Bayesian optimisation (BO) is a powerful framework for global optimisation of costly functions, using predictions from Gaussian process models (GPs). In this work, we apply BO to functions that exhibit invariance to a known group of transformations. We show that vanilla and constrained BO algorithms are inefficient when optimising such invariant objectives, and provide a method for incorporating group invariances into the kernel of the GP to produce invariance-aware algorithms that achieve significant improvements in sample efficiency. We derive a bound on the maximum information gain of these invariant kernels, and provide novel upper and lower bounds on the number of observations required for invariance-aware BO algorithms to achieve $\epsilon$-optimality. We demonstrate our method's improved performance on a range of synthetic invariant and quasi-invariant functions. We also apply our method in the case where only some of the invariance is incorporated into the kernel, and find that these kernels achieve similar gains in sample efficiency at significantly reduced computational cost. Finally, we use invariant BO to design a current drive system for a nuclear fusion reactor, finding a high-performance solution where non-invariant methods failed.

Abstract Various types of errors can propagate in networks, and they are usually hard to diagnose. For example, social networks spread rumors as well as useful information. Computer networks can spread Internet worms or malicious packets. In many cases, it is very hard to find the root cause (a.k.a. initial rumor spreader) of such errors without complete knowledge of the error propagation. We aim to find the root cause node when there is limited information about error propagation. We assume that there are very small number of monitor nodes in the network reporting whether error reached them or not. With this assumption, we first propose an algorithm that finds the most probable root cause node. Second, to improve the accuracy of root cause analysis, we propose another algorithm that makes use of timestamp of error reception. Finally, we study how to select monitors effectively so that root cause analysis can be accurate. With real networks from various domains, our algorithms are shown to be very effective. Copyright © 2016 John Wiley & Sons, Ltd.

As sensor mobility becomes more and more universal, Wireless Sensor Network (WSN) configurations that utilize such mobility will become the norm. We consider the problem of maximizing the lifetime of a wireless connection between a transmitter and a receiver using mobile relays. Initially, all relays are positioned arbitrarily on the line between the transmitter and the receiver and have arbitrary battery capacities. Energy is consumed in proportion to the distance traveled for mobility and in proportion to an exponential function of the distance over which information is sent for communication. Relays can move to different locations as long as they have the energy to do so. The objective is to find positions and thus transmission ranges for the nodes that maximize the lifetime of the network. We study two models. The first is more restrictive, and corresponds to the case where relays are allowed to be set once at time zero (single deployment), while the second model corresponds to the case where relays can be adjusted multiple times (multiple deployments). We show how to compute an optimal solution for the case of no movement cost for both models. We consider a discrete version of the single deployment model, in which relays must be deployed on grid points. We provide two algorithms for this case: a dynamic programming algorithm and a binary search algorithm on potential lifetimes. We prove that both algorithms are FPTASs for the non-discrete problem, if batteries are not too small. Based on these algorithms and on additional ideas we develop a number of heuristics for the multiple deployments model. We evaluate them using simulations and compare them with the lower bound of relays not moving at all and the upper bound of cost-free movement. Our simulations - across a range of mobility and transmission costs, sensible starting locations and battery capacities - demonstrate the benefit of moving over remaining at initial locations even for single deployment.

Positioning disk-shaped sensors to optimize certain coverage parameters is a fundamental problem in ad hoc sensor networks. The hexagon lattice arrangement is known to be optimally efficient in the plane, even though 20.9% of the area is unnecessarily covered twice, however, the arrangement is very rigid—any movement of a sensor from its designated grid position (due to, e.g., placement error or obstacle avoidance) leaves some region uncovered, as would the failure of any one sensor. In this article, we consider how to arrange sensors in order to guarantee multiple coverage, that is, k -coverage for some value k > 1. A naive approach is to superimpose multiple hexagon lattices, but for robustness reasons, we may wish to space sensors evenly apart. We present two arrangement methods for k -coverage: (1) optimizing a Riesz energy function in order to evenly distribute nodes, and (2) simply shrinking the hexagon lattice and making it denser. The first method often approximates the second, and so we focus on the latter. We show that a density increase tantamount to k copies of the lattice can yield k ′-coverage, for k ′ > k (e.g., k = 11, k ′ = 12 and k = 21, k ′ = 24), by exploiting the double-coverage regions. Our examples' savings provably converge in the limit to the ≈ 20.9% maximum. We also provide analogous results for the square lattice and its ≈ 57% inefficiency (e.g., k = 3, k ′ = 4 and k =5, k ′ = 7) and show that for multi-coverage for some values of k ′, the square lattice can actually be more efficient than the hexagon lattice. We also explore other benefits of shrinking the lattice: Doing so allows all sensors to move about their intended positions independently while nonetheless guaranteeing full coverage and can also allow us to tolerate probabilistic sensor failure when providing 1-coverage or k -coverage. We conclude by construing the shrinking factor as a budget to be divided among these three benefits.

