Abstract-Sensor nodes may be equipped with a "directional" sensing device (such as a camera) which senses a physical phenomenon in a certain direction depending on the chosen orientation. In this article, we address the problem of selection and orientation of such directional sensors with the objective of maximizing coverage area. Prior works on sensor coverage have largely focused on coverage with sensors that are associated with a unique sensing region. In contrast, directional sensors have multiple sensing regions associated with them, and the orientation of the sensor determines the actual sensing region. Thus, the coverage problems in the context of directional sensors entails selection as well as orientation of sensors needed to activate in order to maximize/ensure coverage.In this article, we address the problem of selecting a minimum number of sensors and assigning orientations such that the given area (or set of target points) is kcovered (i.e., each point is covered k times). The above problem is NP-complete, and even NP-hard to approximate. Thus, we design a simple greedy algorithm that delivers a solution that k-covers at least half of the target points using at most M log(k|C|) sensors, where |C| is the maximum number of target points covered by a sensor and M is the minimum number of sensor required to k-cover all the given points. The above result holds for almost arbitrary sensing regions. We design a distributed implementation of the above algorithm, and study its performance through simulations. In addition to the above problem, we also look at other related coverage problems in the context of directional sensors, and design similar approximation algorithms for them.

Abstract-In this paper, we address several problems that arise in the context of rotating directional sensors. Rotating directional sensors (RDS) have a "directional" coverage region that "rotates" at a certain speed. For RDS with fixed given locations, we address three problems with the objective to minimize different functions of the dark time (i.e., uncovered time) of the given points in the area. In addition, we also consider the problem of placement and orientation of the minimum number of given RDS, so as to reduce the dark time of all given points to zero. Finally, we address the barrier coverage problems wherein we wish to place and/or orient the RDS to ensure "detection" of maximum number of intruders who are attempting to cross the monitored area. We prove the addressed problems to be NP-hard; some of the them are showed to be even NP-hard to approximate. We provide approximation algorithms which are easy to decentralize.

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A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular the limiting probability for no solution is 1/e at the phase transition, over a prime field.We also study the probability of having exactly s solutions, with s ≥ 1. In particular, the probability of a unique solution is asymptotically 1/e if the number of equations equals the number of variables. The probability decreases very rapidly if the number of equations increases or decreases.Our motivation is that many cryptographic systems can be expressed as large multivariate polynomial systems (usually quadratic) over a finite field. Since decoding is unique, the solution of the system must also be unique. Knowing the probability of having exactly one solution may help us to understand more about these cryptographic systems. For example, whether attacks should be evaluated by trying them against random systems depends very much on the likelihood of a unique solution.

Given the exponential increase in broadband cellular traffic it is imperative that scalable traffic measurement and monitoring techniques be developed to aid various resource management methods. In this paper, we use a machine learning technique to learn the underlying conditional dependence and independence structure in the base station traffic loads to show how such probabilistic models can be used to reduce the traffic monitoring efforts. The broad goal is to exploit the model to develop a spatial sampling technique that estimates the loads on all the base stations based on actual measurements only on a small subset of base stations. We take special care to develop a sparse model that focuses on capturing only key dependences. Using trace data collected in a network of 400 base stations we show the effectiveness of this approach in reducing the monitoring effort. To understand the tradeoff between the accuracy and monitoring complexity better, we also study the use of this modeling approach on real applications. Two applications are studied -energy saving and opportunistic scheduling. They show that load estimation via such modeling is quite effective in reducing the monitoring burden.