In this paper, we study strategies for allocating and managing friendly jammers, so as to create virtual barriers that would prevent hostile eavesdroppers from tapping sensitive wireless communication. Our scheme precludes the use of any encryption technique. Applications include domains such as (i) protecting the privacy of storage locations where RFID tags are used for item identification, (ii) secure reading of RFID tags embedded in credit cards, (iii) protecting data transmitted through wireless networks, sensor networks, etc. By carefully managing jammers to produce noise, we show how to reduce the SINR of eavesdroppers to below a threshold for successful reception, without jeopardizing network performance.We present algorithms targeted towards optimizing power allocation and number of jammers needed in several settings. Experimental simulations back up our results.
We perform importance sampling for a randomized matrix multiplication algorithm by Drineas, Kannan, and Mahoney and derive probabilities that minimize the expected value (with regard to the distributions of the matrix elements) of the variance. We compare these optimized probabilities with uniform probabilities and derive conditions under which the actual variance of the optimized probabilities is lower. Numerical experiments with query matching in information retrieval applications illustrate that the optimized probabilities produce more accurate matchings than the uniform probabilities and that they can also be computed efficiently.
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X, d X , μ X ) satisfying a 2-Poincaré inequality. Given a bounded domain ⊂ X with μ X (X \ ) > 0, and a function f in the Besov class B θ 2,2 (X) ∩ L 2 (X), we study the problem of finding a functionWe show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on , and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
We consider the problem of computing k shortest paths in a two-dimensional environment with polygonal obstacles, where the jth path, for 1 j k, is the shortest path in the free space that is also homotopically distinct from each of the first j 1 paths. In fact, we consider a more general problem: given a source point s, construct a partition of the free space, called the kth shortest path map (k-SPM), in which the homotopy of the kth shortest path in a region has the same structure. Our main combinatorial result establishes a tight bound of ⇥(k 2 h + kn) on the worst-case complexity of this map. We also describe an O((k 3 h + k 2 n) log (kn)) time algorithm for constructing the map. In fact, the algorithm constructs the jth map for every j k. Finally, we present a simple visibility-based algorithm for computing the k shortest paths between two fixed points. This algorithm runs in O(m log n + k) time and uses O(m + k) space, where m is the size of the visibility graph. This latter algorithm can be extended to compute k shortest simple (non-self-intersecting) paths, taking O(k 2 m(m + kn) log(kn)) time. We invite the reader to play with our applet demonstrating k-SPMs [10].
We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincaré inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new "thickening" construction, which can be used to enlarge subsets into spaces admitting Poincaré inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincaré inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincaré inequalities for the class of M CP (K, n)-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincaré inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincaré inequalities "self-improve" to classical (1, q)-Poincaré inequalities for some q ∈ [1, ∞), which is related to Keith's and Zhong's theorem on selfimprovement of Poincaré inequalities.
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