19 research outputs found
Stabbing Pairwise Intersecting Disks by Five Points
Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.
This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points
EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails
Algorithms for the Analysis of Spatio-Temporal Data from Team Sports
Modern object tracking systems are able to simultaneously record trajectories—sequences of time-stamped location points—for large numbers of objects with high frequency and accuracy. The availability of trajectory datasets has resulted in a consequent demand for algorithms and tools to extract information from these data. In this thesis, we present several contributions intended to do this, and in particular, to extract information from trajectories tracking football (soccer) players during matches. Football player trajectories have particular properties that both facilitate and present challenges for the algorithmic approaches to information extraction. The key property that we look to exploit is that the movement of the players reveals information about their objectives through cooperative and adversarial coordinated behaviour, and this, in turn, reveals the tactics and strategies employed to achieve the objectives. While the approaches presented here naturally deal with the application-specific properties of football player trajectories, they also apply to other domains where objects are tracked, for example behavioural ecology, traffic and urban planning
Results on geometric networks and data structures
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