Computational geometry

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

/pdf/analysis-of-boolean-functions-4kueie1j8m.pdf

Analysis of Boolean Functions

Analysis of Boolean Functions: List of Notation

Communication Complexity: Basics

We consider the problem of testing whether a Boolean function f : {−1, 1}n → {−1, 1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w1x1 + w2x2 + · · · + wnxn) where the weights wi take values in {−1, 1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±1-weight halfspace versus -far from all such halfspaces we prove that nonadaptive algorithms must make Ω(log n) queries. We complement this lower bound with a sublinear upper bound showing that O( √ n·poly( 1 )) queries suffice.

Testing +/- 1-Weight Halfspaces

We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n 2 ) time. Achieving this running time requires proving several new structural properties of the MAP, and combining them with a rotating technique of Toussaint [10]. We also discuss applications of our result to the problem of computing the maximum area centrallysymmetric convex body (MAC) inside a given convex polygon, and to a “fault tolerant area maximization” problem which we define.

/pdf/finding-the-maximum-area-parallelogram-in-a-convex-polygon-1rogjg5bas.pdf

Finding the Maximum Area Parallelogram in a Convex Polygon

We define a new property testing model for algorithms that do not have arbitrary query access to the input, but must instead traverse it in a manner that respects the underlying data structure in which it is stored. In particular, we consider the case when the underlying data structure is a linked list, and the testing algorithm is allowed to either sample randomly from the list, or walk to nodes that are adjacent to those already visited. We study the well-known monotonicity testing problem in this model, and show that Θ(n) queries are both necessary and sufficient to distinguish whether a list is sorted (monotone increasing) versus a constant distance from sorted. Our bound is strictly greater than the Θ(log n) queries required in the standard testing model, that allows element access indexed by rank, and strictly less than the Θ( √ n) queries required by a weak model that only allows random sampling.

/pdf/property-testing-on-linked-lists-45qj1d9eka.pdf

Property Testing on Linked Lists.

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(wċx-θ). We consider halfspaces over the continuous domain Rn (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {-1, 1}n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are e-far from any halfspace using only poly(1/e) queries, independent of the dimension n.

In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of {-1, 1}-weight halfspaces, we show that a tester must make at least O(log n) queries. We complement this lower bound with an upper bound showing that O(√n queries suffice.

/pdf/testing-subclasses-of-halfspaces-3lmi1so639.pdf

Kevin Matulef

Papers

Testing +/- 1-Weight Halfspaces

Finding the Maximum Area Parallelogram in a Convex Polygon

Property Testing on Linked Lists.

Testing (subclasses of) halfspaces

Testing and learning Boolean functions