Randomness in Computing L ECTURE 15 Last time Poisson approximation - - PowerPoint PPT Presentation

3/19/2020

Randomness in Computing

LECTURE 15

Last time

• Poisson approximation
• Application: max load
• Application: Coupon Collector

Today

• Hashing

Sofya Raskhodnikova;Randomness in Computing

Extra slide for notes

3/19/2020

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.

3/19/2020

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

A random hash function

• Simplifying assumption: hash function ℎ is selected at random:

Pr ℎ 𝑥 = 𝑘 = 1 𝑜 for all 𝑥 ∈ 𝑉, 𝑘 ∈ [𝑜]

• Once ℎ is chosen, every evaluation of ℎ yields the same answer.

Search time:

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

If we set 𝑜 = 𝑛, then

• the expected search time is O(1)
• max time to search is max load: w.p. close to 1, it is Θ

ln 𝑛 ln ln 𝑛

Faster than a search tree, with space still Θ(𝑛).

3/19/2020

Sofya Raskhodnikova; Randomness in Computing

Are we done?

• How many hash functions are there?
• How many bits do we need to store a description of a hash

function?

This is prohibitively expensive!

• Idea: Choose from a smaller family of hash functions.

3/19/2020

Sofya Raskhodnikova; Randomness in Computing

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}

3/19/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem ℋ is universal.

Proof that ℋ is universal

• Define 𝒊𝒃,𝒄 𝒚 =

𝑏𝑦 + 𝑐 𝑛𝑝𝑒 𝑞 𝑛𝑝𝑒 𝑜 ℋ = ℎ𝑏,𝑐 𝑏 ∈ 𝑞 − 1 , 0 ≤ 𝑐 ≤ 𝑞 − 1} Proof: Fix 𝑦1 ≠ 𝑦2 from U.

• Idea: count # of ℎ𝑏,𝑐 in ℋ for which 𝑦1, 𝑦2 collide.
• We will show that

– They can’t collide after performing mod 𝑞. – So, they must map to different values 𝑤1, 𝑤2 at this point – Each (𝑤1, 𝑤2) corresponds to a unique pair (𝑏, 𝑐). – So it suffices to count the number of pairs (𝑤1, 𝑤2) with 𝑤1 ≠ 𝑤2, but 𝑤1 = 𝑤2 𝑛𝑝𝑒 𝑜

3/19/2020

Sofya Raskhodnikova; Randomness in Computing

Proof that ℋ is universal

• Define 𝒊𝒃,𝒄 𝒚 =

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

3/19/2020

Sofya Raskhodnikova; Randomness in Computing

Proof that ℋ is universal

• Define 𝒊𝒃,𝒄 𝒚 =

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

3/19/2020

Sofya Raskhodnikova; Randomness in Computing