Amortized Bounds for Dynamic Orthogonal Range Reporting

We consider the fundamental problem of 2-D dynamic orthogonal range reporting for 2and 3-sided queries in the standard word RAM model. While many previous dynamic data structures use O(log n/ log log n) update time, we achieve faster O(log n) and O(log n) update times for 2and 3-sided queries, respectively. Our data structures have optimalO(log n/ log log n) query time. Only Mortensen [13] had previously lowered the update time convincingly below O(log n), with 3and 4-sided data structures supporting updates in O(log n) and O(log n) time, respectively. In practice, fast updates are often as important as fast queries, so we make a step forward for an important problem that has not seen any progress in recent years. We also obtain new results for the special case of 3-sided insertion-only emptiness, showing that the difference in complexity between fully dynamic and partially dynamic 2-D orthogonal range reporting can be significant (i.e., Ω(polylog n) factor differences). In particular, we achieve O((log n log log n)) update time and O((log n log log n)) query time. At the other end of our update/query trade-off curve, we achieve O(log n/ log log n) update time and O(log log n) query time. In contrast, in the pointer machine model, there are only O(log log n) factor differences between the complexities of fully dynamic and partially dynamic 2-D orthogonal range reporting.