Open Access

Gabriel Moruz, Andrei Laurian Negoescu

• Paging is one of the prominent problems in the field of on-line algorithms. While in the deterministic setting there exist simple and efficient strongly competitive algorithms, in the randomized setting a tradeoff between competitiveness and memory is still not settled. Bein et al. [4] conjectured that there exist strongly competitive randomized paging algorithms, using o(k) bookmarks, i.e. pages not in cache that the algorithm keeps track of. Also in [4] the first algorithm using O(k) bookmarks (2k more precisely), Equitable2, was introduced, proving in the affirmative a conjecture in [7]. We prove tighter bounds for Equitable2, showing that it requires less than k bookmarks, more precisely ≈ 0.62k. We then give a lower bound for Equitable2 showing that it cannot both be strongly competitive and use o(k) bookmarks. Nonetheless, we show that it can trade competitiveness for space. More precisely, if its competitive ratio is allowed to be (Hk + t), then it requires k/(1 + t) bookmarks. Our main result proves the conjecture that there exist strongly competitive paging algorithms using o(k) bookmarks. We propose an algorithm, denoted Partition2, which is a variant of the Partition algorithm byMcGeoch and Sleator [13]. While classical Partition is unbounded in its space requirements, Partition2 uses θ(k/ log k) bookmarks. Furthermore, we show that this result is asymptotically tight when the forgiveness steps are deterministic.