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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 supportreporting all $k$ points inside an axis-aligned query rectangle. This is one ofthe most fundamental data structure problems in databases and computationalgeometry. Despite the importance of the problem its complexity remainsunresolved in the word-RAM. On the upper bound side, three best tradeoffsexists: (1.) Query time $O(\lg \lg n + k)$ with $O(nlg^{\varepsilon}n)$ wordsof space for any constant $\varepsilon>0$. (2.) Query time $O((1 + k) \lg \lgn)$ 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, theonly known query time lower bound is $\Omega(\log \log n +k)$, even for linearspace data structures.  All three current best upper bound tradeoffs are derived by reducing rangereporting to a ball-inheritance problem. Ball-inheritance is a problem thatessentially encapsulates all previous attempts at solving range reporting inthe word-RAM. In this paper we make progress towards closing the gap betweenthe upper and lower bounds for range reporting by proving cell probe lowerbounds for ball-inheritance. Our lower bounds are tight for a large range ofparameters, excluding any further progress for range reporting using theball-inheritance reduction.

Towards Tight Lower Bounds for Range Reporting on the RAM