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Time complexity of practical parallel steiner point insertion algorithms

Daniel A. Spielman, Shang-Hua Teng, Alper Üngör
2004 Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures - SPAA '04  
An effective method in practice to compute quality Delaunay triangulations is to apply parallel refinements that insert Steiner points whose prestars in the triangulation do not overlap.  ...  We show that these algorithms can be implemented in O(log m) time using m processors, where m is the output size. To our knowledge, this is the first such analysis.  ...  These algorithms are practical, however their time complexity remained unanalyzed.  ... 
doi:10.1145/1007912.1007953 dblp:conf/spaa/SpielmanTU04 fatcat:4wnv54dafbfdhfw3tg7rfojq4a

Page 398 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
A method is given to find a shortest Steiner tree for N points in the plane and it is shown to work in O(C”) time, for some constant C.  ...  First of all, an algorithm for finding such optimum Steiner trees is given. According to the author, his algorithm appears to be practical for problems with N < 12.  ... 

A Study on Applying Parallelism for Construction of Steiner Tree Algorithms in VLSI Design

Shyamala G., Latha N.
2016 International Journal of Computer Applications  
As the steiner problem in general is NP-hard, it is difficult to develop a polynomial-time algorithm to solve the problem exactly.  ...  We present a survey of the different approaches that can be parallelized and also the parallel algorithms available today with special concern to Rectilinear steiner tree for VLSI Design and their appropriateness  ...  The time complexity and practical runtime of B1S can be further improved using incremental / dynamic MST update techniques for I1S.  ... 
doi:10.5120/ijca2016911006 fatcat:i7y4vwl2w5fetigheio2tl4l3q

A parallel Steiner tree heuristic for macro cell routing

Christian Fobel, Gary Grewal
2008 2008 IEEE International Conference on Computer Design  
In this paper, a two-phase, parallel (multiprocessor) algorithm is proposed for quickly constructing a diverse pool of high-quality Steiner trees for routing of multiterminal nets.  ...  Global routing of macro cells remains an important but time-consuming step in the VLSI design cycle.  ...  Given n points (terminals) on a graph, a Steiner tree connects these points through some extra (Steiner) points such that the total length (cost of edges) of the tree is minimized.  ... 
doi:10.1109/iccd.2008.4751836 dblp:conf/iccd/FobelG08 fatcat:jjvgnnakjbemjiouzc3hrjd3k4

Three-dimensional delaunay refinement for multi-core processors

Andrey N. Chernikov, Nikos P. Chrisochoides
2008 Proceedings of the 22nd annual international conference on Supercomputing - ICS '08  
Based on these results, we decompose the iteration space of the sequential Delaunay refinement algorithm by selecting independent subsets from the set of the candidate Steiner points without resorting  ...  First, we prove a criterion and a sufficient condition of Delaunay-independence of Steiner points in three dimensions.  ...  This work was performed using computational facilities at the College of William and Mary which were enabled by grants from Sun Microsystems, the National Science Foundation, and Virginia's Commonwealth  ... 
doi:10.1145/1375527.1375560 dblp:conf/ics/ChernikovC08 fatcat:zuavhbdq3vg55n6qssdeczdqwi

Quality triangulations made smaller

Alper Üngör
2005 European Workshop on Computational Geometry  
We study alternative types of Steiner points (to circumcenters) for computing quality guaranteed Delaunay triangulations in three dimensions.  ...  We show through experiments that their effective use results in smaller (in the number of tetrahedra) triangulations than the output of the traditional circumcenter refinement methods.  ...  Acknowledgements I am thankful to Jonathan Shewchuk for using the off-center insertion algorithm in the Triangle software and for providing the Pyramid software.  ... 
dblp:conf/ewcg/Ungor05 fatcat:pb366665xffopop4oe2kp7lsnm

Efficient Multilayer Obstacle-Avoiding Rectilinear Steiner Tree Construction Based on Geometric Reduction

