The Unified Segment Tree and its Application to the Rectangle Intersection Problem

In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the data structure. We give some new definitions to previously well-defined concepts that arise naturally in this variation, and we show some properties concerning the relationships between the nodes, and the regions those nodes represent. We think these properties will enable the data to be utilized in new situations, beyond those previously studied. As an example, we show that the data structure can be used to solve the Rectangle Intersection Problem in a more straightforward and natural way than had be done in the past.Comment: 14 pages, 6 figure

Rectilinear minimum link paths in two and higher dimensions

The thesis discusses algorithms for the minimum link path problem, which is a well known geometric path finding problem. The goal is to find a path that does the minimum number of turns amidst obstacles in a continuous space. We focus on the most classical variant, the rectilinear minimum link path problem, where the path and the obstacles are restricted to the directions of the coordinate axes. We study the rectilinear minimum link path problem in the plane and in the three-dimensional space, as well as in higher dimensional domains. We present several new algorithms for solving the problem in domains of varying dimension. For the planar case we develop a simple method that has the optimal O(n log n) time complexity. For three-dimensional domains we present a new algorithm with running time O(n^2 log^2 n), which is an improvement over the best previously known result O(n^2.5 log n). The algorithm can also be generalized to higher dimensions, leading to an O(n^(D-1) log^(D-1) n) time algorithm in D-dimensional domains. We describe the new algorithms as well as the data structures used. The algorithms work by maintaining a reachable region that is gradually expanded to form a shortest path map from the starting point. The algorithms rely on several efficient data structures: the reachable region is tracked by using a simple recursive space decomposition, and the region is expanded by a sweep plane method that uses a multidimensional segment tree

Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model

In this paper, we develop a new communication model to prove a data structure lower bound for the dynamic interval union problem. The problem is to maintain a multiset of intervals $\mathcal{I}$ over $[0, n]$ with integer coordinates, supporting the following operations: - insert(a, b): add an interval $[a, b]$ to $\mathcal{I}$, provided that $a$ and $b$ are integers in $[0, n]$; - delete(a, b): delete a (previously inserted) interval $[a, b]$ from $\mathcal{I}$; - query(): return the total length of the union of all intervals in $\mathcal{I}$. It is related to the two-dimensional case of Klee's measure problem. We prove that there is a distribution over sequences of operations with $O(n)$ insertions and deletions, and $O(n^{0.01})$ queries, for which any data structure with any constant error probability requires $\Omega(n\log n)$ time in expectation. Interestingly, we use the sparse set disjointness protocol of H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of nondeterministic communication games, for which we prove lower bounds. For applications, we prove lower bounds for several dynamic graph problems by reducing them from dynamic interval union