We study scheduling problems in battery-operated computing devices, aiming at schedules with low total energy consumption. While most of the previous work has focused on finding feasible schedules in deadline-based settings, in this article we are interested in schedules that guarantee good response times. More specifically, our goal is to schedule a sequence of jobs on a variable-speed processor so as to minimize the total cost consisting of the energy consumption and the total flow time of all jobs. We first show that when the amount of work, for any job, may take an arbitrary value, then no online algorithm can achieve a constant competitive ratio. Therefore, most of the article is concerned with unit-size jobs. We devise a deterministic constant competitive online algorithm and show that

The following result is shown: On an n-node splay tree, the amortized cost of an access at distance d from the preceding access is O(log(d + 1)). In addition, there is an O(n) initialization cost. The accesses include searches, insertions and deletions. 1 Introduction The reader is advised that this paper quotes results from the companion Part I paper [CMSS93]; in addition, the Part I paper introduces a number of the techniques used here, but in a somewhat less involved way. The splay tree is a self-adjusting binary search tree devised by Sleator and Tarjan [ST85]. They showed that it is competitive with many of the balanced search tree schemes for maintaining a dictionary. Specifically, Sleator and Tarjan showed that a sequence of m accesses performed on a splay tree takes time O(m log n), where n is the maximum size attained by the tree (n m). They also showed that in an amortized sense, up to a constant factor, on sufficiently long sequences of searches, the splay tree has as ...

. We survey results on self-organizing data structures for the search problem and concentrate on two very popular structures: the unsorted linear list, and the binary search tree. For the problem of maintaining unsorted lists, also known as the list update problem, we present results on the competitiveness achieved by deterministic and randomized on-line algorithms. For binary search trees, we present results for both on-line and off-line algorithms. Self-organizing data structures can be used to build very effective data compression schemes. We summarize theoretical and experimental results. 1 Introduction This paper surveys results in the design and analysis of self-organizing data structures for the search problem. The general search problem in pointer data structures can be phrased as follows. The elements of a set are stored in a collection of nodes. Each node also contains O(1) pointers to other nodes and additional state data which can be used for navigation and self-organizati...

We present an O(lg lg n)-competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that O(1)-competitive binary search trees exist. 1.

We present a dynamic comparison-based search structure that supports insertions, deletions, and searches within the unified bound. The unified bound specifies that it is quick to access an element that is near a recently accessed element. More precisely, if w(y) distinct elements have been accessed since the last access to element y, and d(x, y) denotes the rank distance between x and y among the current set of elements, then the amortized cost to access element x is O(miny log[w(y) + d(x, y) + 2]). This property generalizes the working-set and dynamic-finger properties of splay trees. Preprint submitted to Elsevier Science 31 January 2007 1

Abstract Splay trees are self-organizing binary search trees that were introduced by Sleator andTarjan [12]. In this paper we present a randomized variant of these trees. The new algorithm for reorganizing the tree is both simple and easy to implement. We prove that our randomizedsplaying scheme has the same asymptotic performance as the original deterministic scheme but improves constants in the expected running time. This is interesting in practice becausethe search time in splay trees is typically higher than the search time in skip lists and AVLtrees. We present a detailed experimental study of our algorithm. On request sequencesgenerated by fixed probability distributions, we can achieve improvements of up to 25 % over deterministic splaying. On request sequences that exhibit high locality of reference, theimprovements are minor.

On the surface, the three on-line machine learning problems analyzed in this thesis may seem unrelated. The first is an on-line investment strategy introduced by Tom Cover. We begin with a simple analysis that extends to the case of fixed-percentage transaction costs. We then describe an efficient implementation that runs in time polynomial in the number of stocks. The second problem is k-fold cross validation, a popular technique in machine learning for estimating the error of a learned hypothesis. We show that this is a valid technique by comparing it to the hold-out estimate. Finally, we discuss work towards a dynamically-optimal adaptive binary search tree algorithm. To my mother, Marilyn Kalai. May her PBSCT be as easy on her as my committee was on me. Acknowledgments It should be no surprise that my biggest thanks go to my parents, who somehow created me and gave me a very happy childhood. For as long as I can remember, my father has been teaching me about problem solving and research through puzzles and questions. If I end up with a fraction of his creativity and accomplishments, I will feel very lucky. Since I was a baby, I couldn't have asked for a better role model than my mother. Even if I could have talked at that age, I still wouldn't have asked for one. I came to CMU in large part because of Avrim Blum. After three advisors, I can say with full confidence that Avrim is the best advisor and teacher at CMU. I don't think I would have finished with anyone else. They often say that, by the time you're ready to graduate, you should know your area better than your advisor. If that was a requirement, I would never graduate. I'm moving from one great advisor to another. Next year I'll be at MIT under the supervision of Santosh Vempala. Many thanks to Santosh...

We propose a set of array primitives, based on experience from structural functional programming. We argue that these primitives provide a right level of abstraction for array computation. These primitives are derived from various perspectives of arrays, with each perspective imposing a particular algebraic structure and demanding specific efficiency requirements. We follow Bird--Meertens formalism [7] [22], in particular the approach used by Meijer, Fokkinga, and Paterson [23], when designing these primitives, but also take into consideration efficient issues in their implementations. 1. Motivation Functional programming languages --- with the exceptions of APL and SISAL --- are not well known for large scale scientific computation. We identify inadequate language support for array operations --- again with apology to APL and SISAL --- a major reason why they are not widely used for scientific applications. Some functional languages provide little support for array operations, which...

Sleator and Tarjan [39] conjectured that splay trees are dynamically optimal binary search trees (BST). In this context, we study the skip list data structure introduced by Pugh [35]. We prove that for a class of skip lists that satisfy a weak balancing property, the working-set bound is a lower bound on the time to access any sequence. Furthermore, we develop a deterministic self-adjusting skip list whose running time matches the working-set bound, thereby achieving dynamic optimality in this class. Finally, we highlight the implications our bounds for skip lists have on multi-way branching search trees such as B-trees, (ab)-trees, and other variants as well as their binary tree representations. In particular, we show a self-adjusting B-tree that is dynamically optimal both in internal and external memory.

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