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We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and ac-tuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our construction entails a discrete abstraction of configurations using… (More)

This paper introduces and solves the problem of cluster-hierarchy-invariant particle navigation in Conf (R<sup>d</sup>, J). Namely, we are given a desired goal configuration, x* ϵ Conf (R<sup>d</sup>, J) and τ, a specified cluster hierarchy that the goal supports. We build a hybrid closed loop controller guaranteed to bring any other… (More)

- Dan P. Guralnik
- 2007

In this work we introduce a new combinatorial notion of boundary C of an ω-dimensional cubing C. C is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation –… (More)

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity d nav counts the steps along a " combing " of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of " edit length… (More)

We examine overlapping clustering schemes with functorial constraints , in the spirit of Carlsson–Mémoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. We work in… (More)

We propose a self-organizing database for perceptual experience capable of supporting autonomous goal-directed planning. The main contributions are: (i) a formal demonstration that the database is complex enough in principle to represent the homotopy type of the sensed environment; (ii) some initial steps toward a formal demonstration that the database… (More)

This work draws its inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carls-son and Mémoli to partition-based clustering, and the… (More)

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… (More)

- Dan P. Guralnik
- 2008

Let X be a proper CAT(0) space. A halfspace system (or cubulation) of X is a set H of open halfspaces closed under h → X h and such that every x ∈ X has a neighbourhood intersecting only finitely many walls of H. Given a cubulation H, one uses the Sageev-Roller construction to form a cubing C(H). One setting in which cubulations were studied is that of a… (More)

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… (More)