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.

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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.

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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.

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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.

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Metaheuristics: From Design to Implementation

Using the Borsuk-Ulam Theorem

We prove a theorem on partitioning point sets inEd (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n logn) deterministic preprocessing time, andO(n1?1/d(logn)O(1)) query time. WithO(nlogn) preprocessing time, where ? is an arbitrary positive constant, a more complicated data structure yields query timeO(n1?1/d(log logn)O(1)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelleet al. [CSW].

The partition result implies that, forrd≤n1??, a (1/r)-approximation of sizeO(rd) with respect to simplices for ann-point set inEd can be computed inO(n logr) deterministic time. A (1/r)-cutting of sizeO(rd) for a collection ofn hyperplanes inEd can be computed inO(n logr) deterministic time, provided thatr≤n1/(2d?1).

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Efficient partition trees

What Is It, and What For?- Examples- Integer Programming and LP Relaxation- Theory of Linear Programming: First Steps- The Simplex Method- Duality of Linear Programming- Not Only the Simplex Method- More Applications- Software and Further Reading

Understanding and Using Linear Programming

We consider the halfspace itrange itreporting problem: given a finite set P of points in R d, preprocess it so that given a query halfspace γ, the points of P ∩ γ can be reported efficiently. We show that with almost linear storage, this problem can be solved substantially more efficiently that the more general simplex range searching problem. We give a data structure for halfspace range reporting in dimensions d ⩾ 4 with O(n log log n) space, O(n log n) deterministic preprocessing time and O(n 1 − 1 ⌊ d 2 ⌋ (logn) c + k) query time, where c = c(d) is a constant and k = |P ∩ γ| (efficient solutions were known for d = 2, 3). For the halfspace emptiness problem, where we only want to know whether P ∩ γ = O, we can achieve query time O(n 1 − 1 ⌊ d 2 ⌋ 2 c'log ∗ n ) with a linear space and O(n1 + δ) preprocessing (c' = c'(d) is a constant and γ > 0 is arbitrarily small but fixed).

Jiří Matoušek

Papers

Using the Borsuk-Ulam Theorem

Efficient partition trees

Understanding and Using Linear Programming

Reporting points in halfspaces

On the L 2 -discrepancy for anchored boxes