Abstract-We introduce the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. Traditionally an unsupervised learning method, hierarchical clustering offers a formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions by relating the continuous space of configurations to the combinatorial space of trees. We formalize and exploit this relation, developing computationally effective reactive algorithms for navigating through the combinatorial space in concert with geometric realizations for a particular choice of hierarchical clustering method. These constructions yield computationally effective vector field planners for both hierarchically invariant as well as transitional navigation in the configuration space. We apply these methods to the centralized coordination and control of n perfectly sensed and actuated Euclidean spheres in a d-dimensional ambient space (for arbitrary n and d). Given a desired configuration supporting a desired hierarchy, we construct a hybrid controller which is quadratic in n and algebraic in d and prove that its execution brings all but a measure zero set of initial configurations to the desired goal with the guarantee of no collisions along the way.

This paper introduces and solves the problem of cluster-hierarchy-invariant particle navigation in Conf R d , J . Namely, we are given a desired goal configuration, x * ∈ Conf R d , J and τ , a specified cluster hierarchy that the goal supports. We build a hybrid closed loop controller guaranteed to bring any other configuration that supports τ to the desired goal, x * ∈ Conf R d , J , through a transient motion whose each configuration along the way also supports that hierarchy.

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group G acting geometrically on a CAT(0) space X we show there is a flat F ⊂ X of maximal dimension (denote it by d), whose boundary sphere intersects every minimal G-invariant subset of ∂∞X. As applications we obtain an improved dimension-dependent bound diam∂ T X ≤ 2π − arccos − 1 d + 1 on the Tits-diameter of ∂X for non-rank-one groups, a necessary and sufficient dynamical condition for G to be virtually-Abelian, and we formulate a new approach to Ballmann's rank rigidity conjectures.

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesting is that in this context the connected components of the edge-induced subgraph of exposed edges are 2-edge connected. We use this latter fact in the weighted case to give a modified version of Kruskal's second algorithm for finding a minimum spanning tree in a weighted chordal graph. This modified algorithm benefits from being local in an important sense.