Dynamic Ordered Sets with Exponential Search Trees

We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM with w-bit words, we show how to perform updates in O(lg w) time and answer queries in O(lglg w) time. The update time is identical to the van Emde Boas structure, but the query time...

It is widely assumed that $O(m+\lg \sigma)$ is the best one can do for finding a pattern of length $m$ in a compacted trie storing strings over an alphabet of size $\sigma$, if one insists on linear-size data structures and deterministic worst-case running times [Cole et al., ICALP'06]. In this article, we first show that a rather straightforward combination of well-known ideas yields $O(m+\lg\lg \sigma)$ deterministic worst-case searching time for static tries. Then we mov...

In the particular case we have insertions/deletions at the tail of a given set S of $n$ one-dimensional elements, we present a simpler and more concrete algorithm than that presented in [Anderson, 2007] achieving the same (but also amortized) upper bound of $O(\sqrt{logd/loglogd})$ for finger searching queries, where $d$ is the number of sorted keys between the finger element and the target element we are looking for. Furthermore, in general case we have insertions/deletions ...

The representation of a dynamic ordered set of $n$ integer keys drawn from a universe of size $m$ is a fundamental data structuring problem. Many solutions to this problem achieve optimal time but take polynomial space, therefore preserving time optimality in the \emph{compressed} space regime is the problem we address in this work. For a polynomial universe $m = n^{\Theta(1)}$, we give a solution that takes $\textsf{EF}(n,m) + o(n)$ bits, where $\textsf{EF}(n,m) \leq n\lceil...

The problem of Text Indexing is a fundamental algorithmic problem in which one wishes to preprocess a text in order to quickly locate pattern queries within the text. In the ever evolving world of dynamic and on-line data, there is also a need for developing solutions to index texts which arrive on-line, i.e. a character at a time, and still be able to quickly locate said patterns. In this paper, a new solution for on-line indexing is presented by providing an on-line suffix ...

A static binary search tree where every search starts from where the previous one ends (lazy finger) is considered. Such a search method is more powerful than that of the classic optimal static trees, where every search starts from the root (root finger), and less powerful than when rotations are allowed---where finding the best rotation based tree is the topic of the dynamic optimality conjecture of Sleator and Tarjan. The runtime of the classic root-finger tree can be expre...

We present a data structure representing a dynamic set S of w-bit integers on a w-bit word RAM. With |S|=n and w > log n and space O(n), we support the following standard operations in O(log n / log w) time: - insert(x) sets S = S + {x}. - delete(x) sets S = S - {x}. - predecessor(x) returns max{y in S | y< x}. - successor(x) returns min{y in S | y >= x}. - rank(x) returns #{y in S | y< x}. - select(i) returns y in S with rank(y)=i, if any. Our O(log n/log w) bound is opt...

This paper presents a general technique for optimally transforming any dynamic data structure that operates on atomic and indivisible keys by constant-time comparisons, into a data structure that handles unbounded-length keys whose comparison cost is not a constant. Examples of these keys are strings, multi-dimensional points, multiple-precision numbers, multi-key data (e.g.~records), XML paths, URL addresses, etc. The technique is more general than what has been done in prev...

We present highly optimized data structures for the dynamic predecessor problem, where the task is to maintain a set $S$ of $w$-bit numbers under insertions, deletions, and predecessor queries (return the largest element in $S$ no larger than a given key). The problem of finding predecessors can be viewed as a generalized form of the membership problem, or as a simple version of the nearest neighbour problem. It lies at the core of various real-world problems such as internet...

We present a general method for de-amortizing essentially any Binary Search Tree (BST) algorithm. In particular, by transforming Splay Trees, our method produces a BST that has the same asymptotic cost as Splay Trees on any access sequence while performing each search in O(log n) worst case time. By transforming Multi-Splay Trees, we obtain a BST that is O(log log n) competitive, satisfies the scanning theorem, the static optimality theorem, the static finger theorem, the wor...