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.