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.

/pdf/an-efficient-k-means-clustering-algorithm-analysis-and-1vgn3zr7cg.pdf

An efficient k-means clustering algorithm: analysis and implementation

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.

/pdf/the-galois-complexity-of-graph-drawing-why-numerical-7lplbnoc2o.pdf

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

We study balanced circle packings and circle-contact representations for planar graphs, where the ratio of the largest circle's diameter to the smallest circle's diameter is polynomial in the number of circles. We provide a number of positive and negative results for the existence of such balanced configurations.

Balanced Circle Packings for Planar Graphs

We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle's theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.

Tracking Paths in Planar Graphs

Given an off-line sequence S of n set-manipulation operations, we investigate the parallel complexity of evaluating S (i.e. finding the response to every operation in S and returning the resulting set). We show that the problem of evaluating S is in NC for various combinations of common set-manipulation operations. Once we establish membership in NC (or, if membershp in NC is obvious), we develop techniques for improving the time and/or processor complexity.

/pdf/parallel-algorithms-for-evaluating-sequences-of-set-3ck0uzxfda.pdf

Parallel algorithms for evaluating sequences of set manipulation operations (preliminary version)

A dictionary (or map) is a key-value store that requires all keys be unique, and a multimap is a key-value store that allows for multiple values to be associated with the same key. We design hashing-based indexing schemes for dictionaries and multimaps that achieve worst-case optimal performance for lookups and updates, with a small or negligible probability the data structure will require a rehash operation, depending on whether we are working in the the external-memory (I/O) model or one of the well-known versions of the Random Access Machine (RAM) model. One of the main features of our constructions is that they are \emph{fully de-amortized}, meaning that their performance bounds hold without one having to tune their constructions with certain performance parameters, such as the constant factors in the exponents of failure probabilities or, in the case of the external-memory model, the size of blocks or cache lines and the size of internal memory (i.e., our external-memory algorithms are cache oblivious). Our solutions are based on a fully de-amortized implementation of cuckoo hashing, which may be of independent interest. This hashing scheme uses two cuckoo hash tables, one "nested" inside the other, with one serving as a primary structure and the other serving as an auxiliary supporting queue/stash structure that is super-sized with respect to traditional auxiliary structures but nevertheless adds negligible storage to our scheme. This auxiliary structure allows the success probability for cuckoo hashing to be very high, which is useful in cryptographic or data-intensive applications.

Michael T. Goodrich

Papers

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

Balanced Circle Packings for Planar Graphs

Tracking Paths in Planar Graphs

Parallel algorithms for evaluating sequences of set manipulation operations (preliminary version)

Fully De-Amortized Cuckoo Hashing for Cache-Oblivious Dictionaries and Multimaps