随机算法 (Spring 2013)/Introduction and Probability Space

Introduction

This course will study Randomized Algorithms, the algorithms that use randomness in computation.

Why do we use randomness in computation?

• Randomized algorithms can be simpler than deterministic ones.

(median selection, load balancing, etc.)

• Randomized algorithms can be faster than the best known deterministic algorithms.

(min-cut, checking matrix multiplication, primality testing, etc.)

• Randomized algorithms can do things that deterministic algorithms cannot do.

(routing, volume estimation, communication complexity, data streams, etc.)

• Randomized algorithms may lead us to smart deterministic algorithms.

(hashing, derandomization, SL=L, Lovász Local Lemma, etc.)

• Randomness is presented in the input.

(average-case analysis, smoothed analysis, learning, etc.)

• Some deterministic problems are random in nature.

(counting, inference, etc.)

• ...

How is randomness used in computation?

• To hit a witness/certificate.

(identity testing, fingerprinting, primality testing, etc.)

• To avoid worst case or to deal with adversaries.

(randomized quick sort, perfect hashing, etc.)

• To simulate random samples.

(random walk, Markov chain Monte Carlo, approximate counting etc.)

• To enumerate/construct solutions.

(the probabilistic method, min-cut, etc.)

• ...

Principles in probability theory

The course is organized by the advancedness of the probabilistic tools. We do this for two reasons: First, for randomized algorithms, analysis is usually more difficult and involved than the algorithm itself; and second, getting familiar with these probability principles will help you understand the true reasons for which the smart algorithms are designed.

• Basic probability theory: probability space, events, the union bound, independence, conditional probability.
• Moments and deviations: random variables, expectation, linearity of expectation, Markov's inequality, variance, second moment method.
• The probabilistic method: averaging principle, threshold phenomena, Lovász Local Lemma.
• Concentrations: Chernoff-Hoeffding bound, martingales, Azuma's inequality, bounded difference method.
• Markov chains and random walks: Markov chians, random walks, hitting/cover time, mixing time.

Probability Space

The axiom foundation of probability theory is laid by Kolmogorov, one of the greatest mathematician of the 20th century, who advanced various very different fields of mathematics.

Definition (Probability Space)

A probability space is a triple [math]\displaystyle{ (\Omega,\Sigma,\Pr) }[/math]. [math]\displaystyle{ \Omega }[/math] is a set, called the sample space. [math]\displaystyle{ \Sigma\subseteq 2^{\Omega} }[/math] is the set of all events, satisfying: (K1). [math]\displaystyle{ \Omega\in\Sigma }[/math] and [math]\displaystyle{ \empty\in\Sigma }[/math]. (The certain event and the impossible event.) (K2). If [math]\displaystyle{ A,B\in\Sigma }[/math], then [math]\displaystyle{ A\cap B, A\cup B, A-B\in\Sigma }[/math]. (Intersection, union, and diference of two events are events). A probability measure [math]\displaystyle{ \Pr:\Sigma\rightarrow\mathbb{R} }[/math] is a function that maps each event to a nonnegative real number, satisfying (K3). [math]\displaystyle{ \Pr(\Omega)=1 }[/math]. (K4). If [math]\displaystyle{ A\cap B=\emptyset }[/math] (such events are call disjoint events), then [math]\displaystyle{ \Pr(A\cup B)=\Pr(A)+\Pr(B) }[/math]. (K5*). For a decreasing sequence of events [math]\displaystyle{ A_1\supset A_2\supset \cdots\supset A_n\supset\cdots }[/math] of events with [math]\displaystyle{ \bigcap_n A_n=\emptyset }[/math], it holds that [math]\displaystyle{ \lim_{n\rightarrow \infty}\Pr(A_n)=0 }[/math].

Remark

• In general, the set [math]\displaystyle{ \Omega }[/math] may be continuous, but we only consider discrete probability in this lecture, thus we assume that [math]\displaystyle{ \Omega }[/math] is either finite or countably infinite.
• Sometimes it is convenient to assume [math]\displaystyle{ \Sigma=2^{\Omega} }[/math], i.e. the events enumerates all subsets of [math]\displaystyle{ \Omega }[/math]. But in general, a probability space is well-defined by any [math]\displaystyle{ \Sigma }[/math] satisfying (K1) and (K2). Such [math]\displaystyle{ \Sigma }[/math] is called a [math]\displaystyle{ \sigma }[/math]-algebra defined on [math]\displaystyle{ \Omega }[/math].
• The last axiom (K5*) is redundant if [math]\displaystyle{ \Sigma }[/math] is finite, thus it is only essential when there are infinitely many events. The role of axiom (K5*) in probability theory is like Zorn's Lemma (or equivalently the Axiom of Choice) in axiomatic set theory.

