The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Introduction to algorithms
Algebraic optimization: the Fermat-Weber location problem
Mathematical Programming: Series A and B
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Solving the robots gathering problem
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Robot networks with homonyms: the case of patterns formation
SSS'11 Proceedings of the 13th international conference on Stabilization, safety, and security of distributed systems
On the Fermat-Weber Point of a Polygonal Chain and its Generalizations
Fundamenta Informaticae
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A set of n points in the plane is in equiangular configuration if there exist a center and an ordering of the points such that the angle of each two adjacent points w.r.t. the center is $\frac{360^{\circ}}{n}$, i.e., if all angles between adjacent points are equal. We show that there is at most one center of equiangularity, and we give a linear time algorithm that decides whether a given point set is in equiangular configuration, and if so, the algorithm outputs the center. A generalization of equiangularity is σ-angularity, where we are given a string σ of n angles and we ask for a center such that the sequence of angles between adjacent points is σ. We show that σ-angular configurations can be detected in time O(n4 log n).