566 research outputs found
Set It and Forget It: Approximating the Set Once Strip Cover Problem
We consider the Set Once Strip Cover problem, in which n wireless sensors are
deployed over a one-dimensional region. Each sensor has a fixed battery that
drains in inverse proportion to a radius that can be set just once, but
activated at any time. The problem is to find an assignment of radii and
activation times that maximizes the length of time during which the entire
region is covered. We show that this problem is NP-hard. Second, we show that
RoundRobin, the algorithm in which the sensors simply take turns covering the
entire region, has a tight approximation guarantee of 3/2 in both Set Once
Strip Cover and the more general Strip Cover problem, in which each radius may
be set finitely-many times. Moreover, we show that the more general class of
duty cycle algorithms, in which groups of sensors take turns covering the
entire region, can do no better. Finally, we give an optimal O(n^2 log n)-time
algorithm for the related Set Radius Strip Cover problem, in which all sensors
must be activated immediately.Comment: briefly announced at SPAA 201
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
Average Case Network Lifetime on an Interval with Adjustable Sensing Ranges
Given n sensors on an interval, each of which is equipped with an adjustable sensing radius and a unit battery charge that drains in inverse linear proportion to its radius, what schedule will maximize the lifetime of a network that covers the entire interval? Trivially, any reasonable algorithm is at least a 2-approximation for this Sensor Strip Cover problem, so we focus on developing an efficient algorithm that maximizes the expected network lifetime under a random uniform model of sensor distribution. We demonstrate one such algorithm that achieves an expected network lifetime within 12 % of the theoretical maximum. Most of the algorithms that we consider come from a particular family of RoundRobin coverage, in which sensors take turns covering predefined areas until their battery runs out
Dynamic Windows Scheduling with Reallocation
We consider the Windows Scheduling problem. The problem is a restricted
version of Unit-Fractions Bin Packing, and it is also called Inventory
Replenishment in the context of Supply Chain. In brief, the problem is to
schedule the use of communication channels to clients. Each client ci is
characterized by an active cycle and a window wi. During the period of time
that any given client ci is active, there must be at least one transmission
from ci scheduled in any wi consecutive time slots, but at most one
transmission can be carried out in each channel per time slot. The goal is to
minimize the number of channels used. We extend previous online models, where
decisions are permanent, assuming that clients may be reallocated at some cost.
We assume that such cost is a constant amount paid per reallocation. That is,
we aim to minimize also the number of reallocations. We present three online
reallocation algorithms for Windows Scheduling. We evaluate experimentally
these protocols showing that, in practice, all three achieve constant amortized
reallocations with close to optimal channel usage. Our simulations also expose
interesting trade-offs between reallocations and channel usage. We introduce a
new objective function for WS with reallocations, that can be also applied to
models where reallocations are not possible. We analyze this metric for one of
the algorithms which, to the best of our knowledge, is the first online WS
protocol with theoretical guarantees that applies to scenarios where clients
may leave and the analysis is against current load rather than peak load. Using
previous results, we also observe bounds on channel usage for one of the
algorithms.Comment: 6 figure
Minimum Sum Edge Colorings of Multicycles
In the minimum sum edge coloring problem, we aim to assign natural numbers to
edges of a graph, so that adjacent edges receive different numbers, and the sum
of the numbers assigned to the edges is minimum. The {\em chromatic edge
strength} of a graph is the minimum number of colors required in a minimum sum
edge coloring of this graph. We study the case of multicycles, defined as
cycles with parallel edges, and give a closed-form expression for the chromatic
edge strength of a multicycle, thereby extending a theorem due to Berge. It is
shown that the minimum sum can be achieved with a number of colors equal to the
chromatic index. We also propose simple algorithms for finding a minimum sum
edge coloring of a multicycle. Finally, these results are generalized to a
large family of minimum cost coloring problems
Maximizing Barrier Coverage Lifetime with Mobile Sensors
Sensor networks are ubiquitously used for detection and tracking and as a
result covering is one of the main tasks of such networks. We study the problem
of maximizing the coverage lifetime of a barrier by mobile sensors with limited
battery powers, where the coverage lifetime is the time until there is a
breakdown in coverage due to the death of a sensor. Sensors are first deployed
and then coverage commences. Energy is consumed in proportion to the distance
traveled for mobility, while for coverage, energy is consumed in direct
proportion to the radius of the sensor raised to a constant exponent. We study
two variants which are distinguished by whether the sensing radii are given as
part of the input or can be optimized, the fixed radii problem and the variable
radii problem. We design parametric search algorithms for both problems for the
case where the final order of the sensors is predetermined and for the case
where sensors are initially located at barrier endpoints. In contrast, we show
that the variable radii problem is strongly NP-hard and provide hardness of
approximation results for fixed radii for the case where all the sensors are
initially co-located at an internal point of the barrier
Efficiently Realizing Interval Sequences
We consider the problem of realizable interval-sequences. An interval
sequence comprises of integer intervals such that , and is said to be graphic/realizable if there exists a
graph with degree sequence, say, satisfying the condition
, for each . There is a characterisation
(also implying an verifying algorithm) known for realizability of
interval-sequences, which is a generalization of the Erdos-Gallai
characterisation for graphic sequences. However, given any realizable
interval-sequence, there is no known algorithm for computing a corresponding
graphic certificate in time.
In this paper, we provide an time algorithm for computing a
graphic sequence for any realizable interval sequence. In addition, when the
interval sequence is non-realizable, we show how to find a graphic sequence
having minimum deviation with respect to the given interval sequence, in the
same time. Finally, we consider variants of the problem such as computing the
most regular graphic sequence, and computing a minimum extension of a length
non-graphic sequence to a graphic one.Comment: 19 pages, 1 figur
The Generalized Microscopic Image Reconstruction Problem
This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat [Andrea Frosini and Maurice Nivat, 2007; Nivat, 2002]. Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value l_x, which we would like to determine. However, it might be that obtaining these values precisely (by a surgical probe) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods.
In this paper we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G, and the vector l in R^n assigns a value l_v to each node v. A probe centered at a vertex v will capture a window encompassing its entire neighborhood N[v], i.e., the outcome of a probe centered at v is P_v = sum_{w in N[v]} l_w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector l from the collection of probes, P^- = {P_v | v in V}.
We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector l yielding P). Let us assume that surgical probes (whose outcome at vertex v is the exact value of l_v) are technically available to us (yet are expensive or risky, and must be used sparingly). We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of standard probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP).
Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviors of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k-ary trees, paths, cycles, grids, tori and tubes
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