Dynamic Ordered Sets with Exponential Search Trees

We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any $n$-node static tree can be represented in $2n + o(n)$ bits and a number of operations on the tree can be supported in constant time under the word-RAM model. However the data structures are complicated and difficult to dynamize. We propose a simple and flexible data structure, called the range min-max tree, that reduces the large number of relevant tree operati...

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{\displaystyl...

We give the first data structure for the problem of maintaining a dynamic set of n elements drawn from a partially ordered universe described by a tree. We define the Line-Leaf Tree, a linear-sized data structure that supports the operations: insert; delete; test membership; and predecessor. The performance of our data structure is within an O(log w)-factor of optimal. Here w <= n is the width of the partial-order---a natural obstacle in searching a partial order.

In this paper we describe a dynamic data structure that answers one-dimensional stabbing-max queries in optimal $O(\log n/\log\log n)$ time. Our data structure uses linear space and supports insertions and deletions in $O(\log n)$ and $O(\log n/\log \log n)$ amortized time respectively. We also describe a $O(n(\log n/\log\log n)^{d-1})$ space data structure that answers $d$-dimensional stabbing-max queries in $O((\log n/\log\log n)^{d})$ time. Insertions and deletions are s...

In this paper we present two versions of a parallel finger structure FS on p processors that supports searches, insertions and deletions, and has a finger at each end. This is to our knowledge the first implementation of a parallel search structure that is work-optimal with respect to the finger bound and yet has very good parallelism (within a factor of O( (log p)^2 ) of optimal). We utilize an extended implicit batching framework that transparently facilitates the use of FS...

We develop a new technique for proving cell-probe lower bounds for static data structures. Previous lower bounds used a reduction to communication games, which was known not to be tight by counting arguments. We give the first lower bound for an explicit problem which breaks this communication complexity barrier. In addition, our bounds give the first separation between polynomial and near linear space. Such a separation is inherently impossible by communication complexity. ...

We introduce the lazy search tree data structure. The lazy search tree is a comparison-based data structure on the pointer machine that supports order-based operations such as rank, select, membership, predecessor, successor, minimum, and maximum while providing dynamic operations insert, delete, change-key, split, and merge. We analyze the performance of our data structure based on a partition of current elements into a set of gaps $\{\Delta_i\}$ based on rank. A query falls...

In this paper we consider the following modification of the iterative search problem. We are given a tree $T$, so that a dynamic catalog $C(v)$ is associated with every tree node $v$. For any $x$ and for any node-to-root path $\pi$ in $T$, we must find the predecessor of $x$ in $\cup_{v\in \pi} C(v)$. We present a linear space dynamic data structure that supports such queries in $O(t(n)+|\pi|)$ time, where $t(n)$ is the time needed to search in one catalog and $|\pi|$ denotes...

Uniquely represented data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of $n$ keys from a totally ordered universe in this context. We introduce a two-layer data structure called $(\alpha,\varepsilon)$-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory. Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze...

A dynamic dictionary is a data structure that maintains sets of cardinality at most $n$ from a given universe and supports insertions, deletions, and membership queries. A filter approximates membership queries with a one-sided error that occurs with probability at most $\epsilon$. The goal is to obtain dynamic filters that are space-efficient (the space is $1+o(1)$ times the information-theoretic lower bound) and support all operations in constant time with high probability....