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• #### 12 Missing Pieces, Derandomization, and Concluding Remarks 175-178

incorporates randomness into the data processed by an algorithm. Properly speaking, we are considering the pair (algorithm, problem instance) and probabilistically exploring the algorithm behavior over a large variety of problem instances. Typically, the analyst can make statements about the probability of selecting a particular instance, or focus attention on the distribution of suitable variables that describe the problem instance. The task is then to relate the algorithm performance to these variables.

This introductory chapter provides several simple illustrations of the distinction between the design of probabilistic algorithms and the probabilistic analysis of algorithms. An excellent examples-oriented survey of probabilistic algorithms is that of Karp (1991).

### 1.1Probabilistic Algorithms

#### 1.1.1Everyday Examples

Before developing more serious examples that require detailed mathematical description, it seems useful to provide a couple of everyday analogies. Mathematics often speaks best for itself, and one should not read too much into analogies, but because there are important uses of probabilistic algorithms that can be explained without any technical prerequisites, it seems appropriate to look at them early on.

We have all had the experience of walking along a narrow path and encountering someone who is approaching on the same side of the path. As the individuals draw closer, one moves to the other side of the path in order to let the other person pass, but often the other person does the same simultaneously. This little shuffle continues one or two more times before the two people end up on opposite sides of the path and can pass each other.

Although people resolve such difficulties with little more cost than a moment's embarrassment, the analogous situation is more serious when packets of information end up vying for the use of the same place at the same time in a communication channel. When one tries to program machines to avoid such deadlocks, there are decided drawbacks to most deterministic rules. The dogged consistency of machines is such that the shuffling from side to side can go on unabated until an outside action halts the dance. This is an unacceptable state of affairs that one ought to be able to avoid through thoughtful design.

One theme of this report is that a natural course of action when one is faced by the disadvantages of purely deterministic rules is to introduce some

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