We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer. AbstractWe give a short exposition of new and known results on the "standard method"of identifying a hidden subgroup of a nonabelian group using a quantum computer.

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, derive some consequences regarding sparsity and coloring, and survey some algorithmic results.

We present a cubic-time algorithm for the following problem: Given a simple graph, decide whether it is realized by adjacencies of countries in a map without holes, in which at most four countries meet at any point.

Abstract. Let ~ be a simple polygon with n vertices. We present a simple decomposition scheme that partitions the interior of ~ into O(n) so-called geodesic triangles, so that any line segment interior to crosses at most 2 log n of these triangles. This decomposition can be used to preprocess ~ in a very simple manner, so that any ray-shooting query can be answered in time O(log n). The data structure requires O(n) storage and O(n log n) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time to O(n). We also extend our general technique to the case of ray shooting amidst k polygonal obstacles with a total of n edges, so that a query can be answered in O(,,fk log n) time.

We consider the special case of the traveling salesman problem (TSP) an which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.