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### Euclidean minimum spanning trees and bichromatic closest pairs

Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, Emo Welzl
1991 Discrete & Computational Geometry
In d > 4 dimensions we obtain expected time C((nm) 1-1/trd/2~+l)+~ + m log n + n log m) for the bichromatic closest pair problem and (6(N 2-2/~fd/21+1)+~) for the Euclidean minimum spanning tree problem  ...  We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of N points in ~_d in time C (,~-a(N, N) log d N), where J~(n, m) is the time required to compute a bichromatic closest  ...  Introduction Given a set S of N points in Euclidean d-dimensional space ~d, a Euclidean minimum spanning tree (EMST) is a spanning tree of S whose edges have a minimum total length among all spanning trees  ...

### Extreme Distances in Multicolored Point Sets [chapter]

2006 Graph Algorithms and Applications 5
Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors.  ...  We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color.  ...  We wish to thank the anonymous referee for a thorough reading and useful suggestions that have lead to improved results in Section 2.2.  ...

### Extreme Distances in Multicolored Point Sets

2004 Journal of Graph Algorithms and Applications
Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors.  ...  We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color.  ...  We wish to thank the anonymous referee for a thorough reading and useful suggestions that have lead to improved results in Section 2.2.  ...

### Extreme Distances in Multicolored Point Sets [chapter]

2002 Lecture Notes in Computer Science
We present efficient algorithms to compute a bichromatic closest pair and a bichromatic farthest pair. We consider both static, and dynamic versions with respect to color flips.  ...  Given a set of n points in some d-dimensional Euclidean space, each point colored with one of k( ≥ 2) colors, a bichromatic closest (resp., farthest) pair is a closest (resp., farthest) pair of points  ...  Design an efficient algorithm to maintain a minimum bichromatic Euclidean spanning tree when colors change dynamically.  ...

### Dynamic Euclidean minimum spanning trees and extrema of binary functions

D. Eppstein
1995 Discrete & Computational Geometry
, , , smallest k-gons , , and minimum spanning trees (MSTs) .  ...  We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized time O(n ~/2 log 2 n) per update operation.  ...  on half-space range searching and nearest-neighbor queries.  ...

### Page 411 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews
trees and bichromatic closest pairs.  ...  /((4/21+)+e + mlogn+nlogm) for the bichromatic closest pair problem and @(N2-2/([4/21+1)+e) for the Euclidean minimum spanning tree problem, for any positive ¢.” 93a:68147 68U05 Aggarwal, Alok (1-IBM);  ...

### Spanning Trees in Multipartite Geometric Graphs [article]

Ahmad Biniaz, Prosenjit Bose, David Eppstein, Anil Maheshwari, Pat Morin, Michiel Smid
2016 arXiv   pre-print
We present Θ(n n)-time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. 3.  ...  The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length. 1.  ...  Open Problem We presented O(n log n log k)-time algorithms for the Min-k-ST and Max-k-ST problems. Presenting faster algorithms for these problems or providing a matching lower bound is open.  ...

### Spanning Trees and Spanners [chapter]

David Eppstein
2000 Handbook of Computational Geometry
We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.  ...  find minimum spanning trees by solving a collection of bichromatic closest pair problems for the pairs of a well-separated pair decomposition, then compute a minimum spanning tree on the resulting graph  ...  (Callahan does use this decomposition in a more complicated way to reduce minimum spanning trees to bichromatic closest pairs).  ...

### Geometric Minimum Spanning Trees with GeoFilterKruskal [chapter]

Samidh Chatterjee, Michael Connor, Piyush Kumar
2010 Lecture Notes in Computer Science
We propose GEOFILTERKRUSKAL, an algorithm that computes the minimum spanning tree of P using well separated pair decomposition in combination with a simple modification of Kruskal's algorithm.  ...  When P is sampled from uniform random distribution, we show that our algorithm takes one parallel sort plus a linear number of additional steps, with high probability, to compute the minimum spanning tree  ...  Bichromatic Closest Pair Algorithm: Given a set of points in R d , the Bichromatic Closest Pair (Bccp) problem asks to find the closest red-green pair  .  ...

### Page 6572 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews
. 7, (n,m) denotes the time to find a bichromatic closest pair between v red points and m blue points.  ...  Minimum spanning trees in d dimensions. (English summary) Nordic J. Comput. 6 (1999), no. 4, 446-461.  ...

### Geometric Planar Networks on Bichromatic Points [article]

Sayan Bandyapadhyay and Aritra Banik and Sujoy Bhore and Martin Nöllenburg
2019 arXiv   pre-print
We study four classical graph problems -- Hamiltonian path, Traveling salesman, Minimum spanning tree, and Minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points.  ...  These problems have been widely studied for points in the Euclidean plane, and many of them are NP-hard.  ...  Next, we consider the Bichromatic spanning tree problem where the objective is to compute a minimum weight spanning tree of G(R, B, E).  ...

### Spanners for Geometric Intersection Graphs [article]

2006 arXiv   pre-print
For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems.  ...  The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time.  ...  minimum spanning tree (or a bichromatic closest pair) in R k are within constant factors.  ...

### Finding Relevant Points for Nearest-Neighbor Classification [article]

David Eppstein
2021 arXiv   pre-print
We provide a simple algorithm for thinning a training set down to its subset of relevant points, using as subroutines algorithms for finding the minimum spanning tree of a set of points and for finding  ...  In nearest-neighbor classification problems, a set of d-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the  ...  Several reductions from Euclidean minimum spanning trees to bichromatic closest pairs have been given but for the known time bounds for bichromatic closest pairs these reductions all have the same efficiency  ...

### Optimal random matchings, tours, and spanning trees in hierarchically separated trees

Béla Csaba, Thomas A. Plick, Ali Shokoufandeh
2013 Theoretical Computer Science
We consider monochromatic and bichromatic versions of the minimum matching, minimum spanning tree, and traveling salesman problems.  ...  We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size n distributed among the leaves of a balanced hierarchically separated tree.  ...  Bichromatic Minimum Spanning Tree Problem (M ST ′ ): The minimum bichromatic spanning tree cost is given by min F ⊂G e∈F |e|, where the minimum is taken over all bichromatic spanning trees F .  ...

### Aren't we all nearest neighbors: Spatial trees, high dimensional reductions and batch nearest neighbor search [article]

Mark Saroufim
2015 arXiv   pre-print
Finally, we propose a dual tree algorithm for Bichromatic Closest Pair and measure the complexity of batch nearest neighbor search.  ...  In the second chapter, we show reductions between two different classes of geo- metric proximity problems: the nearest neighbor problems to solve the Euclidean minimum spanning tree problem and the farthest  ...  I'd like to first thank my advisor Sanjoy Dasgupta for mentoring and guiding me with patience through the numerous revisions that this thesis has undertaken, this document would not exist if it wasn't  ...
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