Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

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

Dynamic Orthogonal Range Searching on the RAM, Revisited

We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n / log log n + k) optimal query time and O(log^{2/3+o(1)}n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen\u27s previous result [SIAM J. Comput., 2006], which has O(log^{7/8+epsilon}n) update time for an arbitrarily small constant epsilon. In the case of 3-sided queries, our update time reduces to O(log^{1/2+epsilon}n), improving Wilkinson\u27s previous bound [ESA 2014] of O(log^{2/3+epsilon}n)

Four-Dimensional Dominance Range Reporting in Linear Space

In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+?} n + k log^? n) time, where k is the number of reported points, n is the number of points in the data structure, and ? is an arbitrarily small positive constant. Our second data structure uses O(n log^? n) space and answers queries in O(log n+k) time. These are the first data structures for this problem that use linear (resp. O(n log^? n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^? n)-space data structure supports queries in O(n^? + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ? 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+?} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time

Algorithms for continuous queries: A geometric approach

<p>There has been an unprecedented growth in both the amount of data and the number of users interested in different types of data. Users often want to keep track of the data that match their interests over a period of time. A continuous query, once issued by a user, maintains the matching results for the user as new data (as well as updates to the existing data) continue to arrive in a stream. However, supporting potentially millions of continuous queries is a huge challenge. This dissertation addresses the problem of scalably processing a large number of continuous queries over a wide-area network. </p><p>Conceptually, the task of supporting distributed continuous queries can be divided into two components--event processing (computing the set of affected users for each data update) and notification dissemination (notifying the set of affected users). The first part of this dissertation focuses on event processing. Since interacting with large-scale data can easily frustrate and overwhelm the users, top-k queries have attracted considerable interest from the database community as they allow users to focus on the top-ranked results only. However, it is nearly impossible to find a set of common top-ranked data that everyone is interested in, therefore, users are allowed to specify their interest in different forms of preferences, such as personalized ranking function and range selection. This dissertation presents geometric frameworks, data structures, and algorithms for answering several types of preference queries efficiently. Experimental evaluations show that our approaches outperform the previous ones by orders of magnitude.</p><p>The second part of the dissertation presents comprehensive solutions to the problem of processing and notifying a large number of continuous range top-k queries across a wide-area network. Simple solutions include using a content-driven network to notify all continuous queries whose ranges contain the update (ignoring top-k), or using a server to compute only the affected continuous queries and notifying them individually. The former solution generates too much network traffic, while the latter overwhelms the server. This dissertation presents a geometric framework which allows the set of affected continuous queries to be described succinctly with messages that can be efficiently disseminated using content-driven networks. Fast algorithms are also developed to reformulate each update into a set of messages whose number is provably optimal, with or without knowing all continuous queries. </p><p>The final component of this dissertation is the design of a wide-area dissemination network for continuous range queries. In particular, this dissertation addresses the problem of assigning users to servers in a wide-area content-based publish/subscribe system. A good assignment should consider both users' interests and locations, and balance multiple performance criteria including bandwidth, delay, and load balance. This dissertation presents a Monte Carlo approximation algorithm as well as a simple greedy algorithm. The Monte Carlo algorithm jointly considers multiple performance criteria to find a broker-subscriber assignment and provides theoretical performance guarantees. Using this algorithm as a yardstick, the greedy algorithm is also concluded to work well across a wide range of workloads.</p>Dissertatio