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Hash Table Implementations: Chaining and Open Addressing, Study notes of Data Structures and Algorithms

University of Maryland Data Structures and Algorithms

An overview of hash table implementations using chaining and open addressing techniques. It covers the concepts of hash functions, hash table size, collision handling, and retrieval methods for both chaining and open addressing. The document also includes examples of hash table implementations and their respective probe sequences.

Typology: Study notes

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hs0

These notes may not be reproduced by any means (mechanical or elec-

tronic or any other) without the express written permission of Hanan Samet

HASHING METHODS

Hanan Samet

Computer Science Department and

Center for Automation Research and

Institute for Advanced Computer Studies

University of Maryland

College Park, Maryland 20742

e-mail: [email protected]

Discover Study notes of Data Structures and Algorithms University of Maryland

Partial preview of the text

Download Hash Table Implementations: Chaining and Open Addressing and more Study notes Data Structures and Algorithms in PDF only on Docsity!

hs

These notes may not be reproduced by any means (mechanical or elec- tronic or any other) without the express written permission of Hanan Samet

HASHING METHODS

Hanan Samet

Computer Science Department and Center for Automation Research and Institute for Advanced Computer Studies University of Maryland College Park, Maryland 20742 e-mail: [email protected]

hs

HASHING OVERVIEW

Task: compare the value of a key with a set of key values in a table
Conventional solutions:
1. use a comparison on key values (tree-based)
2. branching process governed by the digits comprising the key value (trie-based)
Alternative solution is to find a 1-1 mapping (i.e., function) from set of possible key values to a memory address and use table lookup methods to retrieve the

record —O (1) process

Problem: the set of possible key values is much larger than the number of available memory addresses

1. developing the 1-1 functionh is time-consuming as it

requires puzzle-solving abilities

result is called a perfect hashing function

2. onceh is found, addition of a single key value may

render the function meaningless

need to develop it anew

3. can replaceh by a program, which may itself be time-

consuming to compute

Result: usually abandon goal of finding 1-1 mapping and use a special method to resolve any ambiguity (i.e., when more than one key value is mapped to

the same address — termed acollision)

hs

• Hash table of sizem

• One chain (linked list) for each ofm hash values

containing all elements that hash to that location (known

as acollision list )

• Hash chains are known asbuckets

• Hash table locations are known asbucket addresses

• Forn key values, average chain size isn/m

• One chain (linked list) for each ofm hash values

Retrieval
1. use sequential search through chain
2. speed up unsuccessful search by sorting chain by key value
3. speed up successful search by self-organizing methods
  - move key value to start of chain each time it is accessed
Ex:

1 SEPARATE CHAINING b

h(k) NAME k=KEY NEXT 0 JIM 49 Λ 1 JOHN 22 Λ 2 RAY 30 Λ 3 SUZY 3 Λ 4 5 6

hs

• Hash table of sizem

• One chain (linked list) for each ofm hash values

containing all elements that hash to that location (known

as acollision list )

• Hash chains are known asbuckets

• Hash table locations are known asbucket addresses

• Forn key values, average chain size isn/m

• One chain (linked list) for each ofm hash values

Retrieval
1. use sequential search through chain
2. speed up unsuccessful search by sorting chain by key value
3. speed up successful search by self-organizing methods
  - move key value to start of chain each time it is accessed
Ex:

1 SEPARATE CHAINING b

h(k) NAME k=KEY NEXT 0 JIM 49 Λ 1 JOHN 22 Λ 2 RAY 30 Λ 3 SUZY 3 Λ 4 5 6

(^2) hs r

JANE 14 Λ

add JANE(14)→ 0

hs

• Whenm is large, many of the chains are empty

Use empty locations in table for the chain
Must be able to distinguish between free and occupied locations
Insertion algorithm:
1. if key value not present, then allocate a free location
2. link location to chain which was unsuccessfully searched
Ex:

1 IN-PLACE CHAINING b

h(k) NAME k=KEY NEXT 0 JIM 49 Λ 1 JOHN 22 Λ 2 RAY 30 Λ 3 SUZY 3 Λ 4 5 6

hs

• Whenm is large, many of the chains are empty

Use empty locations in table for the chain
Must be able to distinguish between free and occupied locations
Insertion algorithm:
1. if key value not present, then allocate a free location
2. link location to chain which was unsuccessfully searched
Ex:

