Results on geometric networks and data structures

Abstract

This thesis discusses four problems in computational geometry.

In traditional colored range-searching problems, one wants to store a set of n objects with m distinct colors for the following queries: report all colors such that there is at least one object of that color intersecting the query range. Such an object, however, could be an `outlier' in its color class. We consider a variant of this problem where one has to report only those colors such that at least a fraction t of the objects of that color intersects the query range, for some parameter t. Our main results are on an approximate version of this problem, where we are also allowed to report those colors for which a fraction (1-epsilon)t intersects the query range, for some fixed epsilon > 0. We present efficient data structures for such queries with orthogonal query ranges in sets of colored points, and for point stabbing queries in sets of colored rectangles.

A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. R-trees are box-trees with nodes of high degree. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal.

The geometric minimum-diameter spanning tree (MDST) of a set of n points is a tree that spans the set and minimizes the Euclidian length of the longest path in the tree. So far, the MDST can only be found in slightly subcubic time. We give two fast approximation schemes for the MDST, i.e. factor-(1+epsilon) approximation algorithms. One algorithm uses a grid and takes time O*(1/epsilon^(5 2/3) + n), where the O*-notation hides terms of type O(log^O(1) 1/epsilon). The other uses the well-separated pair decomposition and takes O(1/epsilon^3 n + (1/epsilon) n log n) time. A combination of the two approaches runs in O*(1/epsilon^5 + n) time.

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the L1-distance. We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain the following results: for arbitrary graphs, the problem is NP-hard; for trees, we can solve the problem by linear programming on O(n^2) variables and constraints; for paths, we can solve the problem in time O(n^3 log n); for rectangles sorted vertically along a path, the problem can be solved in O(n^2) time.