Laws for probability can be deduced from the above axiom system. Denote that [math]\displaystyle{ \bar{A}=\Omega-A }[/math].

Proposition

Denote that [math]\displaystyle{ \bar{A}=\Omega-A }[/math]. [math]\displaystyle{ \Pr(\bar{A})=1-\Pr(A) }[/math].

Proof. Due to Axiom (K4), [math]\displaystyle{ \Pr(\bar{A})+\Pr(A)=\Pr(\Omega) }[/math] which is equal to 1 according to Axiom (K3), thus [math]\displaystyle{ \Pr(\bar{A})+\Pr(A)=1 }[/math]. The proposition follows. [math]\displaystyle{ \square }[/math]

Exercise: Deduce other useful laws for probability from the axioms. For example, [math]\displaystyle{ A\subseteq B\Longrightarrow\Pr(A)\le\Pr(B) }[/math].

Notation

An event [math]\displaystyle{ A\subseteq\Omega }[/math] can be represented as [math]\displaystyle{ A=\{a\in\Omega\mid \mathcal{E}(a)\} }[/math] with a predicate [math]\displaystyle{ \mathcal{E} }[/math].

The predicate notation of probability is

[math]\displaystyle{ \Pr[\mathcal{E}]=\Pr(\{a\in\Omega\mid \mathcal{E}(a)\}) }[/math].

During the lecture, we mostly use the predicate notation instead of subset notation.

Independence

Definition (Independent events)

Two events [math]\displaystyle{ \mathcal{E}_1 }[/math] and [math]\displaystyle{ \mathcal{E}_2 }[/math] are independent if and only if [math]\displaystyle{ \begin{align} \Pr\left[\mathcal{E}_1 \wedge \mathcal{E}_2\right] &= \Pr[\mathcal{E}_1]\cdot\Pr[\mathcal{E}_2]. \end{align} }[/math]

This definition can be generalized to any number of events:

Definition (Independent events)

Events [math]\displaystyle{ \mathcal{E}_1, \mathcal{E}_2, \ldots, \mathcal{E}_n }[/math] are mutually independent if and only if, for any subset [math]\displaystyle{ I\subseteq\{1,2,\ldots,n\} }[/math], [math]\displaystyle{ \begin{align} \Pr\left[\bigwedge_{i\in I}\mathcal{E}_i\right] &= \prod_{i\in I}\Pr[\mathcal{E}_i]. \end{align} }[/math]

Note that in probability theory, the "mutual independence" is not equivalent with "pair-wise independence", which we will learn in the future.

Checking Matrix Multiplication

Let [math]\displaystyle{ \mathbb{F} }[/math] be a feild (you may think of it as the filed [math]\displaystyle{ \mathbb{Q} }[/math] of rational numbers, or the finite field [math]\displaystyle{ \mathbb{Z}_p }[/math] of integers modulo prime [math]\displaystyle{ p }[/math]). We suppose that each field operation (addition, subtraction, multiplication, division) has unit cost. This model is called the unit-cost RAM model, which is an ideal abstraction of computers.

Consider the following problem:

• Input: Three [math]\displaystyle{ n\times n }[/math] matrices [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math] over the field [math]\displaystyle{ \mathbb{F} }[/math].
• Output: "yes" if [math]\displaystyle{ C=AB }[/math] and "no" if otherwise.

A naive method is to multiply [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] and compare the result with [math]\displaystyle{ C }[/math]. The Strassen's algorithm discovered in 1969 now implemented by many numerical libraries runs in time [math]\displaystyle{ O(n^{\log_2 7})\approx O(n^{2.81}) }[/math]. Strassen's algorithm starts the search for fast matrix multiplication algorithms. The Coppersmith–Winograd algorithm discovered in 1987 runs in time [math]\displaystyle{ O(n^{2.376}) }[/math] but is only faster than Strassens' algorithm on extremely large matrices due to the very large constant coefficient. This has been the best known for decades, until recently Stothers got an [math]\displaystyle{ O(n^{2.3737}) }[/math] algorithm in his PhD thesis in 2010, and independently Vassilevska Williams got an [math]\displaystyle{ n^{2.3727} }[/math] algorithm in 2012. Both these improvements are based on generalization of Coppersmith–Winograd algorithm. It is unknown whether the matrix multiplication can be done in time [math]\displaystyle{ O(n^{2+o(1)}) }[/math].

Freivalds Algorithm

The following is a very simple randomized algorithm due to Freivalds, running in [math]\displaystyle{ O(n^2) }[/math] time:

Algorithm (Freivalds, 1979)

pick a vector [math]\displaystyle{ r \in\{0, 1\}^n }[/math] uniformly at random; if [math]\displaystyle{ A(Br) = Cr }[/math] then return "yes" else return "no";

The product [math]\displaystyle{ A(Br) }[/math] is computed by first multiplying [math]\displaystyle{ Br }[/math] and then [math]\displaystyle{ A(Br) }[/math]. The running time of Freivalds algorithm is [math]\displaystyle{ O(n^2) }[/math] because the algorithm computes 3 matrix-vector multiplications.

