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
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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.
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