# Semi-dynamic Connectivity in the Plane

@inproceedings{Cabello2015SemidynamicCI,
title={Semi-dynamic Connectivity in the Plane},
author={Sergio Cabello and Michael Kerber},
year={2015}
}
• Published in WADS 12 February 2015
• Mathematics
Motivated by a path planning problem we consider the following procedure. Assume that we have two points $s$ and $t$ in the plane and take $\mathcal{K}=\emptyset$. At each step we add to $\mathcal{K}$ a compact convex set that does not contain $s$ nor $t$. The procedure terminates when the sets in $\mathcal{K}$ separate $s$ and $t$. We show how to add one set to $\mathcal{K}$ in $O(1+k\alpha(n))$ amortized time plus the time needed to find all sets of $\mathcal{K}$ intersecting the newly added…
3 Citations

### Soft Subdivision Search in Motion Planning, II: Axiomatics

This paper addresses the issue of subdivision in non-Euclidean configuration spaces, and how exact algorithms can be recovered using soft methods, and provides a general axiomatic theory underlying these results.

### Soft Subdivision Search in Motion Planning, II: Axiomatics

This paper addresses the issue of subdivision in non-Euclidean configuration spaces, and how exact algorithms can be recovered using soft methods, and provides a general axiomatic theory underlying these results.

## References

SHOWING 1-10 OF 11 REFERENCES

### The Complexity of Separating Points in the Plane

• Mathematics
SoCG '13
• 2013
The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear, and it is shown that the generalized version is NP-hard for natural families of curves, like segments in two directions or unit circles.

### Algorithms for connected component labeling based on quadtrees

• Computer Science
Int. J. Imaging Syst. Technol.
• 2009
An algorithm of linear time complexity to label connected components of a binary image by a quadtree that has shown the best performance in large images.

### Efficiency of a Good But Not Linear Set Union Algorithm

It is shown that, if t(m, n) is seen as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m), n is shown to be related to a functional inverse of Ackermann's functmn and as very slow-growing.

### Planning algorithms

This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.

### Graphs on Surfaces

• Mathematics
Johns Hopkins series in the mathematical sciences
• 2001
This chapter discusses Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem, which focuses on planar graphs and the Jordan Curve Theorem, and colorings of Graphs on Surfaces, which are 5-choosable.

### Top-Down Analysis of Path Compression

• Computer Science
SIAM J. Comput.
• 2005
A new analysis of the worst-case cost of path compression is presented, which is an operation that is used in various well-known "union-find" algorithms, and yields recurrence relations from which the various bounds arise naturally.

### Computational geometry: algorithms and applications

This introduction to computational geometry focuses on algorithms as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems.

### Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms

• Computer Science
SCG '88
• 1988
A general purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms, and it is believed that this technique will become a standard tool in writing geometric software.

### Algorithms

• Computer Science
• 1992
Most of the articles appearing in this column are oriented toward Common Lisp. However, a wider community of Lisp dialects still exists. One that is of particular interest is GNU Emacs Lisp---the