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Lecture 9

Notes below are meant to supplement the scribed notes.

Universal Hash Functions (Section 9.2)

Introduction

  • \(\mathcal{H}\) is a family of deterministic hash functions mapping a universe \(U\) to the indices of a hash table \(T\) of size \(n\).
  • We want to hash a subset \(S \subset U\) (may be chosen adversarially) into \(T\).
  • To get a good expected performance regardless of \(S\), for each set \(S\), we sample a function \(h\in\mathcal{H}\) uniformly at random.
  • Use hash function \(h\) to hash elements in \(S\) into table \(T\).

Interpretation of Different Hash Function Properties

The universal hash function property (Definition 2) requires a family \(\mathcal{H}\) to have an expected collision bound, even for adversarially chosen sets \(S\). The (strongly) 2-universal hash function property (Definition 3) indicates for any distinct elements \(x,y\in U\),

\(\mathrm{Pr}_{h\in\mathcal{H}}[h(x) = a] = \frac{1}{n}\) for any \(a\in [n]\),(1)

  1. By \(\mathrm{Pr}_{h\in\mathcal{H}}[h(x) = a] = \sum_{b\in [n]} \mathrm{Pr}_{h\in\mathcal{H}}[h(x) = a \wedge h(y)=b]\).

\(h(x)\) and \(h(y)\) are independent random variables when \(h\) is sampled uniformly from \(\mathcal{H}\).(1)

  1. For any \(a,b\in [n]\), \(\mathrm{Pr}[h(x) = a \wedge h(y)=b]= \frac{1}{n^2} = \mathrm{Pr}[h(x) = a] \cdot \mathrm{Pr}[h(y) = b]\).

Note that pairwise independence DOES NOT imply mutual independence for three or more random variables.(1)

  1. For example, consider three random variables \(X,Y,Z\) such that each takes values in \(\{0,1\}\) with equal probability, and \(Z = X \oplus Y\) (XOR operation). Then any pair of them are independent, but all three are not mutually independent since knowing any two determines the third.

The inequality in the definition of the \(k\)-wise independent property (Definition 4) actually holds as an equality(1) for any distinct \(x_1,x_2,\dots,x_k\in U\) and any \(\alpha_1,\alpha_2,\dots,\alpha_k\in [n]\). Therefore, the \(k\)-wise independent property generalizes the independence property in the 2-universal property to \(k\) elements, and the above two properties extend naturally to the \(k\)-wise independent property.(2)

  1. Since \(\sum_{\alpha_1,\alpha_2,\dots,\alpha_k\in [n]} \mathrm{Pr}_{h\in\mathcal{H}}[h(x_1) = \alpha_1 \wedge \dots \wedge h(x_k) = \alpha_k] = 1\), we have each term must equal \(\frac{1}{n^k}\). Some authors write “\(\leq\)” out of habit or to emphasize a collision bound rather than strict independence.

  2. That is, each \(h(x_i)\) is uniformly distributed over \([n]\), and any subset of \(\{h(x_1),h(x_2),\dots,h(x_k)\}\) are mutually independent random variables when \(h\) is sampled uniformly from \(\mathcal{H}\).

Perfect Hashing (Section 9.5)

Bound on the Sum of Squared Bucket Sizes

Let \(L_i\) be the length of a chain at index \(i\), then by Markov's inequality, we have the total number of collisions is bounded by \(N\) with probability at least \(1/2\):

\[ \sum_{i=0}^{n-1} \binom{L_i}{2} \leq n. \]

This implies, with probability at least \(1/2\),

\[ \sum_{i=0}^{n-1} L_i^2 = \sum_{i=0}^{n-1} \left(2\binom{L_i}{2} + L_i\right) = 2\sum_{i=0}^{n-1} \binom{L_i}{2} + \sum_{i=0}^{n-1} L_i \leq 2n + n = 3n. \]

Bloom Filters (Section 9.6)

Each time we add an item, we set \(k\) bits to 1 using \(k\) different hash functions, and the probability that a specific bit is set to 1 by one hash function for one item is \(\frac{1}{n}\). Therefore, the probability that any individual bit is still 0 after adding \(N\) items is

\[ \left(1 - \frac{1}{n}\right)^{kN} \approx e^{-\frac{kN}{n}}. \]