The (maximum receiver-centric) interference of a geometric graph (von (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^ 1 3 ); (2) no connected graph with vertex set V has interference o((log n)^ 1 4 ); and (3) the minimum spanning tree of @math has interference Theta(( n)^ 1 2 ).
log suggests that point sets with very high interference are unlikely to occur in practice. This intuition is born out by the results of Kranakis al @cite_7 , who show that high interference is unlikely to occur in random point sets in one dimension: Note that, in this one-dimensional case, the minimum spanning tree, @math , is simply a path that connects the points of @math in order, from left to right. Taken together, Part 1 of Theorems and generalize sqrtlogn to arbitrary constant dimensions @math .
@cite_7
1942343061
(MATH) Let P be a set of n points in @math k (P) denote the minimum over all k-flats @math of max peP Dist(p, ). We present an algorithm that computes, for any 0 k (P) from each point of P. The running time of the algorithm is dnO(k e5log(1 e)). The crucial step in obtaining this algorithm is a structural result that says that there is a near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and e but is independent of the dimension.This approach also extends to the case where we want to find a k-flat that is close to a prescribed fraction of the entire point set, and to the case where we want to find j flats, each of dimension k, that are close to the point set. No efficient approximation schemes were known for these problems in high-dimensions, when k>1 or j>1. The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and @math . By employing a subroutine that solves the post office problem, we show that, for fixed k @math 3, such a minimum spanning tree can be found in time O( @math ), where a(k) = @math . The bound can be improved to O( @math ) for points in the 3-dimensional Euclidean space. We also obtain o( @math ) algorithms for finding a farthest pair in a set of n points and for other related problems. Given @math wireless transceivers located in a plane, a fundamental problem in wireless communications is to construct a strongly connected digraph on them such that the constituent links can be scheduled in fewest possible time slots, assuming the SINR model of interference. In this paper, we provide an algorithm that connects an arbitrary point set in @math slots, improving on the previous best bound of @math due to Moscibroda. This is complemented with a super-constant lower bound on our approach to connectivity. An important feature is that the algorithms allow for bi-directional (half-duplex) communication. One implication of this result is an improved bound of @math on the worst-case capacity of wireless networks, matching the best bound known for the extensively studied average-case. We explore the utility of oblivious power assignments, and show that essentially all such assignments result in a worst case bound of @math slots for connectivity. This rules out a recent claim of a @math bound using oblivious power. On the other hand, using our result we show that @math slots suffice, where @math is the ratio between the largest and the smallest links in a minimum spanning tree of the points. Our results extend to the related problem of minimum latency aggregation scheduling, where we show that aggregation scheduling with @math latency is possible, improving upon the previous best known latency of @math . We also initiate the study of network design problems in the SINR model beyond strong connectivity, obtaining similar bounds for biconnected and @math -edge connected structures. A dynamic geometric data stream is a sequence of m ADD REMOVE operations of points from a discrete geometric space 1,…, Δ d ?. ADD (p) inserts a point p from 1,…, Δ d into the current point set P, REMOVE(p) deletes p from P. We develop low-storage data structures to (i) maintain e-nets and e-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + e)-approximation of the weight of the Euclidean minimum spanning tree of P. Our data structure for e-nets uses bits of memory and returns with probability 1 – δ a set of points that is an e-net for an arbitrary fixed finite range space with VC-dimension . Our data structure for e-approximations uses bits of memory and returns with probability 1 – δ a set of points that is an e-approximation for an arbitrary fixed finite range space with VC-dimension . The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O(log(1 δ)(log Δ e)O(d)) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P. The interference at a wireless node s can be modelled by the number of wireless nodes whose transmission ranges cover s. Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (equivalently, a power level) to each node such that the resulting communication graph is connected, while minimizing the maximum interference. We consider the model introduced by von (2005), in which each transmission range is represented by a ball and edges in the communication graph are symmetric. The problem is NP-complete in two dimensions (Buchin 2008) and no polynomial-time approximation algorithm is known. Furthermore, even in one dimension (the highway model), the problem's complexity is unknown and the maximum interference of a set of n wireless nodes can be as high as Theta(sqrt(n)) (von 2005). In this paper we show how to solve the problem efficiently in settings typical for wireless ad hoc networks. In particular, we show that if node positions are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, for any fixed d, then the topology given by the closure of the Euclidean minimum spanning tree of P has maximum interference O(log n) with high probability. We extend this bound to a general class of communication graphs over a broad set of probability distributions. Next we present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on