Speaker: Sofya Raskhodnikova

Abstract:

How quickly can we determine if an object satisfies some basic geometric property? For example, is the object a half-plane? Is it convex? Is it connected? If we need to answer such a question exactly, it requires at least as much time as it takes to read the object. In this talk, we will focus on approximate versions of these questions and will discuss how to solve them in time that depends only on the approximation parameter, but not the size of the input.

Specifically, an algorithm is given access to a discretized image represented by an n x n matrix of 0/1 pixel values. Another input to the algorithm is an approximation parameter, epsilon. The algorithm is required to accept images with the desired property and reject (with high probability) images that are far from having the desired property. An image is far if at least an epsilon fraction of its pixels has to be changed to get an image with the required property. For example, in this model, if the algorithm is allowed to read pixels of its choice, the half-plane property and convexity can be tested in time O(1/epsilon). If the algorithm receives access to pixels chosen uniformly and independently at random, then the half-plane property still takes O(1/epsilon) time, but for convexity the (optimal) bound on the running time is O(1/epsilon^(4/3)).

Based on joint work with Piotr Berman and Meiram Murzabulatov.

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