By Dennis Komm
This textbook explains on-line computation in several settings, with specific emphasis on randomization and suggestion 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 acceptable for undergraduate and graduate scholars of laptop technological know-how, assuming a easy wisdom in algorithmics and discrete arithmetic. additionally researchers will locate this a invaluable reference for the hot box of recommendation complexity.
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Extra resources for An Introduction to Online Computation: Determinism, Randomization, Advice
To prove the claim, we show that, for every ????, there is an instance ???? of paging of length ???? such that cost(Lifo(????))/cost(Opt(????)) grows proportionally with ????. To this end, we give an instance of length ???? that always requests the same two pages; again, it suffices to choose ???? = ???? + 1. The adversary again first requests ????????+1 , and since all pages ????1 , ????2 , . . , ???????? are in the cache at the beginning, Lifo removes some fixed page from the cache, say ???????? . Since the adversary knows that Lifo chooses ???????? , it requests it in time step ????2 and Lifo removes ????????+1 , which is now the page that was last loaded into the cache.
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.
7 the cache is initialized as ????1 ???? 2 ????3 ????4 ???? 5 ????6 . Now suppose that we are given an instance ???? = (????4 , ????7 , ????5 , ????1 , . ). In time step ????1 , page ????4 is requested, which is already in the cache; thus, any online algorithm outputs “0” and there is no cost caused in this time step. The next request is page ????7 , and therefore some page needs to be removed from the cache to make room. Assume page ????1 gets chosen to be replaced by ????7 , which leads to the situation ????7 ???? 2 ????3 ????4 ???? 5 ????6 .