CS 225 Data Structures Oc October 26 26 Ha Hashing G G Carl - - PowerPoint PPT Presentation

CS 225

Data Structures

Oc October 26 26 – Ha Hashing

G G Carl Evans

What if 𝑃(𝑚𝑝𝑕!𝑜) is not fast enough?

Do you feel lucky?

A H A Hash T Table b based D Dictionary

A Hash Table consists of three things:

• 1. A hash function, f(k)
• 2. An array
• 3. Something to handle chaos when it occurs!

Dictionary<KeyType, ValueType> d; d[k] = v; 1 2

Client Code:

A P A Perf rfect H Hash F Function

(Angrave, CS 241) (Beckman, CS 421) (Challen, CS 125) (Davis, CS 101) (Evans, CS 126) (Fagen-Ulmschneider, CS 107) (Gunter, CS 422) (Herman, CS 233)

Hash function Key Value

Ha Hash h Func Function

Our hash function consists of two parts:

• A hash:
• A compression:

Choosing a good hash function is tricky…

• Don’t create your own (yet*)
• Very smart people have created very bad hash functions

Ha Hash h Func Function

Characteristics of a good hash function:

• 1. Computation Time:
• 2. Deterministic:
• 3. Satisfy the SUHA:

Gen Gener eral al Purpose e Has ash Fu Functi tion

… Keyspaces Easy to create if: |KeySpace| N ~

Gen Gener eral al Purpose e Has ash Fu Functi tion

… Keyspaces Easy to create if: |KeySpace| N ~ … Difficult to Create:

Gen Gener eral al Purpose e Has ash Fu Functi tion

… Keyspaces Easy to create if: |KeySpace| N ~ … Difficult to Create:

Ha Hash h Func Function

In CS 225, we focus on general purpose hash functions. Other hash functions exists with different properties (eg: cryptographic hash functions)

Col Collision

• n H

Handling: Se : Separate Ch Chaining

S = { 16, 8, 4, 13, 29, 11, 22 } |S| = n h(k) = k % 7 |Array| = N

1 2 3 4 5 6 Worst Case SUHA Insert Remove/Find (Example of open hashing)

Col Collision

• n H

Handling: P : Prob

• be-ba

base sed d Ha Hashi shing ng

S = { 16, 8, 4, 13, 29, 11, 22 } |S| = n h(k) = k % 7 |Array| = N

1 2 3 4 5 6 (Example of closed hashing)

Col Collision

• n H

Handling: Li : Linear P r Prob

• bing

S = { 16, 8, 4, 13, 29, 11, 22 } |S| = n h(k) = k % 7 |Array| = N Try h(k) = (k + 0) % 7, if full… Try h(k) = (k + 1) % 7, if full… Try h(k) = (k + 2) % 7, if full… Try …

1 2 3 4 5 6 (Example of closed hashing) Worst Case SUHA Insert Remove/Find

A P A Problem w w/ L / Linear P r Probing

Primary clustering: Description: Remedy:

Col Collision

• n H

Handling: D : Dou

• uble h

hashing

S = { 16, 8, 4, 13, 29, 11, 22 } |S| = n h(k) = k % 7 |Array| = N Try h(k) = (k + 0*h2(k)) % 7, if full… Try h(k) = (k + 1*h2(k)) % 7, if full… Try h(k) = (k + 2*h2(k)) % 7, if full… Try … h(k, i) = (h1(k) + i*h2(k)) % 7

1 2 3 4 5 6 (Example of closed hashing)

Ru Running T Times

Linear Probing:

• Successful: ½(1 + 1/(1-α))
• Unsuccessful: ½(1 + 1/(1-α))2

Double Hashing:

• Successful: 1/α * ln(1/(1-α))
• Unsuccessful: 1/(1-α)

Separate Chaining:

• Successful: 1 + α/2
• Unsuccessful: 1 + α

The expected number of probes for find(key) under SUHA

(Don’t memorize these equations, no need.) Instead, observe:

• As α increases:
• If α is constant:

Ru Running T Times

Linear Probing:

• Successful: ½(1 + 1/(1-α))
• Unsuccessful: ½(1 + 1/(1-α))2

Double Hashing:

• Successful: 1/α * ln(1/(1-α))
• Unsuccessful: 1/(1-α)

The expected number of probes for find(key) under SUHA

Re ReHashing

What if the array fills?

Which collision resolution strategy is better?

• Big Records:
• Structure Speed:

What structure do hash tables replace? What constraint exists on hashing that doesn’t exist with BSTs? Why talk about BSTs at all?

Ru Running T Times

Hash Table AVL Linked List

Find

Amortized: Worst Case:

Insert

Amortized: Worst Case:

Storage Space

st std da data struc uctur ures

std::map

st std da data struc uctur ures

std::map ::operator[] ::insert ::erase ::lower_bound(key) è Iterator to first element ≤ key ::upper_bound(key) è Iterator to first element > key

st std da data struc uctur ures

std::unordered_map ::operator[] ::insert ::erase ::lower_bound(key) è Iterator to first element ≤ key ::upper_bound(key) è Iterator to first element > key

st std da data struc uctur ures

std::unordered_map ::operator[] ::insert ::erase ::lower_bound(key) è Iterator to first element ≤ key ::upper_bound(key) è Iterator to first element > key ::load_factor() ::max_load_factor(ml) è Sets the max load factor