Models and Techniques for Proving Data Structure Lower Bounds

In this dissertation, we present a number of new techniques and tools for proving lower bounds on the operational time of data structures. These techniques provide new lines of attack for proving lower bounds in both the cell probe model, the group model, the pointer machine model and the I/O-model. In all cases, we push the frontiers further by proving lower bounds higher than what could possibly be proved using previously known techniques. For the cell probe model, our results have the following consequences: • The first Ω(lg n) query time lower bound for linear space static data structures. The highest previous lower bound for any static data structure problem peaked at Ω(lg n/ lg lg n). • An Ω((lg n/ lg lgn)2) lower bound on the maximum of the update time and the query time of dynamic data structures. This is almost a quadratic improvement over the highest previous lower bound of Ω(lg n). In the group model, we establish a number of intimate connections to the fields of combinatorial discrepancy and range reporting in the pointer machine model. These connections immediately allow us to translate decades of research in discrepancy and range reporting to very high lower bounds on the update time tu and query time tq of dynamic group model data structures. We have listed a few in the following: • For d-dimensional halfspace range searching, we get a lower bound of tutq = Ω(n 1−1/d). This comes within a lg lg n factor of the best known upper bound. • For orthogonal range searching, we get a lower bound of tutq = Ω(lgd−1 n). • For ball range searching, we get a lower bound of tutq = Ω(n1−1/d). The highest previous lower bound proved in the group model does not exceed Ω((lg n/ lg lg n)2) on the maximum of tu and tq. Finally, we present a new technique for proving lower bounds for range reporting problems in the pointer machine and the I/O-model. With this technique, we tighten the gap between the known upper bound and lower bound for the most fundamental range reporting problem, orthogonal range reporting.