Analog Parallel Computational Geometry

We introduce a novel approach to Parallel Computational Geometry by using networks of analog components, referred to as analog networks or analog circuits. The analog network we study here is the Analog Hoppeld Net which was origially introduced by Hoppeld (1983) as a simpliied electronic model of human brain cells. Massively parallel Analog Hoppeld Nets with large numbers of processing elements (neurons) exist in hardware and have proven to be eecient architectures for important problems (e.g. for constructing an associative memory). We demonstrate how Computational Geometry problems can be solved by exploiting the features of such analog parallel architectures. Using massively parallel analog networks requires a radically diierent approach to traditional parallel geometric problem solving because (i) time is continuous instead of the discretized time step used for traditional parallel (or sequential) processing, and (ii) geometric data is represented by analog components (e.g. voltages at certain positions of the circuit) instead of the usual digital representation. We present analog parallel algorithms for several geometrical problems including: minimum weight triangulation of planar point sets or of polygons with holes, determining for a given line segment set a subset of non-intersecting line segments of maximum total length, nding the smallest " so that two given point sets are "-congruent via translation, and partitioning a set of points into k clusters such that the maximum diameter of the clusters is minimized. Our proofs provide ranges for the network parameters for which the networks are guaranteed to produce a feasible solution. Such 1 analysis has previously been unavailable for Hoppeld-type circuits. Our proof techniques yield a considerable improvement over previous methods for applying Hoppeld nets to other problem areas. In fact, by applying our proof techniques back to Hoppeld's original Traveling Salesman network, we obtain a much improved network behaviour. The paper also includes experimental results which demonstrate the performance of our analog parallel algorithms for Computational Geometry problems.

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