By Dennis Komm
This textbook explains on-line computation in numerous settings, with specific emphasis on randomization and suggestion complexity. those settings are analyzed for numerous 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 e-book is suitable for undergraduate and graduate scholars of machine technology, assuming a simple wisdom in algorithmics and discrete arithmetic. additionally researchers will locate this a invaluable reference for the new box of recommendation complexity.
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Extra resources for An Introduction to Online Computation: Determinism, Randomization, Advice
8 (????-Phase Partition). Let ???? = (????1 , ????2 , . . , ???????? ) be an arbitrary instance of paging. A ????-phase partition of ???? assigns the requests from ???? to consecutive disjoint phases ????1 , ????2 , . . , ???????? such that • phase ????1 starts with the first request for a page that is not initially in the cache. Then, ????1 contains a maximum-length subsequence of ???? that contains at most ???? distinct pages; • for any ???? with 2 ≤ ???? ≤ ???? , phase ???????? is a maximum-length subsequence of ???? that starts right after ????????−1 and again contains at most ???? distinct pages.
But this means that the randomized online algorithm Rand may also use fewer bits. Don’t we have to incorporate this fact? If so, the behavior of Rand can be different for all binary strings up to a length of ????(????). As a consequence, there would be ????(????) ∑︁ 2???? = 2????(????)+1 − 1 ????=0 possibilities and not just 2????(????) . Argue why our reasoning is nevertheless correct, and why it is wrong to treat Rand as a set of 2????(????)+1 − 1 deterministic algorithms. 2 is somewhat easier to see when speaking about offline algorithms instead of online algorithms.
1 , ???????? , ???????? , . . , ???????? , . ) ⏟ ⏞ ⏟ ⏞ ℓ requests ℓ requests of the input ????, which again does not help. Continuing in this fashion, the adversary can ensure that Algℓ causes a page fault every ℓ + 1 time steps. 5, Opt(????) causes at most one page fault every ????(ℓ + 1) time steps. For such inputs of length ????, Algℓ causes ????/(ℓ+1) page faults, while Opt(????) causes at most ????/(????(ℓ+1)). As a result, the competitive ratio of Algℓ has a lower bound of ????. 10. No online algorithm with lookahead ℓ for paging is better than ????-competitive.
An Introduction to Online Computation: Determinism, Randomization, Advice by Dennis Komm