283 research outputs found
Conflict-free Chromatic Art Gallery Coverage
We consider a chromatic variant of the art gallery problem, where each
guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen
by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that
ensure a conflict-free covering of all n-vertex polygons? We call this
the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor
networks where colors indicate the wireless frequencies assigned to a
set of covering "landmarks" in the environment so that a mobile robot
can always communicate with at least one landmark in its line-of-sight
range without interference.
Our main result shows that k(n) is O(log n) for orthogonal and for
monotone polygons, and O(log^2 n) for arbitrary simple polygons. By
contrast, if all guards visible from each point must have distinct
colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011]
Computing geodesic furthest neighbors in simple polygons
AbstractAn algorithm is presented for computing geodesic furthest neighbors for all the vertices of a simple polygon, where geodesic denotes the fact that distance between two points of the polygon is defined as the length of an Euclidean shortest path connecting them within the polygon. The algorithm runs in O(n log n) time and uses O(n) space; n being the number of vertices of the polygon. As a corollary, the geodesic diameter of the polygon also can be computed within, the same time and space bounds
The Maximum Exposure Problem
Given a set of points P and axis-aligned rectangles R in the plane, a point p in P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k^2) rectangles, we can expose at least Omega(1/k) of the optimal number of points
Power Aware Routing for Sensor Databases
Wireless sensor networks offer the potential to span and monitor large
geographical areas inexpensively. Sensor network databases like TinyDB are the
dominant architectures to extract and manage data in such networks. Since
sensors have significant power constraints (battery life), and high
communication costs, design of energy efficient communication algorithms is of
great importance. The data flow in a sensor database is very different from
data flow in an ordinary network and poses novel challenges in designing
efficient routing algorithms. In this work we explore the problem of energy
efficient routing for various different types of database queries and show that
in general, this problem is NP-complete. We give a constant factor
approximation algorithm for one class of query, and for other queries give
heuristic algorithms. We evaluate the efficiency of the proposed algorithms by
simulation and demonstrate their near optimal performance for various network
sizes
Observability of Lattice Graphs
We consider a graph observability problem: how many edge colors are needed
for an unlabeled graph so that an agent, walking from node to node, can
uniquely determine its location from just the observed color sequence of the
walk?
Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional
(directed or undirected) lattice of size n^d = n * n * ... * n. We say that
G(n,d) is t-observable if an agent can uniquely determine its current position
in the graph from the color sequence of any t-dimensional walk, where the
dimension is the number of different directions spanned by the edges of the
walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but
a directed walk can have dimension between 1 and 2d because of two different
orientations for each axis.
We derive bounds on the number of colors needed for t-observability. Our main
result is that Theta(n^(d/t)) colors are both necessary and sufficient for
t-observability of G(n,d), where d is considered a constant.
This shows an interesting dependence of graph observability on the ratio
between the dimension of the lattice and that of the walk. In particular, the
number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed
case, and Theta(n) in the undirected case, independent of the lattice
dimension.
All of our results extend easily to non-square lattices: given a lattice
graph of size N = n_1 * n_2 * ... * n_d, the number of colors for
t-observability is Theta (N^(1/t))
Medians and Beyond: New Aggregation Techniques for Sensor Networks
Wireless sensor networks offer the potential to span and monitor large
geographical areas inexpensively. Sensors, however, have significant power
constraint (battery life), making communication very expensive. Another
important issue in the context of sensor-based information systems is that
individual sensor readings are inherently unreliable. In order to address these
two aspects, sensor database systems like TinyDB and Cougar enable in-network
data aggregation to reduce the communication cost and improve reliability. The
existing data aggregation techniques, however, are limited to relatively simple
types of queries such as SUM, COUNT, AVG, and MIN/MAX. In this paper we propose
a data aggregation scheme that significantly extends the class of queries that
can be answered using sensor networks. These queries include (approximate)
quantiles, such as the median, the most frequent data values, such as the
consensus value, a histogram of the data distribution, as well as range
queries. In our scheme, each sensor aggregates the data it has received from
other sensors into a fixed (user specified) size message. We provide strict
theoretical guarantees on the approximation quality of the queries in terms of
the message size. We evaluate the performance of our aggregation scheme by
simulation and demonstrate its accuracy, scalability and low resource
utilization for highly variable input data sets
- …