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Euclidean minimum spanning trees and bichromatic closest pairs

1991
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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

doi:10.1007/bf02574698
fatcat:fbexydetjvhuponkwa4vlnfnji
*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*...##
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Extreme Distances in Multicolored Point Sets
[chapter]

2006
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Graph Algorithms and Applications 5
*

Given a set of n colored points in some d-dimensional

doi:10.1142/9789812773289_0003
fatcat:snrojxoy7rhqzp76praebp5rje
*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. ...##
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Extreme Distances in Multicolored Point Sets

2004
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Journal of Graph Algorithms and Applications
*

Given a set of n colored points in some d-dimensional

doi:10.7155/jgaa.00080
fatcat:ifx4tkwmc5f63eafchegfuw2ba
*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. ...##
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Extreme Distances in Multicolored Point Sets
[chapter]

2002
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Lecture Notes in Computer Science
*

We present efficient algorithms to compute a

doi:10.1007/3-540-47789-6_2
fatcat:3t6wfw7jpjdwhgotxo3lxajtye
*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. ...##
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Dynamic Euclidean minimum spanning trees and extrema of binary functions

1995
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Discrete & Computational Geometry
*

[2], [9], [18], smallest k-gons [6], [11],

doi:10.1007/bf02574030
fatcat:4spdvj3vjrhobjgqrqtjwqss4q
*and**minimum**spanning**trees*(MSTs) [8]. ... 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. ...##
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Page 411 of Mathematical Reviews Vol. , Issue 93a
[page]

1993
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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); ...

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Spanning Trees in Multipartite Geometric Graphs
[article]

2016
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arXiv
*
pre-print

We present Θ(n n)-time algorithms that solve the

arXiv:1611.01661v1
fatcat:mklikketuza4zibcf4vwkjqgr4
*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. ...##
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Spanning Trees and Spanners
[chapter]

2000
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Handbook of Computational Geometry
*

We survey results in geometric network design theory, including algorithms for constructing

doi:10.1016/b978-044482537-7/50010-3
fatcat:gitonpgfozgfribivszd6gf5cy
*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*). ...##
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Geometric Minimum Spanning Trees with GeoFilterKruskal
[chapter]

2010
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Lecture Notes in Computer Science
*

We propose GEOFILTERKRUSKAL, an algorithm that computes the

doi:10.1007/978-3-642-13193-6_41
fatcat:3tozu5w5e5frxluiy5vpyo36zm
*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*[25] . ...##
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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. ...##
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Geometric Planar Networks on Bichromatic Points
[article]

2019
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arXiv
*
pre-print

We study four classical graph problems -- Hamiltonian path, Traveling salesman,

arXiv:1911.08924v1
fatcat:qh25huacmffuhhcfi7tbllnzre
*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). ...##
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Spanners for Geometric Intersection Graphs
[article]

2006
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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

arXiv:cs/0605029v1
fatcat:2udawjmaivdbxmrl5a7fjumpm4
*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. ...##
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Finding Relevant Points for Nearest-Neighbor Classification
[article]

2021
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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

arXiv:2110.06163v1
fatcat:gwp56fg2fbd4zpaug3dldetlje
*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 ...##
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Optimal random matchings, tours, and spanning trees in hierarchically separated trees

2013
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Theoretical Computer Science
*

We consider monochromatic

doi:10.1016/j.tcs.2013.05.021
fatcat:lfi5mslmybatznghne53d7kc6e
*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 . ...##
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Aren't we all nearest neighbors: Spatial trees, high dimensional reductions and batch nearest neighbor search
[article]

2015
*
arXiv
*
pre-print

Finally, we propose a dual

arXiv:1507.03338v1
fatcat:64jyxgqrbra5pouvdwoobmm6h4
*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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