the expected maximum interference. Finally, we discuss an empirical evaluation of our algorithm with a suite of simulation results. We give the first efficient (1−e)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ℝ d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order ( ( 1 n ) ). Our algorithm finds an empty axis-aligned box whose volume is at least (1−e) of the maximum in O((8ede −2) d ⋅nlog d n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant. This paper discusses the topic of dimensionality reduction for @math -means clustering. We prove that any set of @math points in @math dimensions (rows in a matrix @math ) can be projected into @math dimensions, for any @math , in @math time, such that with constant probability the optimal @math -partition of the point set is preserved within a factor of @math . The projection is done by post-multiplying @math with a @math random matrix @math having entries @math or @math with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results. A t-spanner is a graph on a set of points Swith the following property: Between any pair of points there is a path in the spanner whose total length is at most ttimes the actual distance between the points. In this paper, we consider points residing in a metric space equipped with doubling dimension i¾?, and show how to construct a dynamic (1 + i¾?)-spanner with degree i¾?i¾? O(i¾?)in @math update time. When i¾?and i¾?are taken as constants, the degree and update times are optimal. Given a set of @math n points, each is painted by one of the @math k given colors, we want to choose @math k points with distinct colors to form a color spanning set. For each color spanning set, we can construct the convex hull and the smallest axis-aligned enclosing rectangle, etc. Assuming that each point is chosen independently and identically from the subset of points of the same color, we propose an @math O ( n 2 ) time algorithm to compute the expected area of convex hulls of the color spanning sets and an @math O ( n 2 ) time algorithm to compute the expected perimeter of convex hulls of the color spanning sets. For the expected perimeter (resp. area) of the smallest perimeter (resp. area) axis-aligned enclosing rectangles of the color spanning sets, we present an @math O ( n log n ) (resp. @math O ( n 2 ) ) time algorithm. We also propose a simple approximation algorithm to compute the expected diameter of the color spanning sets. For the expected distance of the closest pair, we show that it is @math # P-complete to compute and there exists no polynomial time @math 2 n 1 - ? approximation algorithm to compute the probability that the closest pair distance of all color spanning sets equals to a given value @math d unless @math P = N P , even in one dimension and when each color paints two points. We give processor-allocation algorithms for grid architectures, where the objective is to select processors from a set of available processors to minimize the average number of communication hops. The associated clustering problem is as follows: Given n points in ℜ d, find a size-k subset with minimum average pairwise L 1 distance. We present a natural approximation algorithm and show that it is a @math -approximation for two-dimensional grids; in d dimensions, the approximation guarantee is @math , which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d, and we report on experimental results. We consider the functions @math defined as the @math th partial derivative of Lebesgue's singular function @math with respect to @math at @math . This sequence includes a multiple of the Takagi function as the case @math . We show that @math is continuous but nowhere differentiable for each @math , and determine the Holder order of @math . From this, we derive that the Hausdorff dimension of the graph of @math is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for @math using the binary expansion of @math , and use this to find the sets of points where @math and @math take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of @math and @math . We consider a fundamental problem in unsupervised learning called : given a collection of @math points in @math , if many but not necessarily all of these points are contained in a @math -dimensional subspace @math can we find it? The points contained in @math are called inliers and the remaining points are outliers . This problem has received considerable attention in computer science and in statistics. Yet efficient algorithms from computer science are not robust to adversarial outliers, and the estimators from robust statistics are hard to compute in high dimensions. Are there algorithms for subspace recovery that are both robust to outliers and efficient? We give an algorithm that finds @math when it contains more than a @math fraction of the points. Hence, for say @math this estimator is both easy to compute and well-behaved when there are a constant fraction of outliers. We prove that it is Small Set Expansion hard to find @math when the fraction of errors is any larger, thus giving evidence that our estimator is an optimal compromise between efficiency and robustness. As it turns out, this basic problem has a surprising number of connections to other areas including small set expansion, matroid theory and functional analysis that we make use of here. We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by in the case of a two-dimensional Poisson point process. It is a version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by Hirsch, Jahnel and Cali. We study the SINR graph model for a stationary Cox point process in two or higher dimensions. We show that under suitable moment or boundedness conditions on the path-loss function and the intensity measure, this graph has an infinite connected component if the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds in all dimensions larger than 1 if the intensity measure is asymptotically essentially connected, and in two dimensions also if the intensity measure is only stabilizing but the connection radius is large. A prominent example of the intensity measure is the two-dimensional Poisson-Voronoi tessellation. We show that its total edge length in a given square has some exponential moments. We conclude that its SINR graph has an infinite cluster if the path-loss function is bounded and has a power-law decay of exponent at least 3. We show that the greedy spanner algorithm constructs a @math -spanner of weight @math for a point set in metrics of doubling dimension @math , resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in @math -dimension Euclidean space has a @math -spanner of weight at most @math . Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space. In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a . In the SINR model, the resulting partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. We consider the general case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques as well as the theoretical challenge of understanding the geometry of SINR diagrams. We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line. In addition, one of our key results concerns the behavior of a @math -dimensional map. Specifically, although the @math -dimensional map might be highly fractured, drawing the map in one dimension higher "heals" the zones, which become connected. In addition, as a step toward establishing a weaker form of convexity for the @math -dimensional map, we study the interference function and show that it satisfies the maximum principle. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location. Given a set of n points in d-dimensional Euclidean space, S ⊂ E, and a query point q ∈ E, we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as efficiently as possible. We assume that the dimension d is a constant independent of n. Although reasonably good solutions to this problem exist when d is small, as d increases the performance of these algorithms degrades rapidly. We present a randomized algorithm for approximate nearest neighbor searching. Given any set of n points S ⊂ E, and a constant ǫ > 0, we produce a data structure, such that given any query point, a point of S will be reported whose distance from the query point is at most a factor of (1 + ǫ) from that of the true nearest neighbor. Our algorithm runs in O(log n) expected time and requires O(n log n) space. The data structure can be built in O(n) expected time. The constant factors depend on d and ǫ. Because of the practical importance of nearest neighbor searching in higher dimensions, we have implemented a practical variant of this algorithm, and show empirically that for many point distributions this variant of the algorithm finds the nearest neighbor in moderately large dimension significantly faster than existing practical approaches. We study the asymptotic behavior of the persistent homology of i.i.d. samples from a @math -Ahlfors regular measure --- one that satisfies uniform bounds of the form for some @math all @math in the support of @math and all sufficiently small @math Our main result is that if @math are sampled from a @math -Ahlfors regular measure on @math and @math denotes the @math -weight of the minimal spanning tree on @math [E_ (x_1, ,x_n )= e T (x_1, ,x_n ) |e|^ ] then [E_ (x_1, ,x_n ) n^ d- d ] with high probability as @math We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology. As an application, we exhibit hypotheses under which the fractal dimension of a measure can be computed from the persistent homology of i.i.d. samples from that space, in a manner similar to that proposed in the experimental work of (2018). We prove lower bounds on the complexity of finding @math -stationary points (points @math such that @math ) of smooth, high-dimensional, and potentially non-convex functions @math . We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of @math at a query point @math . We show that for any (potentially randomized) algorithm @math , there exists a function @math with Lipschitz @math th order derivatives such that @math requires at least @math queries to find an @math -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized @math th order regularization are worst-case optimal within their natural function classes. We study an interference network where equally-numbered transmitters and receivers lie on two parallel lines, each transmitter opposite its intended receiver. We consider two short-range interference models: the "asymmetric network," where the signal sent by each transmitter is interfered only by the signal sent by its left neighbor (if present), and a "symmetric network," where it is interfered by both its left and its right neighbors. Each transmitter is cognizant of its own message, the messages of the @math transmitters to its left, and the messages of the @math transmitters to its right. Each receiver decodes its message based on the signals received at its own antenna, at the @math receive antennas to its left, and the @math receive antennas to its right. For such networks we provide upper and lower bounds on the multiplexing gain, i.e., on the high-SNR asymptotic logarithmic growth of the sum-rate capacity. In some cases our bounds meet, e.g., for the asymmetric network. Our results exhibit an equivalence between the transmitter side-information parameters @math and the receiver side-information parameters @math in the sense that increasing decreasing @math or @math by a positive integer @math has the same effect on the multiplexing gain as increasing decreasing @math or @math by @math . Moreover---even in asymmetric networks---there is an equivalence between the left side-information parameters @math and the right side-information parameters @math .