By Dennis Komm
This textbook explains on-line computation in numerous settings, with specific emphasis on randomization and recommendation complexity. those settings are analyzed for varied on-line difficulties akin to the paging challenge, the k-server challenge, activity store scheduling, the knapsack challenge, the bit guessing challenge, and difficulties on graphs.
This booklet is acceptable for undergraduate and graduate scholars of machine technology, assuming a easy wisdom in algorithmics and discrete arithmetic. additionally researchers will locate this a beneficial reference for the new box of recommendation complexity.
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Additional resources for An Introduction to Online Computation: Determinism, Randomization, Advice
However, we want guarantees in the following sense. Our worst-case instances may seem artificial from a practical point of view, but maybe they are actually very natural for certain environments. In such a situation, there may exist a few hard inputs that always cause a given online algorithm to fail, although it performs a lot better on all other instances. The way we measure the solution quality of algorithms, such an algorithm is considered bad. In other words, we do not want that there are some instances of the given problem that always cause an online algorithm to perform poorly; even if our feeling is that these inputs do not occur very often.
Let ???? : N → N be a function that gives the maximum number of random binary decisions (figuratively speaking, the “coin tosses”) of some given randomized online algorithm Rand for a given input length. As we have just discussed, for any natural number ????, ????(????) is well defined, that is, ????(????) is some natural number as well. Therefore, we may say that Rand behaves in at most 2????(????) different ways when reading some fixed input of length ????. If we know Rand, we can compute ????(????) for every ????. Furthermore, for every given instance of length ????, we can simply simulate Rand for every possible random string of length ????(????), and thus study the behavior of the deterministic online algorithms we get as a consequence.
An immediate problem is that we cannot always give an upper bound on the number of random bits used with absolute certainty. As an example, consider the simple randomized online algorithm Three that does not get any input and only picks a number 1, 2, or 3 uniformly at random, that is, each with probability 1/3. How do we achieve this? ” Thus, as a straightforward implementation, Three outputs 0 0 . . → “1,” 0 1 . . → “2,” and 1 0 . . → “3” and all these decisions are made with probability 1/4 as the bits on Three’s random tape are 0 or 1 with probability 1/2 each.
An Introduction to Online Computation: Determinism, Randomization, Advice by Dennis Komm