with optimal $O(\log n/\log\log n)$ query time for the four-dimensional orthogonal range reporting problem.

#### Range Minima Queries.

The best previously known trade-offs are listed in Table \[tab:rangemin\]. The study of compact data structures for range searching problems was initiated by Willard [@Willard86] and Chazelle [@Chazelle88]. In the latter work, published over three decades ago, Chazelle [@Chazelle88] described an $O(n)$-space data structure that supports two-dimensional range minimum queries[^2] in $O(\log^{1+{\varepsilon}} n)$ time. The only improvement for an $O(n)$-space data structure was achieved by Farzan et al. [@FarzanMR12] who reduced the query time to $O(\log n \log \log n)$. Better query times for this problem can be achieved at a cost of increasing the space usage. Chan et al. [@ChanLP11] described a data structure that uses $O(n\log^{{\varepsilon}} n)$ space and answers queries in optimal $O(\log \log n)$ time. Another trade-off was achieved by Karpinski and Nekrich[@KarpinskiN09]; combining their data structure with the result of [@Chan13], we can obtain a data structure that uses $O(n (\log \log n)^3)$ space and answers queries in $O((\log\log n)^2)$ time. A five-sided range reporting query is a special case of three-dimensional orthogonal range reporting queries where a query is bounded on five sides. Range minima problem 