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ch13 - Chapter 13 Randomized Algorithms The idea that a...

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Chapter 13 Randomized Algorithms The idea that a process can be “random” is not a modern one; we can trace the notion far back into the history of human thought and certainly see its reflections in gambling and the insurance business, each of which reach into ancient times. Yet, while similarly intuitive subjects like geometry and logic have been treated mathematically for several thousand years, the mathematical study of probability is surprisingly young; the first known attempts to seriously formalize it came about in the 1600s. Of course, the history of computer science plays out on a much shorter time scale, and the idea of randomization has been with it since its early days. Randomization and probabilistic analysis are themes that cut across many areas of computer science, including algorithm design, and when one thinks about random processes in the context of computation, it is usually in one of two distinct ways. One view is to consider the world as behaving randomly: One can consider traditional algorithms that confront randomly generated input. This approach is often termed average-case analysis , since we are studying the behavior of an algorithm on an “average” input (subject to some underlying random process), rather than a worst-case input. A second view is to consider algorithms that behave randomly: The world provides the same worst-case input as always, but we allow our algorithm to make random decisions as it processes the input. Thus the role of randomiza- tion in this approach is purely internal to the algorithm and does not require new assumptions about the nature of the input. It is this notion of a randomized algorithm that we will be considering in this chapter.
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708 Chapter 13 Randomized Algorithms Why might it be useful to design an algorithm that is allowed to make random decisions? A first answer would be to observe that by allowing ran- domization, we’ve made our underlying model more powerful. Efficient de- terministic algorithms that always yield the correct answer are a special case of efficient randomized algorithms that only need to yield the correct answer with high probability; they are also a special case of randomized algorithms that are always correct, and run efficiently in expectation . Even in a worst- case world, an algorithm that does its own “internal” randomization may be able to offset certain worst-case phenomena. So problems that may not have been solvable by efficient deterministic algorithms may still be amenable to randomized algorithms. But this is not the whole story, and in fact we’ll be looking at randomized algorithms for a number of problems where there exist comparably efficient de- terministic algorithms. Even in such situations, a randomized approach often exhibits considerable power for further reasons: It may be conceptually much simpler; or it may allow the algorithm to function while maintaining very little internal state or memory of the past. The advantages of randomization seem
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