IN this chapter we study several ideas that are basic to the design and analysis of randomized algorithms. All the topics in this chapter share a game-theoretic viewpoint, which enables us to think of a randomized algorithm as a probability distribution on deterministic algorithms. This leads to the Yao ‘s Minimax Principle, which can be used to establish a lower bound on the performance of a randomized algorithm.

Game Tree Evaluation

We begin with another simple illustration of linearity of expectation, in the setting of game tree evaluation. This example will demonstrate a randomized algorithm whose expected running time is smaller than that of any deterministic algorithm. It will also serve as a vehicle for demonstrating a standard technique for deriving a lower bound on the running time of any randomized algorithm for a problem.

A game tree is a rooted tree in which internal nodes at even distance from the root are labeled MIN and internal nodes at odd distance are labeled MAX. Associated with each leaf is a real number, which we call its value. The evaluation of the game tree is the following process. Each leaf returns the value associated with it. Each MAX node returns the largest value returned by its children, and each MIN node returns the smallest value returned by its children. Given a tree with values at the leaves, the evaluation problem is to determine the value returned by the root.

IN Chapters 1 and 2, we bounded the expected running times of several randomized algorithms. While the expectation of a random variable (such as a running time) may be small, it may frequently assume values that are far higher. In analyzing the performance of a randomized algorithm, we often like to show that the behavior of the algorithm is good almost all the time. For example, it is more desirable to show that the running time is small with high probability, not just that it has a small expectation. In this chapter we will begin the study of general methods for proving statements of this type. We will begin by examining a family of stochastic processes that is fundamental to the analysis of many randomized algorithms: these are called occupancy problems. This motivates the study (in this chapter and the next) of general bounds on the probability that a random variable deviates far from its expectation, enabling us to avoid such custom-made analyses. The probability that a random variable deviates by a given amount from its expectation is referred to as a tail probability for that deviation. Readers wishing to review basic material on probability and distributions may consult Appendix C.

Occupancy Problems

We begin with an example of an occupancy problem. In such problems we envision each of m indistinguishable objects ("balls") being randomly assigned to one of n distinct classes ("bins"). In other words, each ball is placed in a bin chosen independently and uniformly at random.

IN this chapter we will study some basic principles of the probabilistic method, a combinatorial tool with many applications in computer science. This method is a powerful tool for demonstrating the existence of combinatorial objects. We introduce the basic idea through several examples drawn from earlier chapters, and follow that by a detailed study of the maximum satisfiability (MAX-SAT) problem. We then introduce the notion of expanding graphs and apply the probabilistic method to demonstrate their existence. These graphs have powerful properties that prove useful in later chapters, and we illustrate these properties via an application to probability amplification.

In certain cases, the probabilistic method can actually be used to demonstrate the existence of algorithms, rather than merely combinatorial objects. We illustrate this by showing the existence of efficient non-uniform algorithms for the problem of oblivious routing. We then present a particular result, the Lovász Local Lemma, which underlies the successful application of the probabilistic method in a number of settings. We apply this lemma to the problem of finding a satisfying truth assignment in an instance of the SAT problem where each variable occurs in a bounded number of clauses. While the probabilistic method usually yields only randomized or non-uniform deterministic algorithms, there are cases where a technique called the method of conditional probabilities can be used to devise a uniform, deterministic algorithm; we conclude the chapter with an exposition of this method for derandomization.

IN this chapter we present some general bounds on the tail of the distribution of the sum of independent random variables, with some extensions to the case of dependent or correlated random variables. These bounds are derived via the use of moment generating functions and result in “Chernoff-type” or “exponential" tail bounds. These Chernoff bounds are applied to the analysis of algorithms for global wiring in chips and routing in parallel communications networks. For applications in which the random variables of interest cannot be modeled as sums of independent random variables, martingales are a powerful probabilistic tool for bounding the divergence of a random variable from its expected value. We introduce the concept of conditional expectation as a random variable, and use this to develop a simplified definition of martingales. Using measure theoretic ideas, we provide a more general description of martingales. Finally, we present an exponential tail bound for martingales and apply it to the analysis of an occupancy problem.

The Chernoff Bound

In Chapter 3 we initiated the study of techniques for bounding the probability that a random variable deviates far from its expectation. In this chapter we focus on techniques for obtaining considerably sharper bounds on such tail probabilities.

The random variables we will be most concerned with are sums of independent Bernoulli trials; for example, the outcomes of tosses of a coin. In designing and analyzing randomized algorithms in various settings, it is extremely useful to have an understanding of the behavior of this sum.

In the previous chapter we introduced a selection of the more popular inference processes which have been proposed. This raises the question of why to prefer one such process over any other. In this chapter we shall consider this question by presenting a number of properties, or as we shall call them, principles, which it might be deemed desirable that an inference process, N, should satisfy.

For the most part these principles could be said to be based on common sense or rationality or ‘consistency’ in the natural language sense of the word. A justification for assuming that adherence to common sense is a desirable property of an inference process comes from the Watts Assumption given in Chapter 5. For if K genuinely does represent all the expert's knowledge then any conclusion the expert draws from K should be the result of applying what, by consensus, we consider correct reasoning, i.e. of common sense.

So our plan now is to present a list of such principles. We shall limit ourselves to the case where Bel is a probability function, although the same criteria could be applied to inference processes for DS-belief functions, possibility functions etc. In what follows N stands for an inference process for L. Here L is to be thought of as variable. If we wish to consider a principle for a particular language L we shall insert ‘for L’.

Equivalence Principle

If K1, K2 ∈ CL are equivalent in the sense that VL(K1) = VL(K2) then N(K1) = N(K2).

In this chapter, we provide a result which characterizes well-formedness of free-choice nets in a very suitable way for verification purposes. All the conditions of the characterization are decidable in polynomial time in the size of the net. The most interesting feature of the result is that it exhibits a tight relation between the well-formedness of a free-choice net and the rank of its incidence matrix. Accordingly, it is known as the Rank Theorem. It will be an extremely useful lemma in the proof of many results of this chapter and of the next ones.

We also provide a characterization of the live and bounded markings of a well-formed free-choice net. Again, the conditions of the characterization can be checked in polynomial time. Together with the Rank Theorem, this result yields a polynomial time algorithm to decide if a given free-choice system is live and bounded.

In the last section of the chapter we use the Rank Theorem to prove the Duality Theorem. This result states that the class of well-formed free-choice nets is invariant under the transformation that interchanges places and transitions and reverses the arcs of the net.

Characterizations of well-formedness

Using the results of Chapter 4 and Chapter 5, it is easy to obtain the following characterization of well-formed free-choice nets.

Proposition 6.1A first characterization of well-formedness

Let N be a connected free-choice net with at least one place and at least one transition.

1. N is structurally live iff every proper siphon contains a proper trap.

2. […]