Learning Unions of Rectangles with Queries

We investigate the efficient learnability of unions of k rectangles in the discrete plane (1,...,n)[2] with equivalence and membership queries. We exhibit a learning algorithm that learns any union of k rectangles with O(k^3log n) queries, while the time complexity of this algorithm is bounded by O(k^5log n). We design our learning algorithm by finding "corners" and "edges" for rectangles contained in the target concept and then constructing the target concept from those "corners" and "edges". Our result provides a first approach to on-line learning of nontrivial subclasses of unions of intersections of halfspaces with equivalence and membership queries.

On the Performance of Polynomial-time CLIQUE Approximation Algorithms on Very Large Graphs

The performance of a randomized version of the subgraph-exclusion algorithm (called Ramsey) for CLIQUE by Boppana and Halldorsson is studied on very large graphs. We compare the performance of this algorithm with the performance of two common heuristic algorithms, the greedy heuristic and a version of simulated annealing. These algorithms are tested on graphs with up to 10,000 vertices on a workstation and graphs as large as 70,000 vertices on a Connection Machine. Our implementations establish the ability to run clique approximation algorithms on very large graphs. We test our implementations on a variety of different graphs. Our conclusions indicate that on randomly generated graphs minor changes to the distribution can cause dramatic changes in the performance of the heuristic algorithms. The Ramsey algorithm, while not as good as the others for the most common distributions, seems more robust and provides a more even overall performance. In general, and especially on deterministically generated graphs, a combination of simulated annealing with either the Ramsey algorithm or the greedy heuristic seems to perform best. This combined algorithm works particularly well on large Keller and Hamming graphs and has a competitive overall performance on the DIMACS benchmark graphs.

A Short History of Computational Complexity

National Science Foundation (CCR-998310); Army Research Office (DAAD19-02-1-0058)

Universal Quantum Circuits

We define and construct efficient depth universal and almost size universal quantum circuits. Such circuits can be viewed as general purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. We prove that this construction is nearly optimal. Our results apply to a number of well-studied quantum circuit classes.

On Finding Sensitivity of Quantum and Classical Gates

We consider a fault model of Boolean gates, both classical and quantum, where some of the inputs may not be connected to the actual gate hardware. This model is somewhat similar to the stuck-at model which is a very popular model in testing Boolean circuits. We consider the problem of detecting such faults; the detection algorithm can query the faulty gate and its complexity is the number of such queries. This problem is related to determining the sensitivity of Boolean functions. We show how quantum parallelism can be used to detect such faults. Specifically, we show that a quantum algorithm can detect such faults more efficiently than a classical algorithm for a Parity gate and an AND gate. We give explicit constructions of quantum detector algorithms and show lower bounds for classical algorithms. We show that the model for detecting such faults is similar to algebraic decision trees and extend some known results from quantum query complexity to prove some of our results.

A Recurrent Cooperative/Competitive Field for Segmentation of Magnetic Resonance Brain Imagery

The Grey-White Decision Network is introduced as an application of an on-center, off-surround recurrent cooperative/competitive network for segmentation of magnetic resonance imaging (MRI) brain images. The three layer dynamical system relaxes into a solution where each pixel is labeled as either grey matter, white matter, or "other" matter by considering raw input intensity, edge information, and neighbor interactions. This network is presented as an example of applying a recurrent cooperative/competitive field (RCCF) to a problem with multiple conflicting constraints. Simulations of the network and its phase plane analysis are presented.