If [math]\displaystyle{ AB=C }[/math] then [math]\displaystyle{ A(Br) = Cr }[/math] for any [math]\displaystyle{ r \in\{0, 1\}^n }[/math], thus the algorithm will return a "yes" for any positive instance ([math]\displaystyle{ AB=C }[/math]). But if [math]\displaystyle{ AB \neq C }[/math] then the algorithm will make a mistake if it chooses such an [math]\displaystyle{ r }[/math] that [math]\displaystyle{ ABr = Cr }[/math]. However, the following lemma states that the probability of this event is bounded.

Lemma

If [math]\displaystyle{ AB\neq C }[/math] then for a uniformly random [math]\displaystyle{ r \in\{0, 1\}^n }[/math], [math]\displaystyle{ \Pr[ABr = Cr]\le \frac{1}{2} }[/math].

Proof. Let [math]\displaystyle{ D=AB-C }[/math]. The event [math]\displaystyle{ ABr=Cr }[/math] is equivalent to that [math]\displaystyle{ Dr=0 }[/math]. It is then sufficient to show that for a [math]\displaystyle{ D\neq \boldsymbol{0} }[/math], it holds that [math]\displaystyle{ \Pr[Dr = \boldsymbol{0}]\le \frac{1}{2} }[/math]. Since [math]\displaystyle{ D\neq \boldsymbol{0} }[/math], it must have at least one non-zero entry. Suppose that [math]\displaystyle{ D_{ij}\neq 0 }[/math]. We assume the event that [math]\displaystyle{ Dr=\boldsymbol{0} }[/math]. In particular, the [math]\displaystyle{ i }[/math]-th entry of [math]\displaystyle{ Dr }[/math] is [math]\displaystyle{ (Dr)_{i}=\sum_{k=1}^n D_{ik}r_k=0. }[/math] The [math]\displaystyle{ r_j }[/math] can be calculated by [math]\displaystyle{ r_j=-\frac{1}{D_{ij}}\sum_{k\neq j}^n D_{ik}r_k. }[/math] Once all other entries [math]\displaystyle{ r_k }[/math] with [math]\displaystyle{ k\neq j }[/math] are fixed, there is a unique solution of [math]\displaystyle{ r_j }[/math]. Therefore, the number of [math]\displaystyle{ r\in\{0,1\}^n }[/math] satisfying [math]\displaystyle{ Dr=\boldsymbol{0} }[/math] is at most [math]\displaystyle{ 2^{n-1} }[/math]. The probability that [math]\displaystyle{ ABr=Cr }[/math] is bounded as [math]\displaystyle{ \Pr[ABr=Cr]=\Pr[Dr=\boldsymbol{0}]\le\frac{2^{n-1}}{2^n}=\frac{1}{2} }[/math]. [math]\displaystyle{ \square }[/math]

When [math]\displaystyle{ AB=C }[/math], Freivalds algorithm always returns "yes"; and when [math]\displaystyle{ AB\neq C }[/math], Freivalds algorithm returns "no" with probability at least 1/2.

To improve its accuracy, we can run Freivalds algorithm for [math]\displaystyle{ k }[/math] times, each time with an independent [math]\displaystyle{ r\in\{0,1\}^n }[/math], and return "yes" if and only if all running instances returns "yes".

Freivalds' Algorithm (multi-round)

pick [math]\displaystyle{ k }[/math] vectors [math]\displaystyle{ r_1,r_2,\ldots,r_k \in\{0, 1\}^n }[/math] uniformly and independently at random; if [math]\displaystyle{ A(Br_i) = Cr_i }[/math] for all [math]\displaystyle{ i=1,\ldots,k }[/math] then return "yes" else return "no";

If [math]\displaystyle{ AB=C }[/math], then the algorithm returns a "yes" with probability 1. If [math]\displaystyle{ AB\neq C }[/math], then due to the independence, the probability that all [math]\displaystyle{ r_i }[/math] have [math]\displaystyle{ ABr_i=C_i }[/math] is at most [math]\displaystyle{ 2^{-k} }[/math], so the algorithm returns "no" with probability at least [math]\displaystyle{ 1-2^{-k} }[/math]. For any [math]\displaystyle{ 0\lt \epsilon\lt 1 }[/math], choose [math]\displaystyle{ k=\log_2 \frac{1}{\epsilon} }[/math]. The algorithm runs in time [math]\displaystyle{ O(n^2\log_2\frac{1}{\epsilon}) }[/math] and has a one-sided error (false positive) bounded by [math]\displaystyle{ \epsilon }[/math].