Verified Analysis of List Update Algorithms

This paper presents a machine-verified analysis of a number of classical algorithms for the list update problem: 2-competitiveness of move-to-front, the lower bound of 2 for the competitiveness of deterministic list update algorithms and 1.6-competitiveness of the randomized COMB algorithm, the best randomized list update algorithm known to date. The analysis is verified with help of the theorem prover Isabelle; some low-level proofs could be automated

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

Alternative Approaches for Analysis of Bin Packing and List Update Problems

In this thesis we introduce and evaluate new algorithms and models for the analysis of online bin packing and list update problems. These are two classic online problems which are extensively studied in the literature and have many applications in the real world. Similar to other online problems, the framework of competitive analysis is often used to study the list update and bin packing algorithms. Under this framework, the behavior of online algorithms is compared to an optimal offline algorithm on the worst possible input. This is aligned with the traditional algorithm theory built around the concept of worst-case analysis. However, the pessimistic nature of the competitive analysis along with unrealistic assumptions behind the proposed models for the problems often result in situations where the existing theory is not quite useful in practice. The main goal of this thesis is to develop new approaches for studying online problems, and in particular bin packing and list update, to guide development of practical algorithms performing quite well on real-world inputs. In doing so, we introduce new algorithms with good performance (not only under the competitive analysis) as well as new models which are more realistic for certain applications of the studied problems. For many online problems, competitive analysis fails to provide a theoretical justification for observations made in practice. This is partially because, as a worst-case analysis method, competitive analysis does not necessarily reflect the typical behavior of algorithms. In the case of bin packing problem, the Best Fit and First Fit algorithms are widely used in practice. There are, however, other algorithms with better competitive ratios which are rarely used in practice since they perform poorly on average. We show that it is possible to optimize for both cases. In doing so, we introduce online bin packing algorithms which outperform Best Fit and First Fit in terms of competitive ratio while maintaining their good average-case performance. An alternative for analysis of online problems is the advice model which has received significant attention in the past few years. Under the advice model, an online algorithm receives a number of bits of advice about the unrevealed parts of the sequence. Generally, there is a trade-off between the size of the advice and the performance of online algorithms. The advice model generalizes the existing frameworks in which an online algorithm has partial knowledge about the input sequence, e.g., the access graph model for the paging problem. We study list update and bin packing problems under the advice model and answer several relevant questions about the advice complexity of these problems. Online problems are usually studied under specific settings which are not necessarily valid for all applications of the problem. As an example, online bin packing algorithms are widely used for server consolidation to minimize the number of active servers in a data center. In some applications, e.g., tenant placement in the Cloud, often a `fault-tolerant' solution for server consolidation is required. In this setting, the problem becomes different and the classic algorithms can no longer be used. We study a fault-tolerant model for the bin packing problem and analyze algorithms which fit this particular application of the problem. Similarly, the list update problem was initially proposed for maintaining self-adjusting linked lists. However, presently, the main application of the problem is in the data compression realm. We show that the standard cost model is not suitable for compression purposes and study a compression cost model for the list update problem. Our analysis justifies the advantage of the compression schemes which are based on Move-To-Front algorithm and might lead to improved compression algorithms