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

We present an algorithm for the hidden-surface elimination problem for rectangles, which is also known as window rendering. The time complexity of our algorithm is dependent on both the number of input rectangles, n, and on the size of the output, κ. Our algorithm obtains a trade-off between these two components, in that its running time is O(r(n1+1/r+κ)), where 1≤r≤log n is a tunable parameter. By using this method while adjusting the parameter r “on the fly” one can achieve a running time that is O(n log n+κ(log n/log(1+κ/n))). Note that when κ is Θ(n), this achieves an O(n log n) running time, and when κ is Θ(n1+e) for any positive constant ɛ, then this achieves an O(κ) running time, both of which are optimal.

An Input-Size/Output-Size Trade-Off in the Time-Complexity of Rectilinear Hidden Surface Removal (Preliminary Version)

We introduce the notion of Lombardi graph drawings, named after the American abstract artistMark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.

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Lombardi drawings of graphs

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An input-size/output-size trade-off in the time-complexity of rectilinear hidden surface removal

We present two new approaches to improving the integrity of network broadcasts and multicasts with low storage and computation overhead. The first approach is a leapfrog linking protocol for securing the integrity of packets as they traverse a network during a broadcast, such as in the setup phase for link-state routing. This technique allows each router to gain confidence about the integrity of a packet before passing it on to the next router; hence, allows many integrity violations to be stopped immediately in their tracks. The second approach is a novel key predistribution scheme that we use in conjunction with a small number of hashed message authentication codes (HMAC), which allows end-to-end integrity checking as well as improved hop-by-hop integrity checking. Our schemes are suited to environments, such as in ad hoc and overlay networks, where routers can share only a small number of symmetric keys. Moreover, our protocols do not use encryption (which, of course, can be added as an optional security enhancement). Instead, security is based strictly on the use of one-way hash functions; hence, our algorithms are considerably faster than those based on traditional public-key signature schemes. This improvement in speed comes with only modest reductions in the security for broadcasting, as our schemes can tolerate small numbers of malicious routers, provided they do not form significant cooperating coalitions.

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Leap-frog packet linking and diverse key distributions for improved integrity in network broadcasts

We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

Michael T. Goodrich

Papers

An Input-Size/Output-Size Trade-Off in the Time-Complexity of Rectilinear Hidden Surface Removal (Preliminary Version)

Lombardi drawings of graphs

An input-size/output-size trade-off in the time-complexity of rectilinear hidden surface removal

Leap-frog packet linking and diverse key distributions for improved integrity in network broadcasts

Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area