In pursuit of the dynamic optimality conjecture

In 1985, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence---any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any other tree algorithm is known to be dynamically optimal. Here we survey the progress that has been made in the almost thirty years since the conjecture was first formulated, and present a binary search tree algorithm that is dynamically optimal if any binary search tree algorithm is dynamically optimal.Comment: Preliminary version of paper to appear in the Conference on Space Efficient Data Structures, Streams and Algorithms to be held in August 2013 in honor of Ian Munro's 66th birthda

Belga B-trees

We revisit self-adjusting external memory tree data structures, which combine the optimal (and practical) worst-case I/O performances of B-trees, while adapting to the online distribution of queries. Our approach is analogous to undergoing efforts in the BST model, where Tango Trees (Demaine et al. 2007) were shown to be $O(\log\log N)$-competitive with the runtime of the best offline binary search tree on every sequence of searches. Here we formalize the B-Tree model as a natural generalization of the BST model. We prove lower bounds for the B-Tree model, and introduce a B-Tree model data structure, the Belga B-tree, that executes any sequence of searches within a $O(\log \log N)$ factor of the best offline B-tree model algorithm, provided $B=\log^{O(1)}N$. We also show how to transform any static BST into a static B-tree which is faster by a $\Theta(\log B)$ factor; the transformation is randomized and we show that randomization is necessary to obtain any significant speedup

Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry

International audienc

A Distribution-Sensitive Dictionary with Low Space Overhead ⋆

Abstract. The time required for a sequence of operations on a data structure is usually measured in terms of the worst possible such sequence. This, however, is often an overestimate of the actual time required. Distribution-sensitive data structures attempt to take advantage of underlying patterns in a sequence of operations in order to reduce time complexity, since access patterns are non-random in many applications. Unfortunately, many of the distribution-sensitive structures in the literature require a great deal of space overhead in the form of pointers. We present a dictionary data structure that makes use of both randomization and existing space-efficient data structures to yield very low space overhead while maintaining distribution sensitivity in the expected sense.

1,3-Dipolar Cycloaddition Reactions of 1-(4-Phenylphenacyl)-1,10-phenanthrolinium N-Ylide with Activated Alkynes and Alkenes

The 3 2 cycloaddition reaction of 1-(4-phenylphenacyl)-1,10-phenanthrolinium ylide with activated alkynes gave pyrrolo[1,2- 4a][1,10]phenanthrolines 6a-d. The "one pot" synthesis of 6a,b,d from 4, activatedalkenes, Et3N and tetrakis-pyridine cobalt (II) dichromate (TPCD) is described. Thehelical chirality of pyrrolophenanthrolines 6b-d was put in evidence by NMRspectroscopy

Dynamizing Succinct Tree Representations

Abstract. We consider succinct, or space-efficient, representations of ordinal trees. Representations exist that take 2n + o(n) bits to represent a static n-node ordinal tree – close to the information-theoretic minimum – and support navigational operations in O(1) time on a RAM model; and some implementations have good practical performance. The situation is different for dynamic ordinal trees. Although there is theoretical work on succinct dynamic ordinal trees, there is little work on the practical performance of these data structures. Motivated by applications to representing XML documents, in this paper, we report on a preliminary study on dynamic succinct data structures. Our implementation is based on representing the tree structure as a sequence of balanced parentheses, with navigation done using the min-max tree of Sadakane and Navarro (SODA ’10). Our implementation shows promising performance for update and navigation, and our findings highlight two issues that we believe will be important to future implementations: the difference between the finger model of (say) Farzan and Munro (ICALP ’09) and the parenthesis model of Sadakane and Navarro, and the choice of the balanced tree used to represent the min-max tree.