Narrow Results

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems.

Simplex range reporting is the problem of storing n points in the d-dimensional space in a data structure such that the k points that lie inside a query simplex can be found efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using space in which the notation either hides a polylogarithmic or an n^ε factor for any constant ε > 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who showed any pointer machine data structure that can answer queries in O(n^γ + k) time must use Ω(n^d-ε-dγ) space. Observe that this bound is a polynomial factor away from the best known data structures.

In this article, we improve the space lower bound to . Not only this bridges the gap from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our space lower bound almost equals Ω((n/Q(n))^d); the latter is generally believed to be the “right” bound.

By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we study the external memory model and offer a new simple framework for proving lower bounds in this model. We show that answering simplex range reporting queries with Q(n)+O(k/B) I/Os requires ) space or blocks, in which B is the block size.

Orthogonal range search is ubiquitous nowadays, with natural applications in databases, data mining, and text indexing. Very recently, yet another application was discovered, which is to maintain a DFS forest in a dynamic graph. In this paper, we want to extend the above recent study, by applying orthogonal range search to efficient maintenance of a BFS-like forest, called no-back-edge-traversal (NBET) forest, which refers to a spanning forest obtained from a traversal that does not create any back edge. The study of such a problem is motivated by the fact that NBET forest can be used as a strong certificate of 2-connectivity of an undirected graph, which is more general than a spanning forest obtained from a scan-first search traversal.

Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O(log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n²). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O((log n)²) time. In this dynamic data structure, a query with output size k can be solved in O(log n+k(log n)²) time. The second dynamic data structure, which allows only semi-online updates, has O((log n)²) amortized update time and O(log n+k) query time.