Weighted dynamic finger in binary search trees

It is shown that the online binary search tree data structure GreedyASS performs asymptotically as well on a sufficiently long sequence of searches as any static binary search tree where each search begins from the previous search (rather than the root). This bound is known to be equivalent to assigning each item $i$ in the search tree a positive weight $w_i$ and bounding the search cost of an item in the search sequence $s_1,\ldots,s_m$ by $$O\left(1+ \log \frac{\displaystyle \sum_{\min(s_{i-1},s_i) \leq x \leq \max(s_{i-1},s_i)}w_x}{\displaystyle \min(w_{s_i},w_{s_{i-1}})} \right)$$ amortized. This result is the strongest finger-type bound to be proven for binary search trees. By setting the weights to be equal, one observes that our bound implies the dynamic finger bound. Compared to the previous proof of the dynamic finger bound for Splay trees, our result is significantly shorter, stronger, simpler, and has reasonable constants.Comment: An earlier version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithm

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

Combining Binary Search Trees

We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any “well-behaved” bound on the running time of the given BSTs, for any online access sequence. (A BST has a well-behaved bound with f(n) overhead if it spends at most O(f(n)) time per access and its bound satisfies a weak sense of closure under subsequences.) In particular, we obtain a BST data structure that is O(loglogn) competitive, satisfies the working set bound (and thus satisfies the static finger bound and the static optimality bound), satisfies the dynamic finger bound, satisfies the unified bound with an additive O(loglogn) factor, and performs each access in worst-case O(logn) time

Competitive Online Search Trees on Trees

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(\log \log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths

The Log-Interleave Bound: Towards the Unification of Sorting and the BST Model

We study the connections between sorting and the binary search tree model, with an aim towards showing that the fields are connected more deeply than is currently known. The main vehicle of our study is the log-interleave bound, a measure of the information-theoretic complexity of a permutation $\pi$. When viewed through the lens of adaptive sorting -- the study of lists which are nearly sorted according to some measure of disorder -- the log-interleave bound is comparable to the most powerful known measure of disorder. Many of these measures of disorder are themselves virtually identical to well-known upper bounds in the BST model, such as the working set bound or the dynamic finger bound, suggesting a connection between BSTs and sorting. We present three results about the log-interleave bound which solidify the aforementioned connections. The first is a proof that the log-interleave bound is always within a $\lg \lg n$ multiplicative factor of a known lower bound in the BST model, meaning that an online BST algorithm matching the log-interleave bound would perform within the same bounds as the state-of-the-art $\lg \lg n$-competitive BST. The second result is an offline algorithm in the BST model which uses $O(\text{LIB}(\pi))$ accesses to search for any permutation $\pi$. The technique used to design this algorithm also serves as a general way to show whether a sorting algorithm can be transformed into an offline BST algorithm. The final result is a mergesort algorithm which performs work within the log-interleave bound of a permutation $\pi$. This mergesort also happens to be highly parallel, adding to a line of work in parallel BST operations

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

Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition

Greedy BST (or simply Greedy) is an online self-adjusting binary search tree defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon, Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan 1985), Greedy is considered the most promising candidate for being dynamically optimal, i.e., starting with any initial tree, their access costs on any sequence is conjectured to be within $O(1)$ factor of the offline optimal. However, in the past four decades, the question has remained elusive even for highly restricted input. In this paper, we prove new bounds on the cost of Greedy in the ''pattern avoidance'' regime. Our new results include: The (preorder) traversal conjecture for Greedy holds up to a factor of $O(2^{\alpha(n)})$, improving upon the bound of $2^{\alpha(n)^{O(1)}}$ in (Chalermsook et al., FOCS 2015). This is the best known bound obtained by any online BSTs. We settle the postorder traversal conjecture for Greedy. The deque conjecture for Greedy holds up to a factor of $O(\alpha(n))$, improving upon the bound $2^{O(\alpha(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{\alpha(n)})$. Key to all these results is to partition (based on the input structures) the execution log of Greedy into several simpler-to-analyze subsets for which classical forbidden submatrix bounds can be leveraged. Finally, we show the applicability of this technique to handle a class of increasingly complex pattern-avoiding input sequences, called $k$-increasing sequences. As a bonus, we discover a new class of permutation matrices whose extremal bounds are polynomially bounded. This gives a partial progress on an open question by Jacob Fox (2013).Comment: Accepted to SODA 202

Selectable Heaps and Their Application to Lazy Search Trees

We show the O(log n) time extract minimum function of efficient priority queues can be generalized to the extraction of the k smallest elements in O(k log(n/k)) time. We first show the heap-ordered tree selection of Kaplan et al. can be applied on the heap-ordered trees of the classic Fibonacci heap to support the extraction in O(k \log(n/k)) amortized time. We then show selection is possible in a priority queue with optimal worst-case guarantees by applying heap-ordered tree selection on Brodal queues, supporting the operation in O(k log(n/k)) worst-case time. Via a reduction from the multiple selection problem, Ω(k log(n/k)) time is necessary. We then apply the result to the lazy search trees of Sandlund & Wild, creating a new interval data structure based on selectable heaps. This gives optimal O(B+n) lazy search tree performance, lowering insertion complexity into a gap Δi to O(log(n/|Δi|))$ time. An O(1)-time merge operation is also made possible under certain conditions. If Brodal queues are used, all runtimes of the lazy search tree can be made worst-case. The presented data structure uses soft heaps of Chazelle, biased search trees, and efficient priority queues in a non-trivial way, approaching the theoretically-best data structure for ordered data