In the weighted paging problem there is a weight (cost) for fetching each page into the cache. We design a randomized O(log k)-competitive online algorithm for the weighted paging problem, where k is the cache size. This is the first randomized o(k)-competitive algorithm and its competitiveness matches the known lower bound on the problem. More generally, we design an O(log(k/(k − h + 1)))-competitive online algorithm for the version of the problem where the online algorithm has cache size k and the offline algorithm has cache size h ≤ k. Weighted paging is a special case (weighted star metric) of the well known k-server problem for which it is a major open question whether randomization can be useful in obtaining sublinear competitive algorithms. Therefore, abstracting and extending the insights from paging is a key step in the resolution of the k-server problem. Our solution for the weighted paging problem is based on a two-step approach. In the first step we obtain an O(log k)-competitive fractional algorithm which is based on a novel online primal-dual approach. In the second step we obtain a randomized algorithm by rounding online the fractional solution to an actual distribution on integral cache solutions. We conclude with a randomized O(log N)competitive algorithm for the well studied Metrical Task System problem (MTS) on a metric defined by a weighted star on N leaves, improving upon a previous O(log 2 N)competitive algorithm of Blum et al. [9].

Recently, Coté et al. [10] proposed an approach for solving the k-server problem on Hierchically Separated Trees (HSTs). In particular, they define a problem on a uniform metric, and show that if an algorithm with a certain refined guarantee exists for it, then one can obtain polylogarithmic (in diameter) competitive factors for the k-server problem on HSTs by solving this problem recursively. By designing such an algorithm for a two point metric, they obtained a logarithmic competitive algorithm for well-separated binary HSTs. Extending their result to uniform metrics on arbitrarily many points would imply a poly-logarithmic competitive algorithm for k-server on general HSTs (and hence general metrics) and is thus of major interest. Here, we design such an algorithm for any uniform metric, provided the instance satisfies a certain “convexity” property. Even though this does not give a result for k-server, convexity seems to be a very natural property, and we give evidence that instances arising in the Coté et al. [10] reduction from k-server essentially possess this property, suggesting that this might be a promising approach. Already, our setting is general enough to model the finely competitive paging problem proposed by Blum et al. [4], who motivated it as a first step towards achieving a polylog(k) competitive algorithm for k-server. Our result implies an r + O(log k)competitive algorithm for finely competitive paging, resolving the main open problem of [4]. Our results are based on an extension of the primaldual framework for online algorithms developed by Buchbinder and Naor [7]. The original approach works for problems whose offline version can be expressed as

We design a randomized online algorithm for k-server on binary trees with hierarchical edge lengths, with expected competitive ratio O(log ∆), where ∆ is the diameter of the metric. This is one of the first k-server algorithms with competitive ratio poly-logarithmic in the natural problem parameters, and represents substantial progress on the randomized k-server conjecture. Extending the algorithm to trees of higher degree would give a competitive ratio of O(log 2 ∆ log n) for the k-server problem on general metrics with n points and diameter ∆.

OF THE DISSERTATION On-line Algorithms and Fast Digital Identity Revocation by Sachin P Lodha Dissertation Director: Endre Szemeredi The k--server problem is a generalized model of certain scheduling problems such as, for instance, multi--level memory paging, disk caching and head motion planning of multi--headed disks. The problem can be stated as follows: We are given k mobile servers that occupy k points of a metric space M . We assume that M is finite, |M | = n > k, and k > 1. At each time step a request, a point of M , appears. An algorithm must serve this request by moving one of its servers to the requested point (if that is vacant). The algorithm is charged a cost which is equal to the distance moved by the server. The algorithm A is on--line if it serves the request without knowing what the future requests will be. We present a O(n 2 3 ln n)--competitive randomized on--line algorithm when the underlying metric space is given by n equally spaced points on a line. This algorithm is o(k)-competitive for n = k + o(k 3/2 / ln k). Let A be an n-uniform hypergraph. Assume that the maximum degree of A is D = D(A) (local condition), and |A| = N = N (A) (global condition). By the Lovasz ii Local Lemma (L.L.L.), if D < 2 n 8n , then A has a proper 2-coloring (i.e. there is no monochromatic edge) independently of the value of N (N can be infinite). Unfortunately L.L.L. is a pure existence argument which does not give any clue of how to find a proper 2-coloring. A hypergraph with the property that any two edges have at most one point in common is called almost disjoint (e.g. a family of lines). Assume that A is an n-uniform almost disjoint hypergraph. In this special case, we provide a polynomial time (in terms of N) algorithm to find a prope...