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 ...

Abstract. We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized Ω(lg n) lower bound per operation for several data structural problems on n elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of Ω(lg n/lg lg n) for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan’s dynamic trees. We also prove the first Ω(log B n) lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update. Key words. Cell-probe complexity, lower bounds, data structures, dynamic graph problems, partial-sums problem AMS subject classification. 68Q17 1. Introduction. The

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 Move-toFront 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 decision-making costs are not considered.

. 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 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 Davenport-Schinzel 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 inverse-Ackermann 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.

We present lower and upper bounds on adaptive heuristics for maintaining binary search trees using a constant number of link or pointer changes for each operation (constant linkage cost (CLC)). We show that no adaptive heuristic with an amortized linkage cost of o(log n) can be competitive. In particular, we show that any heuristic that performs f(n) = o(log n) promotions (rotations) amortized over each access has a competitive ratio of at least \Omega\Gammaast n=f(n)) against an oblivious adversary, and any heuristic that performs f(n) = o(log n) pointer changes amortized over each access has a competitive ratio of at least\Omega\Gamma log n f(n) log(log n=f(n)) ) against an adaptive online adversary. In our investigation of upper bounds we present four adaptive heuristics: ffl A randomized, worst-case-CLC heuristic (R2P) whose expected search time is within a constant factor of the search time using an optimal tree; that is, it is statically competitive ffl A randomized, expecte...

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.

Recall from last lecture that we are looking at the document-retrieval problem. The problem can be stated as follows: Given a set of texts T1, T2,..., Tk and a pattern P, determine the distinct texts in which the patterns occurs. In particular, we are allowed to preprocess the texts in order to be able to answer the query faster. Our preprocessing choice was the use of a single suffix tree, in which all the suffixes of all the texts appear, each suffix ending with a distinct symbol that determines the text in which the suffix appears. In order to answer the query we reduced the problem to range-min queries, which in turn was reduced to the least common ancestor (LCA) problem on the cartesian tree of an array of numbers. The cartesian tree is constructed recursively by setting its root to be the minimum element of the array and recursively constructing its two subtrees using the left and right partitions of the array. The range-min query of an interval [i, j] is then equivalent to finding the LCA of the two nodes of the cartesian tree that correspond to i and j. In this lecture we continue to see how we can solve the LCA problem on any static tree. This will involve a reduction of the LCA problem back to the range-min query problem (!) and then a

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...

Abstract. We present skip-splay, the first binary search tree algorithm known to have a running time that nearly achieves the unified bound. Skip-splay trees require only O(m lg lg n + UB(σ)) time to execute a query sequence σ = σ1...σm. The skip-splay algorithm is simple and similar to the splay algorithm. 1 Introduction and Related Work Although the worst-case access cost for comparison-based dictionaries is Ω(lg n), many sequences of operations are highly nonrandom, allowing tighter, instancespecific running time bounds to be achieved by algorithms that adapt to the input sequence. Splay trees [1] are an example of such an adaptive algorithm