When a sensor network is deployed, it is typically required to support multiple simultaneous missions. Schemes that assign sensing resources to missions thus become necessary. In this article, we formally define the sensor-mission assignment problem and discuss some of its variants. In its most general form, this problem is NP-hard. We propose algorithms for the different variants, some of which include approximation guarantees. We also propose distributed algorithms to assign sensors to missions which we adapt to include energy-awareness to extend network lifetime. Finally, we show comprehensive simulation results comparing these solutions to an upper bound on the optimal solution.

Positioning disk-shaped sensors to optimize certain coverage parameters is a fundamental problem in ad-hoc sensor networks. The hexagon grid lattice is known to be optimally efficient, but the 20.9% of the area covered by two sensors may be considered a waste. Furthermore, any movement of a sensor from its designated grid position or sensor failure, due to placement error or obstacle avoidance, leaves some region uncovered, as would the failure of any one sensor. We explore how shrinking the grid can help to remedy these shortcomings. First, shrinking to obtain a denser hexagonal lattice allows all sensors to move about their intended positions independently while nonetheless guaranteeing full coverage. Second, sufficiently increasing the lattice density will naturally yield k-coverage for k > 1. Moreover, we show that a density increase tantamount to fc copies of the lattice can yield k' -coverage, for kj> k (e.g. k = 11, kj= 12), through the exploitation of the double-coverage regions. Our examples' savings provably converge in the limit to the ap 20.9% maximum. We also provide analogous results for the square lattice and its ap 57% inefficiency, including k = 3, kj= 4, k = 5,kj= 7, indicating that for multi-coverage, the square lattice can actually be more efficient than the hexagon lattice. All these efficiency gains can be used to provide 1-coverage or fc-coverage even in the face of probabilistic sensor failure. We conclude by construing the shrinking factor as a budget to be divided among these three benefits.

We consider a sensor network placement problem where the sensing range of a sensor depends on its location in order to model the effect of terrain features. We study how sensors should be placed in order to maximize the coverage and illustrate how digital halftoning algorithms from the field of image processing can be useful in this respect. In particular, we reduce the sensor placement problem to a corresponding image halftoning problem and then apply two well known halftoning algorithms to the problem: dither mask halftoning and direct binary search. We illustrate our approach with experimental results and show that this approach is also applicable to the problem of preferential coverage.

When a sensor network is deployed in the field, it is typically required to support multiple simultaneous missions, which may start and finish at different times. Schemes that match sensor resources to mission demands thus become necessary. In this paper, we propose centralized and distributed schemes to assign sensors to missions. We also adapt our distributed scheme to make it energy-aware to extend network lifetime. Finally, we show simulation results comparing these solutions. We find that our greedy algorithm frequently performs near-optimally and that the distributed schemes usually perform nearly as well.

An ad-hoc sensor network is composed of sensing devices which can measure or detect features of their environment, communicate with one other and possibly with other devices that perform data fusion. One of the problems motivated by ad-hoc sensor networks is to position sensors in order to maximize coverage, or equivalently to minimize the number of sensors required to cover a given area.

Multiple-instruction multiple-data (MIMD) algorithms that use multiple processors to do median splitting, k-splitting and parallel splitting into t equal sections are presented. Both concurrent read, exclusive write (CREW) and exclusive read, exclusive write (EREW) versions of the algorithms are given. It is shown that a k-splitting problem can be easily converted into a median-splitting problem. Methods for finding multiple split points quickly and application of k-splitting to merging and sorting are discussed.>

~STRACr.An analytic-numeric procedure for fmdmg the response time for a givenjob size ("condiUonal response") Is demonstrated for a wide class of quanta-of-service allocation disciplines The procedure is demonstrated for a discipline that is like standard round robin except that the queue of waiting jobs is ordered by the number of services previously received.Laplace-Stieltjes transforms (LST) are found for this "ordered round robin" for an M/G/I queue allowing allocation of nonidentical quanta of service for a job.The denvauon of the mean is shown explicitly, and higher moments can be derived by differentiation of the LST The mean conditional response is found for the steady state, and tt is shown that the method is capable as well of determining the mean conditional response under specified initial condmons.The methodology can be broadly applied.It applies to a class of disciplines that includes the ordered round robin and feedback to lower priority queues, as well as variants and combinaUons of these.