Optimal External Memory Interval Management

AMS subject classifications. 68P05, 68P10, 68P15 DOI. 10.1137/S009753970240481XIn this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be usedin an optimal solution to the dynamic interval management problem, which is a central problem for object-orientedandtemp oral databases andfor constraint logic programming.Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures

Optimal External Memory Interval Management

This is the published version. Copyright © 2003 Society for Industrial and Applied MathematicsIn this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for object-oriented and temporal databases and for constraint logic programming. Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures

Optimal External Memory Interval Management

This is the publisher's version, which is being shared on KU Scholarworks with permission. The original version may be found at the following link: http://dx.doi.org/10.1137/S009753970240481XIn this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be usedin an optimal solution to the dynamic interval management problem, which is a central problem for object-orientedandtemp oral databases andfor constraint logic programming. Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures

A Dynamic I/O-Efficient Structure for One-Dimensional Top-k Range Reporting

We present a structure in external memory for "top-k range reporting", which uses linear space, answers a query in O(lg_B n + k/B) I/Os, and supports an update in O(lg_B n) amortized I/Os, where n is the input size, and B is the block size. This improves the state of the art which incurs O(lg^2_B n) amortized I/Os per update.Comment: In PODS'1

Optimal Color Range Reporting in One Dimension

Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a \emph{color}. While the answer to an orthogonal point reporting query contains all points in the query range $Q$, the answer to a color reporting query contains only distinct colors of points in $Q$. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal $O(k+1)$ time, where $k$ is the number of colors in the answer and $N$ is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model

Dynamic Range Majority Data Structures

Given a set $P$ of coloured points on the real line, we study the problem of answering range $\alpha$-majority (or "heavy hitter") queries on $P$. More specifically, for a query range $Q$, we want to return each colour that is assigned to more than an $\alpha$-fraction of the points contained in $Q$. We present a new data structure for answering range $\alpha$-majority queries on a dynamic set of points, where $\alpha \in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / \alpha)$ time, and updates in $O((\lg n) / \alpha)$ amortized time. If the coordinates of the points are integers, then the query time can be improved to $O(\lg n / (\alpha \lg \lg n) + (\lg(1/\alpha))/\alpha))$. For constant values of $\alpha$, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for $d \ge 2$, as well as dynamic arrays, in which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201

Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing

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 previous work as no particular exploitation of the underlying structure of is required. The only requirement is that the insertion of a key must identify its predecessor or its successor. Using the proposed technique, online suffix tree can be constructed in worst case time $O(\log n)$ per input symbol (as opposed to amortized $O(\log n)$ time per symbol, achieved by previously known algorithms). To our knowledge, our algorithm is the first that achieves $O(\log n)$ worst case time per input symbol. Searching for a pattern of length $m$ in the resulting suffix tree takes $O(\min(m\log |\Sigma|, m + \log n) + tocc)$ time, where $tocc$ is the number of occurrences of the pattern. The paper also describes more applications and show how to obtain alternative methods for dealing with suffix sorting, dynamic lowest common ancestors and order maintenance