The access graph model for paging, defined by (, 1991) and studied in (, 1992) has a number of troubling aspects. The access graph has to be known in advance to the paging algorithm and the memory required to represent the access graph itself may be very large. We present a truly online strongly competitive paging algorithm in the access graph model that does not have any prior information on the access sequence. We give both strongly competitive deterministic and strongly competitive randomized algorithms. Our algorithms need only O(k log n) bits of memory, where k is the number of page slots available and n is the size of the virtual address space, i.e., no more memory than needed to store the virtual translation tables for pages in memory. In fact, we can reduce this to O(k log k) bits using appropriate probabilistic data structures. We also extend the locality of reference concept captured by the access graph model to allow changes in the behavior of the underlying process. We formalize this by introducing the concept of an "extended access graph". We consider a graph parameter spl Delta that captures the degree of change allowed. We study this new model and give algorithms that are strongly competitive for the (unknown) extended access graph. We can do so for almost all values of spl Delta for which it is possible.
Fiat and Rosen @cite_11 present an access graph based heuristic that is truly online and makes use of a (weighted) dynamic access graph. In this sense we emulate their concept. While the Fiat and Rosen algorithm is experimentally interesting in that it seems to beat , it is certainly not strongly competitive, and is known to have a competitive ratio of @math .
@cite_11
2079669094 1964506478 2011730242 2777690836
In a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1 e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1 e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1 e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1 e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of , removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good "strength" to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis. We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job J has processing length pj, weight wj, and is processed at a rate of lij when scheduled on machine i. The online scheduler knows the values of wj and lij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1+e)-speed O(1 2)-competitive for any constant e > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f-1i(P) when consuming power P, where fi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms. We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 - 1 e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1 2, and that the optimal algorithm presented there has a ratio of 1 - 1 e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auction-style process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking alforithm of [KVV90] has a ratio of 1 - 1 e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio [EQUATION] in expectation on d-regular graphs. In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than [EQUATION]. Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one.