Approximating Tverberg Points in Linear Time for Any Fixed Dimension

Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012. This version removes an incorrect example at the end of Section 3.

LIPIcs

The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2

Approximating Tverberg points in linear time for any fixed dimension

Let P ⊆ R d be a d-dimensional n-point set. A Tverberg partition of P is a partition of P into r sets P1,..., Pr such that the convex hulls conv(P1),..., conv(Pr) have non-empty intersection. A point in ⋂r i=1 conv(Pi) is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size ⌈n/(d + 1)⌉, but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size ⌈n/4(d + 1) 3 ⌉ in time d O(log d) n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result b

Fault-tolerant control policies for multi-robot systems

Throughout the past decade, we have witnessed an active interest in distributed motion coordination algorithms for networked mobile autonomous robots. Often, in multi-robot systems, each robot executing a coordination task is a little cost, a disposable autonomous agent that has ad-hoc sensing or communication capability, and limited mobility. Coordination tasks that a group of multiple mobile robots might perform include formation control, rendezvous, distributed estimation, deployment, flocking, etc. Also, there are challenging tasks that are more suitable for a group of mobile robots than an individual robot, such as surveillance, exploration, or hazardous environmental monitoring. The field has been collectively investigated by many researchers in robotics, control, artificial intelligence, and distributed computing. However, relatively little work has been done on developing algorithms to provide resilience to failures that can occur. The problem is extremely difficult to handle in that any partial failure of a robot is not readily detectable. Some failures in robot resources can have an adverse effect on not only the performance of the robot itself, but also other robots, and the collective task performance as well. This study presents the development of fault-tolerant distributed control policies for multi-robot systems. We consider two problems: rendezvous and coverage. For the former, the goal is to bring all robots to a common location, while for the latter the goal is to deploy robots to achieve optimal coverage of an environment. We consider the case in which each robot is an autonomous decision maker that is anonymous (i.e., robots are indistinguishable to one another), memoryless (i.e., each robot makes decisions based upon only its current information), and dimensionless (i.e., collision checking is not considered). Each robot has a limited sensing range and can directly estimate the state of only those robots within that sensing range, which induces a network topology for the multi-robot system. We assume that it is not possible for the fault-free robots to identify the faulty robots (e.g., due to the anonymous property of the robots). For each problem, we provide an efficient computational framework and analysis of algorithms, all of which converge in the face of faulty robots under a few assumptions on the network topology and sensing abilities. A suite of experiments and simulations confirm our theoretical analysis and demonstrate that our proposed algorithms are useful in fault-prone multi-robot systems