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An $O(\log k)$-competitive algorithm for generalized caching

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Jenny Lam

In the generalized caching problem, we have a set of pages and a cache
of size $k$. Each page $p$ has a size $w_p\ge 1$ and fetching cost $c_p$
for loading the page into the cache. At any point in time, the sum of
the sizes of the pages stored in the cache cannot exceed $k$. The input
consists of a sequence of page requests. If a page is not present in the
cache at the time it is requested, it has to be loaded into the cache
incurring a cost of $c_p$. We give a randomized $O(\log k)$-competitive
online algorithm for the generalized caching problem, improving the
previous bound of $O(\log^2 k)$ by Bansal, Buchbinder, and Naor
(STOC'08). This improved bound is asymptotically tight and of the same
order as the known bounds for the classic problem with uniform weights
and sizes. We follow the LP based techniques proposed by Bansal et
al. and our main contribution are improved and slightly simplified
methods for rounding fractional solutions online.

(From a
paper at SODA 2012
by Anna Adamaszek, Artur Czumaj, Matthias Englert, and Harald Racke.)