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 equivalent to the paging challenge, the k-server challenge, task store scheduling, the knapsack challenge, the bit guessing challenge, and difficulties on graphs.
This e-book is suitable for undergraduate and graduate scholars of machine technological know-how, assuming a uncomplicated wisdom in algorithmics and discrete arithmetic. additionally researchers will locate this a useful reference for the new box of recommendation complexity.
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Extra resources for An Introduction to Online Computation: Determinism, Randomization, Advice
Every marking algorithm is ????/(???? − ℎ + 1)-competitive for (ℎ, ????)paging. Proof. 4. Let Mark be any marking algorithm. Once more, consider the ????-phase partition of a given input ????. 8. To bound the number of page faults that Opt(????) causes, let us again shift the phases by one to obtain a new partition ????1′ , ????2′ , . . , ????????′ ; again, ????????′ may be empty. Let ???? be the first request during the phase ???????? . Then, Opt’s cache contains ℎ − 1 pages at the beginning of ????????′ that are different from ????, and since, for any ???? with 1 ≤ ???? ≤ ???? − 1, ???? distinct pages (that are all different from ????) are requested within ????????′ , Opt(????) has to make ???? − (ℎ − 1) page faults.
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.
Except possibly the last one, a phase of a marking algorithm consists of a maximum-length sequence of requests for ???? different pages. This makes it very easy for us to argue why such an algorithm makes at most ???? page faults in one phase. 25 Chapter 1. 8. Every marking algorithm is strictly ????-competitive for paging. Proof. Let Mark be a fixed marking algorithm; let ???? denote the given input and consider its ????-phase partition into ???? phases ????1 , ????2 , . . 8. 4, we conclude that any optimal algorithm Opt makes at least ???? page faults in total on ????.
An Introduction to Online Computation: Determinism, Randomization, Advice by Dennis Komm