1 IN-PLACE CHAINING b

h(k) NAME k=KEY NEXT 0 JIM 49 Λ 1 JOHN 22 Λ 2 RAY 30 Λ 3 SUZY 3 Λ 4 5 6

(^2) hs r

add JANE(14)→0 which collides with JIM(49)→ 0

6

JANE 14 Λ

hs

• Whenm is large, many of the chains are empty

Use empty locations in table for the chain
Must be able to distinguish between free and occupied locations
Insertion algorithm:
1. if key value not present, then allocate a free location
2. link location to chain which was unsuccessfully searched
Ex:

1 IN-PLACE CHAINING b

h(k) NAME k=KEY NEXT 0 JIM 49 Λ 1 JOHN 22 Λ 2 RAY 30 Λ 3 SUZY 3 Λ 4 5 6

(^2) hs r

add JANE(14)→0 which collides with JIM(49)→ 0

6

JANE 14 Λ

(^3) hs z

add LUCY(41)→6 which collides with JANE(14)→ 0 which is stored at 6
- result in coalescing of chains of JANE and LUCY making unsuccessful search longer as several chains must be searched

LUCY 41 Λ 5

(^4) hs g

Can avoid coalescing by moving JANE just before adding LUCY

LUCY 41

JANE 14

5

Λ

hs

IN-PLACE CHAINING INSERTION ALGORITHM

location procedure CHAINING_WITH_COALESCING_INSERTION(k); begin value key k; integer i; global integer r; /* r is the most recently allocated location */ global hashtable table; i←h(k); if OCCUPIED(table[i]) then begin while NOT(NULL(NEXT(table[i])) do begin if k=KEY(table[i]) then return(i) else i←NEXT(table[i]); end; if k=KEY(table[i]) then return(i); while OCCUPIED(table[r]) do r←r-1; if r≤0 then return(OVERFLOW') else begin NEXT(table[i])←r; i←r; end; end; MARK(table[i],OCCUPIED'); KEY(table[i])←k; NEXT(table[i])←NIL; return(i); end;

hs

Avoid extra space for NEXT field by not storing entire key value with record

• k =m ·q(k) +h(k),q(k) =  k/m ,h(k) =k modm

• Storeq(k) in table instead ofk

• Can computek givenm,q(k), andh(k),

• Ex: 0 ≤ k < 2 32

• Since only compareq(k), all elements in same collision

list must have the same value ofh(k) and thus no

coalescing is allowed

Data structure:
1. circular collision lists
2. flag FIRST denoting if first element on collision list
3. pointer NEXT to next element in circular list with same

h(k) value

Ex:

1 LAMPSON’S IN-PLACE CHAINING b

h(k) NAME k=KEY FIRST 0 JIM 49 T 7 0 1 JOHN 22 T 3 1 2 RAY 30 T 4 2 3 SUZY 3 T 0 3 4 5 6

q(k) NEXT

q(k) h(k)

0 21 22 31

(^2) hs r

add JANE(14)→ 0

JANE 14 F 2 0

6

hs

Avoid extra space for NEXT field by not storing entire key value with record

• k =m ·q(k) +h(k),q(k) =  k/m ,h(k) =k modm

• Storeq(k) in table instead ofk

• Can computek givenm,q(k), andh(k),

• Ex: 0 ≤ k < 2 32

• Since only compareq(k), all elements in same collision

list must have the same value ofh(k) and thus no

coalescing is allowed

Data structure:
1. circular collision lists
2. flag FIRST denoting if first element on collision list
3. pointer NEXT to next element in circular list with same

h(k) value

Ex:

1 LAMPSON’S IN-PLACE CHAINING b

h(k) NAME k=KEY FIRST 0 JIM 49 T 7 0 1 JOHN 22 T 3 1 2 RAY 30 T 4 2 3 SUZY 3 T 0 3 4 5 6

q(k) NEXT

q(k) h(k)

0 21 22 31

(^2) hs r

add JANE(14)→ 0

JANE 14 F 2 0

6

(^3) hs z

add LUCY(41)→6 but 6 contains JANE
- if at least one element of the hash chain starting at 6 exists, then it must be stored there
- must move JANE as it does not belong in 6

