Pattern Matching for sets of segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200

Computational Philosophy: On Fairness in Automated Decision Making

As more and more of our lives are taken over by automated decision making systems (whether it be for hiring, college admissions, criminal justice or loans), we have begun to ask whether these systems are making decisions that humans would consider fair, or non-discriminatory. The problem is that notions of fairness, discrimination, transparency and accountability are concepts in society and the law that have no obvious formal analog. But our algorithms speak the language of mathematics. And so if we want to encode our beliefs into automated decision systems, we must formalize them precisely, while still capturing the natural imprecision and ambiguity in these ideas. In this talk, I\u27ll survey the new field of fairness, accountability and transparency in computer science. I\u27ll focus on how we formalize these notions, how they connect to traditional notions in theoretical computer science, and even describe some impossibility results that arise from this formalization. I\u27ll conclude with some open questions

The Hunting of the Bump: On Maximizing Statistical Discrepancy

Anomaly detection has important applications in biosurveilance and environmental monitoring. When comparing measured data to data drawn from a baseline distribution, merely, finding clusters in the measured data may not actually represent true anomalies. These clusters may likely be the clusters of the baseline distribution. Hence, a discrepancy function is often used to examine how different measured data is to baseline data within a region. An anomalous region is thus defined to be one with high discrepancy. In this paper, we present algorithms for maximizing statistical discrepancy functions over the space of axis-parallel rectangles. We give provable approximation guarantees, both additive and relative, and our methods apply to any convex discrepancy function. Our algorithms work by connecting statistical discrepancy to combinatorial discrepancy; roughly speaking, we show that in order to maximize a convex discrepancy function over a class of shapes, one needs only maximize a linear discrepancy function over the same set of shapes. We derive general discrepancy functions for data generated from a one- parameter exponential family. This generalizes the widely-used Kulldorff scan statistic for data from a Poisson distribution. We present an algorithm running in $O(\smash[tb]{\frac{1}{\epsilon} n^2 \log^2 n})$ that computes the maximum discrepancy rectangle to within additive error $\epsilon$, for the Kulldorff scan statistic. Similar results hold for relative error and for discrepancy functions for data coming from Gaussian, Bernoulli, and gamma distributions. Prior to our work, the best known algorithms were exact and ran in time $\smash[t]{O(n^4)}$.Comment: 11 pages. A short version of this paper will appear in SODA06. This full version contains an additional short appendi

Sensor network localization for moving sensors

pre-printSensor network localization (SNL) is the problem of determining the locations of the sensors given sparse and usually noisy inter-communication distances among them. In this work we propose an iterative algorithm named PLACEMENT to solve the SNL problem. This iterative algorithm requires an initial estimation of the locations and in each iteration, is guaranteed to reduce the cost function. The proposed algorithm is able to take advantage of the good initial estimation of sensor locations making it suitable for localizing moving sensors, and also suitable for the refinement of the results produced by other algorithms. Our algorithm is very scalable. We have experimented with a variety of sensor networks and have shown that the proposed algorithm outperforms existing algorithms both in terms of speed and accuracy in almost all experiments. Our algorithm can embed 120,000 sensors in less than 20 minutes