorners

rule, ignoring overflows, at a fixed value x:

a0+ x (a1+ x (a2+ ... x (an-2+ x an-1) ... ))The choicex = 33, 37, 39, or 41 gives at most 6

collisions on a vocabulary of 50,000 English

words

Why is the component-sum hash code bad

for strings?


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Random Hashing

Random hashing

Uses a simple random number generation

technique Scatters the items randomly throughout

the hash table


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Hash code

static inthashCode(long i) {

return(int)((i >> 32) +(int) i);}


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Linear Probing Hash Table

/** constructor providing the hash comparator and the

capacity

* of the bucket array */

public LinearProbingHashTable(HashComparator hc, int

bN) {

h = hc;

N = bN;

A = newItem[N];

}


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Dictionary

public voidinsertItem (Object key, Object element)

throwsInvalidKeyException {

check(key);

int i = h.

hashValue(key) % N; // division method compression

map

intj = i;


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Hashing Powerpoint

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