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

On some lattice computations related to moduli problems

We show how to solve computationally a combinatorial problem about the possible number of roots orthogonal to a vector of given length in $E_8$. We show that the moduli space of K3 surfaces with polarisation of degree 2d is also of general type for d=52. This case was omitted from the earlier work of Gritsenko, Hulek and the second author. We also apply this method to some related problems. In Appendix A, V. Gritsenko shows how to arrive at the case d=52 and some others directly.Comment: With an appendix by V. Gritsenk

Towards Tight Lower Bounds for Range Reporting on the RAM

In the orthogonal range reporting problem, we are to preprocess a set of $n$ points with integer coordinates on a $U \times U$ grid. The goal is to support reporting all $k$ points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exists: (1.) Query time $O(\lg \lg n + k)$ with $O(nlg^{\varepsilon}n)$ words of space for any constant $\varepsilon>0$. (2.) Query time $O((1 + k) \lg \lg n)$ with $O(n \lg \lg n)$ words of space. (3.) Query time $O((1+k)\lg^{\varepsilon} n)$ with optimal $O(n)$ words of space. However, the only known query time lower bound is $\Omega(\log \log n +k)$, even for linear space data structures. All three current best upper bound tradeoffs are derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction