Randomness in Computing L ECTURE 16 Last time Hashing Universal - - PowerPoint PPT Presentation

3/24/2020

Randomness in Computing

LECTURE 16

Last time

• Hashing
• Universal hash families

Today

• Using universal hash families
• Perfect hashing
• Bloom filters

Sofya Raskhodnikova;Randomness in Computing

Static dictionary problem

Motivating example Password checker to prevent people from using common passwords.

• S is the set of common passwords
• Universe: set 𝑉
• 𝑇 ⊆ 𝑉 and 𝑛 = |𝑇|
• 𝑛 ≪ |𝑉|

Goal: A data structure for storing 𝑻 that supports the search query “Does 𝑥 ∈ 𝑻 ?” for all words 𝑥 ∈ 𝑽.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Solutions

Deterministic solutions

• Store 𝑻 as a sorted array (or as a binary search tree)

Search time: O(log 𝑛), Space: O(𝑛)

• Store an array that for each 𝑥 ∈ 𝑽 has 1 if 𝑥 ∈ 𝑻 and 0 otherwise.

Search time: O(1), Space: O(|𝑽|)

A randomized solution

• Hashing

Chain Hashing

• Hash table: 𝒐 bins, words that fall in the

same bin are chained into a linked list.

• Hash function: ℎ : 𝑉 [𝑜]

To construct the table hash all elements of 𝑇 To search for word 𝒙 check if 𝑥 is in bin ℎ(𝑥) Desiderata for 𝒊:

• O(1) evaluation time.
• O(1) space to store ℎ.

⋮ ⋮ 1 2

𝒐

Elements of 𝑻

Universal hash family

• A set ℋof hash functions is universal if for every pair

𝑥1, 𝑥2 ∈ 𝑉 and for ℎ chosen uniformly from ℋ Pr ℎ 𝑥1 = ℎ 𝑥2 ≤ 1 𝑜 Constructing a universal hash family

• Fix a prime 𝑞 ≥ |𝑉| and think of the range as 0,1, … , 𝑜 − 1 .
• Define 𝒊𝒃,𝒄 𝒚 =

𝑏𝑦 + 𝑐 𝑛𝑝𝑒 𝑞 𝑛𝑝𝑒 𝑜 ℋ = ℎ𝑏,𝑐 𝑏 ∈ 𝑞 − 1 , 0 ≤ 𝑐 ≤ 𝑞 − 1}

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem ℋ is universal.

Using a universal family

As before:

• If 𝑥 ∉ 𝑇, expected number of words in bin ℎ(𝑥) is
• If 𝑥 ∈ 𝑇, expected number of words in bin ℎ(𝑥) is

The previous guarantee on max load no longer holds! Goal: Given 𝑇, find a hash function with no collisions for words in S.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

≤ 𝒏 𝒐 ≤ 𝟐 + 𝒏 − 𝟐 𝒐

Perfect hashing: no collisions

Proof: Let 𝑡1, … , 𝑡𝑛 be elements of 𝑇.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem If ℎ: 𝑉 → {0,1, … , 𝑜 − 1} is chosen uniformly at random from a universal hash family, then ∀𝑇 of size 𝑛, such that 𝑜 ≥ 𝑛2, Pr ℎ is perfect ≥ 1/2.

Perfect hashing

• Select ℎ ∈ ℋ until a perfect ℎ is found.
• Expected number of tries is at most 2.
• Each try takes 𝑃(𝑛) time.
• Drawback: Ω 𝑛2 space.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem If ℎ: 𝑉 → {0,1, … , 𝑜 − 1} is chosen uniformly at random from a universal hash family, then ∀𝑇 of size 𝑛, such that 𝑜 ≥ 𝑛2, Pr ℎ is perfect ≥ 1/2.

2-level scheme for perfect hashing

• Set 𝑛 = 𝑜.
• Select ℎ ∈ ℋ until ℎ with at most 𝑛 collisions is found.
• For each bin 𝑗 with collisions, that is, with 𝑙 > 1 items:

– select a new hash function ℎ𝑗 with 𝑙2 bins from a universal family until ℎ𝑗 has no collisions.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

1 2 . .

.

𝑜 − 1

1 2 . . . 𝑜 − 1

2-level scheme for perfect hashing

• Set 𝑛 = 𝑜.
• Select ℎ ∈ ℋ until ℎ with at most 𝑛 collisions is found.
• For each bin 𝑗 with collisions, that is, with 𝑙 > 1 items:

– select a new hash function ℎ𝑗 with 𝑙2 bins from a universal family until ℎ𝑗 has no collisions.

A solution for static dictionary problem with:

• O(1) worst case guarantee on search time.
• O(𝒏) space.
• Expected O(𝒏) preprocessing time.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem 2-level scheme achieves perfect hashing with 𝑃(𝑛) space.

Proof:

• Let 𝑌 = # of collisions in Stage 1.
• We showed before: Pr 𝑌 >

𝑛2 𝑜

≤

1 2.

• Now 𝑛 = 𝑜: Pr 𝑌 > 𝑛 ≤

1 2 .

• So at least half of ℎ ∈ ℋ have ≤ 𝑛 collisions.
• Assume we found such ℎ.

Analysis of 2-level scheme

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem 2-level scheme achieves perfect hashing with 𝑃(𝑛) space.

Proof (continued):

Analysis of 2-level scheme

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem 2-level scheme achieves perfect hashing with 𝑃(𝑛) space.

Conclusion: 2-level hashing

A solution for static dictionary problem with:

• O(1) worst case guarantee on search time.
• O(𝒏) space.
• Expected O(𝒏) preprocessing time.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Approximate solutions

for static dictionary problem

• False positives: If 𝑥 ∈ 𝑻, our data structure must answer
• correctly. If 𝑥 ∉ 𝑻, we may err with small probability.
• E.g, we prevent all unsuitable passwords and some suitable ones,

too.

Fingerprints

• Use hash function ℎ
• Store sorted list 𝑀 of fingerprints ℎ 𝑦 , 𝑦 ∈ 𝑇.
• To see if 𝑥 ∈ 𝑇, perform binary search for ℎ 𝑥 .

3/24/2020

Sofya Raskhodnikova; Randomness in Computing

Bloom filters

• Trade off between space and false positive probability
• Parameters 𝑙, 𝑜
• Bloom filter: array of 𝑜 bits 𝐵 1 , … , 𝐵[𝑜]

– Initially: all bits are 0 𝑙 independent random hash functions ℎ1, … , ℎ𝑙 with range [𝑜]

• To represent set 𝑇

– For each 𝑦 ∈ 𝑇 and 𝑗 ∈ [𝑙], set bits 𝐵[ℎ𝑗 𝑦 ] to 1.

• To search for 𝑥:

– If for all 𝑗 ∈ 𝑙 , bits 𝐵 ℎ𝑗 𝑥 = 1, accept, o.w. reject.

3/24/2020

Sofya Raskhodnikova; Randomness in Computing