Amortized Bounds for Dynamic Orthogonal Range Reporting

Abstract

We consider the fundamental problem of 2-D dynamic orthogonal range reporting for 2- and 3-sided queries in the standard word RAM model. While many previous dynamic data structures use O(logn /loglogn) update time, we achieve faster O(log1/2 + e n) and O(log2/3 + e n) update times for 2- and 3-sided queries, respectively. Our data structures have optimal O(logn / loglogn) query time. Only Mortensen [14] had previously lowered the update time convincingly below O(logn), with 3- and 4-sided data structures supporting updates in O(log5/6 + e n) and O(log7/8 + e 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((logn loglogn)2/3) update time and O((logn loglogn)1/3) query time. At the other end of our update/query trade-off curve, we achieve O(logn/loglogn) update time and O(loglogn) query time. In contrast, in the pointer machine model, there are only O(loglogn) factor differences between the complexities of fully dynamic and partially dynamic 2-D orthogonal range reporting.