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 show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in O(logn(loglogn)d-l) time using linear work on an EREW PRAM. A key component in these algorithms is a new, efficient parallel method for constructing c-nets and c-approximations (which have wide applicability in computational geometry). In addition, we introduce a new deterministic set approximation for range spaces with finite VC-exponent, which we call the b-relative c-approtimation, and we show how such approximations can be efficiently constructed in parallel.

Fixed-dimensional parallel linear programming via relative e-approximations

We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph embedded in the plane into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D0(G)), where n and m are the number of vertices and edges of the graph G, and D0(G) is the time it takes to obtain an initial embedding of G in the plane. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n×n grid and the running time reduces to O(n log n). Communicated by G. Liotta and S. H. Whitesides: submitted November 1998; revised November 1999. Research supported in part by ARO grant DAAH04–96–1–0013 and NSF grant CCR9732300. Duncan, Goodrich, and Kobourov, BAR Trees , JGAA, 4(3) 19–46 (2000) 20

Balanced aspect ratio trees and their use for drawing large graphs

We describe an approach for visually teaching important proofs in the Junior-Senior level course on the design and analysis of data structures and algorithms (CS7/DS&A). The main idea of this educational paradigm is to justify important claims about data structures and algorithms by using pictures that visualize proofs so clearly that the pictures can qualify as proofs themselves. The advantage of using this approach for DS&A is that it augments or even replaces inductive arguments that many students find difficult. Moreover, this paradigm communicates important algorithmic facts in a compelling way for students who are more visually-oriented. We illustrate this technique by giving examples of visual proofs of several key concepts in DS&A.

Teaching the analysis of algorithms with visual proofs

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

We propose a bandwidth-efficient algorithmic solution for perfectly-secure communication in the absence of secure infrastructure. Our solution involves connecting vertex pairs by a set of k edge-disjoint paths (a structure we call a k-system) where k is a parameter determined by the connectivity of the network. This structure is resilient to adversaries with bounded eavesdropping capability. To ensure that bandwidth is efficiently used we consider connection requests as inputs to the k-Edge Disjoint Path Coloring Problem (k-EDPCOL), a generalization of the Path Coloring Problem, in which each vertex pair is connected by a k-system, and each k-system is assigned a color such that two overlapping k-systems do not have the same color. The objective is to minimize the number of colors. We give a distributed and competitive online algorithm for k-EDPCOL. Additionally, since security applications are our focus we prove that a malicious adversary which attacks the algorithm during the process of construction of a k-system cannot learn anything more than if it had attacked the k-system once it was built.

Michael T. Goodrich

Papers

Fixed-dimensional parallel linear programming via relative e-approximations

Balanced aspect ratio trees and their use for drawing large graphs

Teaching the analysis of algorithms with visual proofs

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Constructing disjoint paths for secure communication