The major challenge in designing wireless sensor networks (WSNs) is the support of the functional, such as data latency, and the non-functional, such as data integrity, requirements while coping with the computation, energy and communication constraints. Careful node placement can be a very effective optimization means for achieving the desired design goals. In this paper, we report on the current state of the research on optimized node placement in WSNs. We highlight the issues, identify the various objectives and enumerate the different models and formulations. We categorize the placement strategies into static and dynamic depending on whether the optimization is performed at the time of deployment or while the network is operational, respectively. We further classify the published techniques based on the role that the node plays in the network and the primary performance objective considered. The paper also highlights open problems in this area of research.

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Strategies and techniques for node placement in wireless sensor networks: A survey

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

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Analysis of Boolean Functions

We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well-known closest vector problem). The approximation factor we obtain is $n \log^{O(1)} n$ for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1-32] up to the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118-169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high-dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged.

Worst-Case to Average-Case Reductions Based on Gaussian Measures

Recent developments in processor, memory and radio technology have enabled wireless sensor networks which are deployed to collect useful information from an area of interest. The sensed data must be gathered and transmitted to a base station where it is further processed for end-user queries. Since the network consists of low-cost nodes with limited battery power, power efficient methods must be employed for data gathering and aggregation in order to achieve long network lifetimes.In an environment where in a round of communication each of the sensor nodes has data to send to a base station, it is important to minimize the total energy consumed by the system in a round so that the system lifetime is maximized. With the use of data fusion and aggregation techniques, while minimizing the total energy per round, if power consumption per node can be balanced as well, a near optimal data gathering and routing scheme can be achieved in terms of network lifetime.So far, besides the conventional protocol of direct transmission, two elegant protocols called LEACH and PEGASIS have been proposed to maximize the lifetime of a sensor network. In this paper, we propose two new algorithms under name PEDAP (Power Efficient Data gathering and Aggregation Protocol), which are near optimal minimum spanning tree based routing schemes, where one of them is the power-aware version of the other. Our simulation results show that our algorithms perform well both in systems where base station is far away from and where it is in the center of the field. PEDAP achieves between 4x to 20x improvement in network lifetime compared with LEACH, and about three times improvement compared with PEGASIS.

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Power efficient data gathering and aggregation in wireless sensor networks

We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators G : FsrarrFn that fool polynomials over a prime field F: (1) a generator that fools degree-2 (i.e., quadratic) polynomials to within error 1/n, with seed length s = O(log n); (2) a generator that fools degree-3 (i.e., cubic) polynomials to within error epsiv, with seed length s = O(Iog|F| n) + f(epsiv, F) where f depends only on epsiv and F (not on n), (3) assuming the "Gowers inverse conjecture," for every d a generator that fools degree-d polynomials to within error epsiv, with seed length, s = O(dldrIog|F| n) + f(d, epsiv, F) where f depends only on d, epsiv, and F (not on n). We stress that the results in (1) and (2) are unconditional, i.e. do not rely on any unproven assumption. Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove. Our generator for degree-d polynomials is the component-wise sum of d generators for degree-l polynomials (on independent seeds). Prior to our work, generators with logarithmic seed length were only known for degree-1 (i.e., linear) polynomials (Naor and Naor; SIAM J. Comput., 1993). In fact, over small fields such as F2 = {0,1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Velickovic and Wigderson (ISTCS1993), whose seed length is much bigger: s = exp (Omega(radiclogn)), even for the case of degree-2 polynomials over F2.

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Pseudorandom Bits for Polynomials

We consider the problem of positioning data collecting base stations in a sensor network. We show that in general, the choice of positions has a marked influence on the data rate, or equivalently, the power efficiency, of the network. In our model, which is partly motivated by an experimental environmental monitoring system, the optimum data rate for a fixed layout of base stations can be found by a maximum flow algorithm. Finding the optimum layout of base stations, however, turns out to be an NP-complete problem, even in the special case of homogeneous networks. Our analysis of the optimum layout for the special case of the regular grid shows that all layouts that meet certain constraints are equally good. We also consider two classes of random graphs, chosen to model networks that might be realistically encountered, and empirically evaluate the performance of several base station positioning algorithms on instances of these classes. In comparison to manually choosing positions along the periphery of the network or randomly choosing them within the network, the algorithms tested find positions, which significantly improve the data rate and power efficiency of the network.

Power-aware base station positioning for sensor networks

We consider the problem of testing 3-colorability in the bounded-degree model. We show that, for small enough /spl epsiv/, every tester for 3-colorability must have query complexity /spl Omega/(n). This is the first linear lower bound for testing a natural graph property in the bounded-degree model. An /spl Omega/(/spl radic/n) lower bound was previously known. For one-sided error testers, we also show an /spl Omega/(n) lower bound for testers that distinguish 3-colorable graphs from graphs that are (1/3 - /spl alpha/)-far from 3-colorable, for arbitrarily small /spl alpha/. In contrast, a polynomial time algorithm by Frieze and Jerrum (1997) distinguishes 3-colorable graphs from graphs that are 1/5-far from 3-colorable. As a by-product of our techniques, we obtain tight unconditional lower bounds on the approximation ratios achievable by sublinear time algorithms for Max E3SAT, Max E3LIN-2 and other problems.

A lower bound for testing 3-colorability in bounded-degree graphs

We survey the average-case complexity of problems in NP.

We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory.

A major open question is whether the existence of hard-on-average problems in NP can be based on the P ≠ NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case complexity. We discuss some of these "hardness amplification" results.

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Average-case complexity

We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm Using this approach, we obtain the following main constructions of explicitly computable generators $G:\mathbb{F}^s\to\mathbb{F}^n$ that fool polynomials over a finite field $\mathbb{F}$: We stress that the results in (1) and (2) are unconditional, ie, do not rely on any unproven assumption Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove

Our generator for degree-$d$ polynomials is the componentwise sum of $d$ generators for degree-1 polynomials (on independent seeds)

Prior to our work, generators with logarithmic seed length were only known for degree-1 (ie, linear) polynomials [J Naor and M Naor, SIAM J Comput, 22 (1993), pp 838-856] In fact, over small fields such as $\mathbb{F}_2=\{0,1\}$, our results constitute the first progress on these problems since the long-standing generator by Luby, Velickovic, and Wigderson [Deterministic approximate counting of depth-2 circuits, in Proceedings of the 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), 1993, pp 18-24], whose seed length is much bigger: $s=\exp\left(\Omega\left(\sqrt{\log n}\right)\right)$, even for the case of degree-2 polynomials over $\mathbb{F}_2$

Andrej Bogdanov

Papers

Pseudorandom Bits for Polynomials

Power-aware base station positioning for sensor networks

A lower bound for testing 3-colorability in bounded-degree graphs

Average-case complexity

Pseudorandom Bits for Polynomials