On-Line Paging against Adversarially Biased Random Inputs

In evaluating an algorithm, worst-case analysis can be overly pessimistic. Average-case analysis can be overly optimistic. An intermediate approach is to show that an algorithm does well on a broad class of input distributions. Koutsoupias and Papadimitriou recently analyzed the least-recently-used (LRU) paging strategy in this manner, analyzing its performance on an input sequence generated by a so-called diffuse adversary -- one that must choose each request probabilitistically so that no page is chosen with probability more than some fixed epsilon>0. They showed that LRU achieves the optimal competitive ratio (for deterministic on-line algorithms), but they didn't determine the actual ratio. In this paper we estimate the optimal ratios within roughly a factor of two for both deterministic strategies (e.g. least-recently-used and first-in-first-out) and randomized strategies. Around the threshold epsilon ~ 1/k (where k is the cache size), the optimal ratios are both Theta(ln k). Below the threshold the ratios tend rapidly to O(1). Above the threshold the ratio is unchanged for randomized strategies but tends rapidly to Theta(k) for deterministic ones. We also give an alternate proof of the optimality of LRU.Comment: Conference version appeared in SODA '98 as "Bounding the Diffuse Adversary

Cross-input Amortization Captures the Diffuse Adversary

Koutsoupias and Papadimitriou recently raised the question of how well deterministic on-line paging algorithms can do against a certain class of adversarially biased random inputs. Such an input is given in an on-line fashion; the adversary determines the next request probabilistically, subject to the constraint that no page may be requested with probability more than a fixed $\epsilon\u3e0$. In this paper, we answer their question by estimating, within a factor of two, the optimal competitive ratio of any deterministic on-line strategy against this adversary. We further analyze randomized on-line strategies, obtaining upper and lower bounds within a factor of two. These estimates reveal the qualitative changes as $\epsilon$ ranges continuously from 1 (the standard model) towards 0 (a severely handicapped adversary). The key to our upper bounds is a novel charging scheme that is appropriate for adversarially biased random inputs. The scheme adjusts the costs of each input so that the expected cost of a random input is unchanged, but working with adjusted costs, we can obtain worst-case bounds on a per-input basis. This lets us use worst-case analysis techniques while still thinking of some of the costs as expected costs

First-Come-First-Served for Online Slot Allocation and Huffman Coding

Can one choose a good Huffman code on the fly, without knowing the underlying distribution? Online Slot Allocation (OSA) models this and similar problems: There are n slots, each with a known cost. There are n items. Requests for items are drawn i.i.d. from a fixed but hidden probability distribution p. After each request, if the item, i, was not previously requested, then the algorithm (knowing the slot costs and the requests so far, but not p) must place the item in some vacant slot j(i). The goal is to minimize the sum, over the items, of the probability of the item times the cost of its assigned slot. The optimal offline algorithm is trivial: put the most probable item in the cheapest slot, the second most probable item in the second cheapest slot, etc. The optimal online algorithm is First Come First Served (FCFS): put the first requested item in the cheapest slot, the second (distinct) requested item in the second cheapest slot, etc. The optimal competitive ratios for any online algorithm are 1+H(n-1) ~ ln n for general costs and 2 for concave costs. For logarithmic costs, the ratio is, asymptotically, 1: FCFS gives cost opt + O(log opt). For Huffman coding, FCFS yields an online algorithm (one that allocates codewords on demand, without knowing the underlying probability distribution) that guarantees asymptotically optimal cost: at most opt + 2 log(1+opt) + 2.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 201

Probabilistic alternatives for competitive analysis

In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: it sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis.operations research and management science;

Simple optimality proofs for Least Recently Used in the presence of locality of reference

It is well known that competitive analysis yields results that do not reflect the observed performance of online paging algorithms. Many deterministic paging algorithms achieve the same competitive ratio, ranging from inefficient strategies as flush-when-full to the well-performing least-recently-used (LRU). In this paper, we study this fundamental online problem from the viewpoint of stochastic dominance. We give simple proofs that whensequences are drawn from distributions modelling locality of reference, LRU stochastically dominates any other online paging algorithm. As a byproduct, we obtain simple proofs of some earlier results.operations research and management science;

Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

In this paper, we study the $k$-forest problem in the model of resource augmentation. In the $k$-forest problem, given an edge-weighted graph $G(V,E)$, a parameter $k$, and a set of $m$ demand pairs $\subseteq V \times V$, the objective is to construct a minimum-cost subgraph that connects at least $k$ demands. The problem is hard to approximate---the best-known approximation ratio is $O(\min\{\sqrt{n}, \sqrt{k}\})$. Furthermore, $k$-forest is as hard to approximate as the notoriously-hard densest $k$-subgraph problem. While the $k$-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the $k$-forest problem can be viewed as to remove at most $m-k$ demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the $k$-forest problem that, for every $\epsilon>0$, removes at most $m-k$ demands and has cost no more than $O(1/\epsilon^{2})$ times the cost of an optimal algorithm that removes at most $(1-\epsilon)(m-k)$ demands

Alternative Measures for the Analysis of Online Algorithms

In this thesis we introduce and evaluate several new models for the analysis of online algorithms. In an online problem, the algorithm does not know the entire input from the beginning; the input is revealed in a sequence of steps. At each step the algorithm should make its decisions based on the past and without any knowledge about the future. Many important real-life problems such as paging and routing are intrinsically online and thus the design and analysis of online algorithms is one of the main research areas in theoretical computer science. Competitive analysis is the standard measure for analysis of online algorithms. It has been applied to many online problems in diverse areas ranging from robot navigation, to network routing, to scheduling, to online graph coloring. While in several instances competitive analysis gives satisfactory results, for certain problems it results in unrealistically pessimistic ratios and/or fails to distinguish between algorithms that have vastly differing performance under any practical characterization. Addressing these shortcomings has been the subject of intense research by many of the best minds in the field. In this thesis, building upon recent advances of others we introduce some new models for analysis of online algorithms, namely Bijective Analysis, Average Analysis, Parameterized Analysis, and Relative Interval Analysis. We show that they lead to good results when applied to paging and list update algorithms. Paging and list update are two well known online problems. Paging is one of the main examples of poor behavior of competitive analysis. We show that LRU is the unique optimal online paging algorithm according to Average Analysis on sequences with locality of reference. Recall that in practice input sequences for paging have high locality of reference. It has been empirically long established that LRU is the best paging algorithm. Yet, Average Analysis is the first model that gives strict separation of LRU from all other online paging algorithms, thus solving a long standing open problem. We prove a similar result for the optimality of MTF for list update on sequences with locality of reference. A technique for the analysis of online algorithms has to be effective to be useful in day-to-day analysis of algorithms. While Bijective and Average Analysis succeed at providing fine separation, their application can be, at times, cumbersome. Thus we apply a parameterized or adaptive analysis framework to online algorithms. We show that this framework is effective, can be applied more easily to a larger family of problems and leads to finer analysis than the competitive ratio. The conceptual innovation of parameterizing the performance of an algorithm by something other than the input size was first introduced over three decades ago [124, 125]. By now it has been extensively studied and understood in the context of adaptive analysis (for problems in P) and parameterized algorithms (for NP-hard problems), yet to our knowledge this thesis is the first systematic application of this technique to the study of online algorithms. Interestingly, competitive analysis can be recast as a particular form of parameterized analysis in which the performance of opt is the parameter. In general, for each problem we can choose the parameter/measure that best reflects the difficulty of the input. We show that in many instances the performance of opt on a sequence is a coarse approximation of the difficulty or complexity of a given input sequence. Using a finer, more natural measure we can separate paging and list update algorithms which were otherwise indistinguishable under the classical model. This creates a performance hierarchy of algorithms which better reflects the intuitive relative strengths between them. Lastly, we show that, surprisingly, certain randomized algorithms which are superior to MTF in the classical model are not so in the parameterized case, which matches experimental results. We test list update algorithms in the context of a data compression problem known to have locality of reference. Our experiments show MTF outperforms other list update algorithms in practice after BWT. This is consistent with the intuition that BWT increases locality of reference