Michelangelo Grigni

Ray shooting in polygons using geodesic triangulations

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

Quantum mechanical algorithms for the nonabelian hidden subgroup problem

We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.

Ray Shooting in Polygons Using Geodesic Triangulations

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

An approximation scheme for planar graph TSP

We consider the special case of the traveling salesman problem (TSP) in 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.

Tight bounds on minimum broadcast networks

A broadcast graph is an n-vertex communication network that supports a broadcast from any one vertex to all other vertices in optimal time

Map graphs

\lceil \lg n\rceil

On the Complexity of the Generalized Block Distribution

, given that each message transmission takes one time unit and a vertex participates in at most one transmission per time step. This paper establishes tight bounds for

Jamming model for the extremal optimization heuristic

B( n )

Monotone Separation of Logarithmic Space from Logarithmic Depth

, the minimum number of edges of a broadcast graph, and

Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

D( n )