We consider the problem of finding low-cost spanning trees for sets of $n$ points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of $m$ barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost $O(\min(m^2,m\sqrt{n}))$; (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost $O(m)$; (iii) ] if the barriers are disjoint convex objects, then there is always a spanning tree of cost $O(n+m)$. All our bounds are worst-case optimal, up to multiplicative constants. [ABSTRACT FROM AUTHOR]

ALGORITHMS, MOBILE communication systems, and APPROXIMATION theory

Abstract

abstract. we propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. this subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. as the nodes move, an event-based kinetic data structure updates the clustering as necessary. this kinetic data structure is shown to be responsive, efficient, local, and compact. the produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. this clustering algorithm is distributed in nature and can enable numerous applications in ad hoc wireless networks, where mobile devices must be interconnected to perform various tasks collaboratively. [ABSTRACT FROM AUTHOR]