There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

This paper provides an overview of the Internet of Things (IoT) with emphasis on enabling technologies, protocols, and application issues. The IoT is enabled by the latest developments in RFID, smart sensors, communication technologies, and Internet protocols. The basic premise is to have smart sensors collaborate directly without human involvement to deliver a new class of applications. The current revolution in Internet, mobile, and machine-to-machine (M2M) technologies can be seen as the first phase of the IoT. In the coming years, the IoT is expected to bridge diverse technologies to enable new applications by connecting physical objects together in support of intelligent decision making. This paper starts by providing a horizontal overview of the IoT. Then, we give an overview of some technical details that pertain to the IoT enabling technologies, protocols, and applications. Compared to other survey papers in the field, our objective is to provide a more thorough summary of the most relevant protocols and application issues to enable researchers and application developers to get up to speed quickly on how the different protocols fit together to deliver desired functionalities without having to go through RFCs and the standards specifications. We also provide an overview of some of the key IoT challenges presented in the recent literature and provide a summary of related research work. Moreover, we explore the relation between the IoT and other emerging technologies including big data analytics and cloud and fog computing. We also present the need for better horizontal integration among IoT services. Finally, we present detailed service use-cases to illustrate how the different protocols presented in the paper fit together to deliver desired IoT services.

Internet of Things: A Survey on Enabling Technologies, Protocols, and Applications

In k-means clustering, we are given a set of n data points in d-dimensional space R/sup d/ and an integer k and the problem is to determine a set of k points in Rd, called centers, so as to minimize the mean squared distance from each data point to its nearest center. A popular heuristic for k-means clustering is Lloyd's (1982) algorithm. We present a simple and efficient implementation of Lloyd's k-means clustering algorithm, which we call the filtering algorithm. This algorithm is easy to implement, requiring a kd-tree as the only major data structure. We establish the practical efficiency of the filtering algorithm in two ways. First, we present a data-sensitive analysis of the algorithm's running time, which shows that the algorithm runs faster as the separation between clusters increases. Second, we present a number of empirical studies both on synthetically generated data and on real data sets from applications in color quantization, data compression, and image segmentation.

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An efficient k-means clustering algorithm: analysis and implementation

Dating back to two famous experiments by the social-psychologist, Stanley Milgram, in the 1960s, the small-world phenomenon is the idea that all people are connected through a short chain of acquaintances that can be used to route messages. Many subsequent papers have attempted to model this phenomenon, with most concentrating on the "short chain" of acquantances rather than their ability to efficiently route messages. For example, a well-known preferential attachment model by Barabási and Albert provides a mathematical explanation of how a social network can have small diameter---hence, short chains between participants--- but this model doesn't explain how they can route messages. A notable exception is a well-known model by Jon Kleinberg, which shows that it is possible for participants in a n × n grid to route a message in O(log2 n) hops by augmenting the grid with a small number of long-range random links and using a simple greedy routing strategy. Although Kleinberg's model is intriguing, it does not take into account the road network of the United States used in the original Milgram experiments and its O(log2 n) number of hops for messages is actually quite far from the average of six hops for successful messages observed by Milgram in his experiments, which gave rise to the "six-degrees-of-separation" expression. In this paper, we study the small-world navigability of the U.S. road network, with the goal of providing a model that explains how messages in the original small-world experiments could be routed along short paths using U.S. roads. To this end, we introduce the Neighborhood Preferential Attachment model, which combines elements from Kleinberg's model and the Barabási-Albert model, such that long-range links are chosen according to both the degrees and (road-network) distances of vertices in the network. We empirically evaluate all three models by running a decentralized routing algorithm, where each vertex only has knowledge of its own neighbors, and find that our model outperforms both of these models in terms of the average hop length. Moreover, our experiments indicate that similar to the Barabási-Albert model, networks generated by our model are scale-free, which could be a more realistic representation of acquaintanceship links in the original small-world experiment.

/pdf/modeling-the-small-world-phenomenon-with-road-networks-3p7xe7sh.pdf

Modeling the small-world phenomenon with road networks

In this paper, we study the parallel query complexity of reconstructing biological and digital phylogenetic trees from simple queries involving their nodes. This is motivated from computational biology, data protection, and computer security settings, which can be abstracted in terms of two parties, a responder, Alice, who must correctly answer queries of a given type regarding a degree-d tree, T, and a querier, Bob, who issues batches of queries, with each query in a batch being independent of the others, so as to eventually infer the structure of T. We show that a querier can efficiently reconstruct an n-node degree-d tree, T, with a logarithmic number of rounds and quasilinear number of queries, with high probability, for various types of queries, including relative-distance queries and path queries. Our results are all asymptotically optimal and improve the asymptotic (sequential) query complexity for one of the problems we study. Moreover, through an experimental analysis using both real-world and synthetic data, we provide empirical evidence that our algorithms provide significant parallel speedups while also improving the total query complexities for the problems we study.

Reconstructing Biological and Digital Phylogenetic Trees in Parallel.

We describe a data structure, called a priority range tree, which accommodates fast orthogonal range reporting queries on prioritized points. Let S be a set of n points in the plane, where each point p in S is assigned a weight w(p) that is polynomial in n, and define the rank of p to be \(r(p)=\lfloor \log w(p) \rfloor\). Then the priority range tree can be used to report all points in a three- or four-sided query range R with rank at least \(\lfloor \log w \rfloor\) in time O(logW/w + k), and report k highest-rank points in R in time O(loglogn + logW/w′ + k), where W = ∑ p ∈ S w(p), w′ is the smallest weight of any point reported, and k is the output size. All times assume the standard RAM model of computation. If the query range of interest is three sided, then the priority range tree occupies O(n) space, otherwise O(nlogn) space is used to answer four-sided queries. These queries are motivated by the Weber–Fechner Law, which states that humans perceive and interpret data on a logarithmic scale.

Priority Range Trees

This paper studies a discrepancy-sensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric.

Michael T. Goodrich

Papers

Modeling the small-world phenomenon with road networks

Reconstructing Biological and Digital Phylogenetic Trees in Parallel.

Priority Range Trees

Discrepancy-Sensitive Dynamic Fractional Cascading, Dominated Maxima Searching, and 2-d Nearest Neighbors in Any Minkowski Metric

Data Structures and Algorithms in Java 6th Edition International Student Version