JANE 14 F 2 0 LUCY 41 T 5 6

5

hs

Like chaining but NEXT link field is open or unspecified
Probe sequence: set of locations comprising collision list of a key
Goal: cycle through all locations with little or no duplication

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

Insertion Algorithm:

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

Ex:

1 OPEN ADDRESSING b

h(k) NAME k=KEY 0 JIM 49 1 JOHN 22 2 RAY 30 3 SUZY 3 4 5 6

hs

Like chaining but NEXT link field is open or unspecified
Probe sequence: set of locations comprising collision list of a key
Goal: cycle through all locations with little or no duplication

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

Insertion Algorithm:

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

Ex:

1 OPEN ADDRESSING b

h(k) NAME k=KEY 0 JIM 49 1 JOHN 22 2 RAY 30 3 SUZY 3 4 5 6

(^2) hs r

adding JANE(14)→0 yields a collision; cyclic probe sequence causes its insertion in 4

JANE 14

hs

Like chaining but NEXT link field is open or unspecified
Probe sequence: set of locations comprising collision list of a key
Goal: cycle through all locations with little or no duplication

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

Insertion Algorithm:

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

Ex:

1 OPEN ADDRESSING b

h(k) NAME k=KEY 0 JIM 49 1 JOHN 22 2 RAY 30 3 SUZY 3 4 5 6

(^2) hs r

adding JANE(14)→0 yields a collision; cyclic probe sequence causes its insertion in 4

JANE 14

(^3) hs z

adding LUCY(41)→ 6

LUCY 41

(^4) hs g

delete RAY(30)→ 2

hs

Like chaining but NEXT link field is open or unspecified
Probe sequence: set of locations comprising collision list of a key
Goal: cycle through all locations with little or no duplication

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

Insertion Algorithm:

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

Ex:

1 OPEN ADDRESSING b

h(k) NAME k=KEY 0 JIM 49 1 JOHN 22 2 RAY 30 3 SUZY 3 4 5 6

(^2) hs r

adding JANE(14)→0 yields a collision; cyclic probe sequence causes its insertion in 4

JANE 14

(^3) hs z

adding LUCY(41)→ 6

LUCY 41

(^4) hs g

delete RAY(30)→ 2

(^5) hs r

problem: if look up JANE then don’t find her since a collision exists at location 0, and probe sequence finds location 2 unoccupied

Hash Table Implementations: Chaining and Open Addressing, Study notes of Data Structures and Algorithms

Related documents

Partial preview of the text

Download Hash Table Implementations: Chaining and Open Addressing and more Study notes Data Structures and Algorithms in PDF only on Docsity!

HASHING METHODS

record —O (1) process

1. developing the 1-1 functionh is time-consuming as it

2. onceh is found, addition of a single key value may

3. can replaceh by a program, which may itself be time-

the same address — termed acollision)

• Hash table of sizem

• One chain (linked list) for each ofm hash values

as acollision list )

• Hash chains are known asbuckets

• Hash table locations are known asbucket addresses

• Forn key values, average chain size isn/m

• One chain (linked list) for each ofm hash values

• Hash table of sizem

• One chain (linked list) for each ofm hash values

as acollision list )

• Hash chains are known asbuckets

• Hash table locations are known asbucket addresses

• Forn key values, average chain size isn/m

• One chain (linked list) for each ofm hash values

• Whenm is large, many of the chains are empty

• Whenm is large, many of the chains are empty

• Whenm is large, many of the chains are empty

• k =m ·q(k) +h(k),q(k) =  k/m ,h(k) =k modm

• Storeq(k) in table instead ofk

• Can computek givenm,q(k), andh(k),

• Ex: 0 ≤ k < 2 32

• Since only compareq(k), all elements in same collision

list must have the same value ofh(k) and thus no

h(k) value

q(k) h(k)

• k =m ·q(k) +h(k),q(k) =  k/m ,h(k) =k modm

• Storeq(k) in table instead ofk

• Can computek givenm,q(k), andh(k),

• Ex: 0 ≤ k < 2 32

• Since only compareq(k), all elements in same collision

list must have the same value ofh(k) and thus no

h(k) value

q(k) h(k)

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE

• Linear probing: h(k),h(k)+1,h(k)+2, …,m–1, 0, 1,h(k)–

1. calculate hash addressi

2. if TABLE(i ) is empty then insert and exit; elsei← i+

modm and repeat step 2 until exhausting TABLE