Journal ArticleDOI
An algorithm for geometric minimum spanning trees requiring nearly linear expected time
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An algorithm for finding a minimum spanning tree of the weighted complete graph induced by a set ofn points in Euclideand-space and the robustness of bucket sorting is demonstrated, which requiresO(n) expected time in this case despite the probability dependency between the edge weights.Abstract:
We describe an algorithm for finding a minimum spanning tree of the weighted complete graph induced by a set ofn points in Euclideand-space. The algorithm requires nearly linear expected time for points that are independently uniformly distributed in the unitd-cube. The first step of the algorithm is the spiral search procedure described by Bentleyet al. [BWY82] for finding a supergraph of the MST that hasO(n) edges. (The constant factor in the bound depends ond.) The next step is that of sorting the edges of the supergraph by weight using a radix distribution, or “bucket,” sort. These steps require linear expected time. Finally, Kruskal's algorithm is used with the sorted edges, requiringO(nα(cn, n)) time in the worst case, withc>6. Since the function α(cn, n) grows very slowly, this step requires linear time for all practical purposes. This result improves the previous bestO(n log log*n), and employs a much simpler algorithm. Also, this result demonstrates the robustness of bucket sorting, which requiresO(n) expected time in this case despite the probability dependency between the edge weights.read more
Citations
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Journal ArticleDOI
Voronoi diagrams—a survey of a fundamental geometric data structure
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Book ChapterDOI
Chapter 5 – Voronoi Diagrams*
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Entropic graphs for registration
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Some new exact solutions of ( 3 + 1 ) $(3+1)$ -dimensional Burgers system via Lie symmetry analysis
TL;DR: In this paper, the authors used Lie symmetry analysis to obtain the geometric vector fields of the $(3+1)$¯¯¯¯ -Burgers system and then applied the 3-dimensional optimal system to reduce the order of the system.
References
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Book
Data Structures and Network Algorithms
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Proceedings ArticleDOI
Fibonacci Heaps And Their Uses In Improved Network Optimization Algorithms
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Journal ArticleDOI
On constructing minimum spanning trees in k-dimensional spaces and related problems
TL;DR: By employing a subroutine that solves the post office problem, it is shown that, for fixed k $\geq$ 3, such a minimum spanning tree can be found in time O($n^{2-a(k)} {(log n)}^{1-a (k)}$), where a(k) = $2^{-(k+1)}$.
Journal ArticleDOI
Optimal Expected-Time Algorithms for Closest Point Problems
TL;DR: Algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees can be implemented to solve practical problems very efficiently.
Journal ArticleDOI
The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees
TL;DR: Sous l'hypothese que trois points d'entree ne forment pas un triangle isocele, le RNG de n points dans un espace dimension r peut etre trouve en un temps O(n 2 ) pour r≥3.