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

Succinct Indices for Range Queries with applications to Orthogonal Range Maxima

We consider the problem of preprocessing $N$ points in 2D, each endowed with a priority, to answer the following queries: given a axis-parallel rectangle, determine the point with the largest priority in the rectangle. Using the ideas of the \emph{effective entropy} of range maxima queries and \emph{succinct indices} for range maxima queries, we obtain a structure that uses O(N) words and answers the above query in $O(\log N \log \log N)$ time. This is a direct improvement of Chazelle's result from FOCS 1985 for this problem -- Chazelle required $O(N/\epsilon)$ words to answer queries in $O((\log N)^{1+\epsilon})$ time for any constant $\epsilon > 0$.Comment: To appear in ICALP 201

Approximating Huffman codes in parallel

AbstractIn this paper we present new results on the approximate parallel construction of Huffman codes. Our algorithm achieves linear work and logarithmic time, provided that the initial set of elements is sorted. This is the first parallel algorithm for that problem with the optimal time and work. Combining our approach with the best known parallel sorting algorithms we can construct an almost optimal Huffman tree with optimal time and work. This also leads to the first parallel algorithm that constructs exact Huffman codes with maximum codeword length H in time O(H) with n/logn processors, if the elements are sorted

ORTHOGONAL POINT LOCATION AND RECTANGLE STABBING QUERIES IN 3-D

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.• Orthogonal point location. We give the first linear-space data structure that sup- ports 3-d point location queries on n disjoint axis-aligned boxes with optimal O (log") query time in the (arithmetic) pointer machine model. This improves the previous 0 (\ogi/2 n^ bound of Rahul \SODA 201o|. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word HAM model with sub-logarithmic query time. Our tech- nique also improves upon the result of de Berg, van Kreveld, and Snoeyink \ Journal of Algorithms I995| for 3-d point location in (space filling) box subdivisions: we obtain a linear-space data structure with G{\og2\ogU) query time. • Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(\ogwn 4- k) time in the word RAM model, where k is the number of rectangles reported and w is the number of bits in a word. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-/: rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing.For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup. Brodal. and Rauhe's grid-based recursive technique \FOCS 2000|, combined with a number of new ideas

The prevalence and progression of astimatism and myopia in children

Optometrists are commonly asked how to manage and reduce progression of refractive errors. Epidemiological studies on refractive error are not representative of optometric practices. Therefore this study aimed to provide knowledge on prevalence and progression of astigmatism and myopia in children, allowing optometrists to use information collected routinely to advise on potential for refractive change, and to more accurately determine when the next examination should be. It also investigated the possible influence of astigmatism and myopia on one another. It was the first to investigate birth season and refractive progression,and in the UK to assess the influence of birth season on astigmatism and myopia in children.This retrospective study analysed 900 subjective refractions of children under 19years of age (mean 11.1 years) from two optometric practices in Liverpool. A subset of 242 of these children, tested longitudinally for a mean 5.97 years, was assessed for progression of astigmatism and myopia, using Decision Tree statistical Analysis to determine a possible association between astigmatism and myopia progression.Cross-sectional data showed that boys were more likely to have astigmatism than girls (p=0.004). Age affected astigmatic axis, with against-the-rule and oblique astigmatism more prevalent in the older children, and with-the-rule prevalence reducing with age (p=0.004). Myopia increased with age (p=<0.0001). Astigmatism increased by 0.04D/year (SD 0.087), and myopia by -0.15D/year (SD0.23D). The presence of astigmatism was related to faster astigmatic progression(p=<0.0001). Being myopic was the sole risk factor for faster myopic progression(p=<0.0001). Birth season was unrelated to prevalence or progression of refractive error.Advice on how sex, age and birth season may influence refraction can be discussed with patients and their families. Progression of astigmatism and myopia were not linked, suggesting that whilst they may share some risk factors, they appear to be independent of one another

Size-constrained weighted ancestors with applications

The weighted ancestor problem on a rooted node-weighted tree T is a generalization of the classic predecessor problem: construct a data structure for a set of integers that supports fast predecessor queries. Both problems are known to require Ω(log log n) time for queries provided O(n poly log n) space is available, where n is the input size. The weighted ancestor problem has attracted a lot of attention by the combinatorial pattern matching community due to its direct application to suffix trees. In this formulation of the problem, the nodes are weighted by string depth. This research has culminated in a data structure for weighted ancestors in suffix trees with O(1) query time and an O(n)-time construction algorithm [Belazzougui et al., CPM 2021]. In this paper, we consider a different version of the weighted ancestor problem, where the nodes are weighted by any function weight that maps each node of T to a positive integer, such that weight(u) ≤ size(u) for any node u and weight(u1) ≤ weight(u2) if node u1 is a descendant of node u2, where size(u) is the number of nodes in the subtree rooted at u. In the size-constrained weighted ancestor (SWA) problem, for any node u of T and any integer k, we are asked to return the lowest ancestor w of u with weight at least k. We show that for any rooted tree with n nodes, we can locate node w in O(1) time after O(n)-time preprocessing. In particular, this implies a data structure for the SWA problem in suffix trees with O(1) query time and O(n)-time preprocessing, when the nodes are weighted by weight. We also show several string-processing applications of this result