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### A Fast and Simple Algorithm for Computing Approximate Euclidean Minimum Spanning Trees

Sunil Arya, David M. Mount
2015 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms
the true minimum spanning tree.  ...  The Euclidean minimum spanning tree (EMST) is a fundamental and widely studied structure.  ...  For example, it is well known that the exact EMST is the minimum bottleneck spanning tree, meaning that it minimizes the maximum edge length.  ...

### Reactive Proximity Data Structures for Graphs [article]

David Eppstein, Michael T. Goodrich, Nil Mamano
2020 arXiv   pre-print
Definition 3 (Reactive bichromatic closest-pair data structure).  ...  In another paper  , we use this data structure as part of an algorithm for Steiner TSP in road networks.  ...

### When crossings count --- approximating the minimum spanning tree

Sariel Har-Peled, Piotr Indyk
2000 Proceedings of the sixteenth annual symposium on Computational geometry - SCG '00
We present an (1+ε)-approximation algorithm for computing the minimum-spanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and  ...  It is easy to verify that if we have triangle inequality then the Steiner tree weight is at least half the weight of the MST.  ...  Lemma 2.6 ([Aga91]) There exists a Steiner tree M of P , so that W opt (P, L) = O(n √ n), and this is tight in the worst case (even for the case the arcs are lines).  ...

### When Crossings Count - Approximating the Minimum Spanning Tree [article]

Sariel Har-Peled, Piotr Indyk
2003 arXiv   pre-print
the spanning tree and the lines.  ...  In the first part of the paper, we present an (1+\mu)-approximation algorithm to the minimum-spanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between  ...  It is easy to verify that if we have triangle inequality then the Steiner tree weight is at least half the weight of the MST.  ...

### Red-Blue-Partitioned MST, TSP, and Matching [article]

Matthew P. Johnson
2018 arXiv   pre-print
: min-sum, min-max, and bottleneck.  ...  Three structures have been investigated by ArkinBCCJKMM17---spanning trees, traveling salesperson tours, and perfect matchings---and the three objectives to optimize for when computing such pairs of structures  ...  First consider a set of 2n points V = V L ∪ V R in a metric space, arranged in the form of two full binary trees T L , T R (with root points L , R , respectively, and each with λ leaf points and n = 2λ  ...

### Euclidean TSP, Motorcycle Graphs, and Other New Applications of Nearest-Neighbor Chains [article]

Nil Mamano, Alon Efrat, David Eppstein, Daniel Frishberg, Michael Goodrich, Stephen Kobourov, Pedro Matias, Valentin Polishchuk
2019 arXiv   pre-print
We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for Euclidean TSP in O(nlog n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n√  ...  Algorithm 4 shows the full pseudo code. Algorithm 4 Nearest-neighbor-chain algorithm for motorcycle graphs.  ...  It is a tree-like structure similar to the medial axis of a polygon, but which consists of straight segments only.  ...

### Disk Intersection Graphs: Models, Data Structures, and Algorithms [article]

Paul Seiferth, Universitätsbibliothek Der FU Berlin, Universitätsbibliothek Der FU Berlin
2016
This list includes computing the chromatic number, a maximum independent set, a maximum connected dominating set, finding hamiltonian cycles and paths, and finding steiner trees.  ...  We show how to maintain a maximal bichromatic matching M between R and B with respect to D(R ∪ B).  ...

### Abstract

1991 Bulletin of the Psychonomic Society
STEINER, University of Basel (read by Gerhard F.  ...  A fmal test evaluated her ability to generalize conceptual use of fractions with liquids (juice presented in glasses), including 1-3 full glasses, or glasses that were 'h or 1,4 full.  ...