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

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

/pdf/finding-community-structure-in-networks-using-the-4em5so8glh.pdf

Finding community structure in networks using the eigenvectors of matrices

We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

/pdf/approximate-nearest-neighbors-towards-removing-the-curse-of-27lmlz5prr.pdf

Approximate nearest neighbors: towards removing the curse of dimensionality

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

Recently, He and Yesha gave an algorithm for recognizing directed series-parallel graphs in time O(log2 n) with linearly many EREW processors We give a new algorithm for this problem, based on a structural characterization of series-parallel graphs in terms of their ear decompositions Our algorithm can recognize undirected as well as directed series-parallel graphs It can be implemented in the CRCW model of parallel computation to take time O(log n) In the EREW model the time is O(log2 n) but the number of processors required improves the bounds of the previous algorithm

/pdf/parallel-recognition-of-series-parallel-graphs-198vrtahfg.pdf

Parallel recognition of series-parallel graphs

Dynamic programming solutions to a number of different recurrence equations for sequence comparison and for RNA secondary structure prediction are considered. These recurrences are defined over a number of points that is quadratic in the input size; however only a sparse set matters for the result. Efficient algorithms for these problems are given, when the weight functions used in the recurrences are taken to be linear. The time complexity of the algorithms depends almost linearly on the number of points that need to be considered; when the problems are sparse this results in a substantial speed-up over known algorithms.

/pdf/sparse-dynamic-programming-i-linear-cost-functions-4725g4ym0h.pdf

Sparse dynamic programming I: linear cost functions

The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n -gon with r reflex vertices in time O(n

 1+e
 + n

 8/11+e
 
r

 9/11+e
 
) , for any fixed e >0 , improving the previous best upper bound of O(nr log n) . Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in R
3 and answer queries asking which triangle is first hit by a query ray, and (2) maintain a changing set of rays in R
3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a lower envelope of triangles in R
3 . The same time bounds apply to constructing non-self-intersecting offset curves with mitered or beveled corners, and similar methods extend to other problems of simulating collisions and other pairwise interactions among sets of moving objects.

https://link.springer.com/content/pdf/10.1007/PL00009479.pdf

Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination. We describe new data structures for quickly finding the rule matching an incoming packet, in near-linear space, and a new algorithm for determining whether a rule set contains any conflicts, in time O(n3/2).

/pdf/internet-packet-filter-management-and-rectangle-geometry-24o2pyogdz.pdf

Internet packet filter management and rectangle geometry

https://www.ics.uci.edu/~eppstein/pubs/Epp-TR-94-11.pdf

David Eppstein

Papers

Parallel recognition of series-parallel graphs

Sparse dynamic programming I: linear cost functions

Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

Internet packet filter management and rectangle geometry

Arboricity and bipartite subgraph listing algorithms