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On the Dynamic Finger Conjecture for Splay Trees. Part II: The Proof
 SIAM Journal on Computing
"... The following result is shown: On an nnode 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 ..."
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Cited by 45 (1 self)
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The following result is shown: On an nnode 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 selfadjusting 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 ...
Static Optimality and Dynamic SearchOptimality in Lists and Trees
, 2002
"... Adaptive data structures form a central topic of online algorithms research, beginning with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that MovetoFront is constant competitive for the list update prob lem [ST85a, ST85b]. This paper is ..."
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Cited by 19 (3 self)
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Adaptive data structures form a central topic of online algorithms research, beginning with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that MovetoFront is constant competitive for the list update prob lem [ST85a, ST85b]. This paper is inspired by the observation that one can in fact achieve a 1 + e ra tio against the best static object in hindsight for a wide range of data structure problems via "weighted experts" techniques from Machine Learning, if computational decisionmaking costs are not considered.
SelfOrganizing Data Structures
 In
, 1998
"... . We survey results on selforganizing 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 competit ..."
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Cited by 18 (0 self)
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. We survey results on selforganizing 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 online algorithms. For binary search trees, we present results for both online and offline algorithms. Selforganizing 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 selforganizing 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 selforganizati...
Splay trees, DavenportSchinzel sequences, and the deque conjecture
, 2007
"... We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct ..."
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Cited by 15 (5 self)
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We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverseAckermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.
Probabilistic and Online Methods in Machine Learning
, 2001
"... On the surface, the three online machine learning problems analyzed in this thesis may seem unrelated. The first is an online investment strategy introduced by Tom Cover. We begin with a simple analysis that extends to the case of fixedpercentage transaction costs. We then describe an efficient i ..."
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Cited by 3 (0 self)
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On the surface, the three online machine learning problems analyzed in this thesis may seem unrelated. The first is an online investment strategy introduced by Tom Cover. We begin with a simple analysis that extends to the case of fixedpercentage transaction costs. We then describe an efficient implementation that runs in time polynomial in the number of stocks. The second problem is kfold 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 holdout estimate. Finally, we discuss work towards a dynamicallyoptimal 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...
Untangling binary trees via rotations
 Comput. J
"... In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination bina ..."
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In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination binary tree must also be degenerate, of a more restricted nature. Previous work on rotation distance has focused on approximation algorithms. Our algorithm is the only known nontrivial polynomial time algorithm for exact rotation distance between special cases of binary trees. 1.
Adaptive Binary Search Trees
, 2009
"... A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the wors ..."
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A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, inputsensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instancespecific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,
Upper Bounds for Maximally Greedy Binary Search Trees
"... Abstract. At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture, an offline BST algorithm that ..."
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Abstract. At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture, an offline BST algorithm that greedily rearranges the search path to minimize the cost of future searches. They showed that GreedyFuture is actually an online algorithm in their geometric view, and that there is a way to turn GreedyFuture into an online BST algorithm with only a constant factor increase in total search cost. Demaine et al. conjectured this algorithm was dynamically optimal, but no upper bounds were given in their paper. We prove the first nontrivial upper bounds for the cost of search operations using GreedyFuture including giving an access lemma similar to that found in Sleator and Tarjan’s classic paper on splay trees. 1
unknown title
"... For the past few lectures we have been studying self organizing binary search trees. In these two lectures we will study a novel connection between binary search trees(BSTs) and points satisfying a simple property from [1]. Using this correspondence, we will restate many results and conjectures rela ..."
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For the past few lectures we have been studying self organizing binary search trees. In these two lectures we will study a novel connection between binary search trees(BSTs) and points satisfying a simple property from [1]. Using this correspondence, we will restate many results and conjectures relating to BSTs and dynamic optimality. 2