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

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