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

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

/pdf/the-quickhull-algorithm-for-convex-hulls-ug134bwaff.pdf

The quickhull algorithm for convex hulls

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

Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(cd, ϵ log n) time, where cd,ϵ≤d ⌈1 + 6d/ϵ⌉d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

/pdf/an-optimal-algorithm-for-approximate-nearest-neighbor-29l6kwjnzm.pdf

An optimal algorithm for approximate nearest neighbor searching fixed dimensions

A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.

/pdf/metaheuristics-from-design-to-implementation-1vqdfv8nk9.pdf

Metaheuristics: From Design to Implementation

Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d >= 2, we have dim X Y and ask whether it extends to a map X -> Y, or computing the Z_2-index---everything in the stable range. Outside the stable range, the extension problem is undecidable.

Computing all maps into a sphere

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝd of cardinality (d+1)t, partitioned into t-point subsets C 1,C 2,…,C d+1 (which we think of as color classes; e.g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint sets R 1,R 2,…,R r ⊆C that are rainbow, meaning that |R i ∩C j |≤1 for every i,j, and whose convex hulls all have a common point.

All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojevic, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.

/pdf/a-geometric-proof-of-the-colored-tverberg-theorem-54x7urwnqt.pdf

A Geometric Proof of the Colored Tverberg Theorem

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

/pdf/zone-diagrams-in-euclidean-spaces-and-in-other-normed-spaces-4oplpwlydw.pdf

Zone diagrams in Euclidean spaces and in other normed spaces

A model for segmentation of an object space by an array of binary, radiation-field sensors and geometric reference structures is described. Given a family of binary, radiation-field sensors and a geometric reference structure, we refer to the set of sensor states induced by a source at point p as the signature of p. We study the segmentation of an object space into signature cells and prove near optimal bounds on the number of distinct signatures induced by a point source, as a function of sensor and reference structure complexity. We also show that almost any family of signatures can be implemented under this model.

https://users.cs.duke.edu/~pankaj/publications/papers/sensor-ref.pdf

Segmenting object space by geometric reference structures

A family of permutations F ⊆ Sn, with a probability distribution on it is called k-restricted rain-wise independent if we have Pr[min π(X) = π(x)] = 1|X| for every subset X ⊆ [n] with |X| ≤ k, every x ∈ X, and π ∈ F chosen at random. We present a simple proof of a result of Norin: every such family has size at least (n-1 k-1/2) . Some features of our method might be of independent interest. The best available upper bound for the size of such family is 1 + σj=2k(J - 1) (n j). We show that this bound is tight if the goal is to imitate not the uniform distribution on Sn, but a distribution given by assigning suitable priorities to the elements of [n] (the stationary distribution of the Tsetlin library, or self-organizing lists). This is analogous to a result of Karloff and Mansour for k-wise independent random variables. We also investigate the cases where the min-wise independence condition is required only for sets X of size exactly k (where we have only an ω(log log n + k) lower bound), or for sets of size k and k - 1 (where we already obtain a lower bound of n - k + 2).

/pdf/on-restricted-min-wise-independence-of-permutations-1h057osgcn.pdf

Jiří Matoušek

Papers

Computing all maps into a sphere

A Geometric Proof of the Colored Tverberg Theorem

Zone diagrams in Euclidean spaces and in other normed spaces

Segmenting object space by geometric reference structures

On restricted min-wise independence of permutations