Chih-Hung Liu, Chun-Xun Lin, I-Che Chen, D. T. Lee, Ting-Chi Wang
2014 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
Last but not least, we also propose an advanced Steiner point selection which potentially selects more proper Steiner points but does not increase the time complexity.  ...  Compared with the Steiner point selection in [4] and [6], our advanced Steiner point selection adopts the concept of Kruskal algorithm [15] instead of Prim algorithm [15] , and performs a novel update  ...  Time Complexity Analysis We give the time complexity of the advanced Steiner point selection in the following analysis.  ... 
doi:10.1109/tcad.2014.2363390 fatcat:krjh4nvkurfgppjnr42gyjgjuy

A template for developing next generation parallel Delaunay refinement methods

Andrey N. Chernikov, Nikos P. Chrisochoides
2010 Finite elements in analysis and design  
We prove that any point placement algorithm which inserts Steiner points inside selection disks of Type I terminates, and any algorithm which inserts Steiner points inside selection disks of Type II produces  ...  Finally, we present a novel parallel algorithm which can be used in conjunction with any sequential point 1 placement strategy that chooses points within the selection disks.  ...  Spielman, Teng, and Ungör [48, 49] presented the first theoretical analysis of the complexity of parallel Delaunay refinement algorithms.  ... 
doi:10.1016/j.finel.2009.06.014 fatcat:yoev7a3fqbbgxktvvbvz3hpxt4

No quadrangulation is extremely odd [chapter]

Prosenjit Bose, Godfried Toussaint
1995 Lecture Notes in Computer Science  
If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n logn) time even in the presence of collinear points.  ...  If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an (n logn) time lower bound for the problem.  ...  Michael for noting a simpli cation of our proof of Lemma 3.1. We thank Anna Lubiw for discussions on models of computation in the proof of the lower bound.  ... 
doi:10.1007/bfb0015443 fatcat:57dyqgsvgrdghnomn7v3hif7jm

Page 1317 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
The objective of the algorithm is to divide each wire into multiple smaller segments and minimize the number of the buffer insertion points (Steiner points) which are only located at the end of each segment  ...  Summary: “This paper presents an approximation algorithm for simultaneously constructing a rectilinear Steiner tree and buffer insertion points into the tree.  ... 

Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations

Alper Üngör
2009 Computational geometry  
In practice, however, the new algorithm inserts fewer Steiner points, runs faster, and generates smaller triangulations than the best previous algorithms.  ...  We propose a new Delaunay refinement algorithm based on iterative insertion of off-centers.  ...  Acknowledgements I would like to thank Jonathan Shewchuk for his feedback on an earlier version of this paper and integrating the offcenter insertion algorithm in the latest releases (versions 1.5 and  ... 
doi:10.1016/j.comgeo.2008.06.002 fatcat:wb55eq6prbbcnlxbbfqsamabcq

Maximizing the Minimum Angle with the Insertion of Steiner Vertices

Shankar Sastry
2015 Canadian Conference on Computational Geometry  
The worst-case complexity of the algorithm remains O(n 3 log n), but it is much easier to implement partly because our algorithm requires the solutions of an (at most) eighth-degree, univariate polynomial  ...  We extend the algorithm by imposing constraints on the location of the Steiner vertex and solving the constrained optimization problem in a similar manner.  ...  The author would also like to thank Christine Pickett, an editor at the University of Utah, for proofreading and finding numerous typos in the paper.  ... 
dblp:conf/cccg/Sastry15 fatcat:6ngxlniplvay5gnglfwf53e2vm

A 3D Surface Reconstruction Method for Large-Scale Point Cloud Data

Baoyun Guo, Jiawen Wang, Xiaobin Jiang, Cailin Li, Benya Su, Zhiting Cui, Yankun Sun, ChangLei Yang
2020 Mathematical Problems in Engineering  
Then, the Delaunay triangulation algorithm of multicore parallel computing was used to construct the point cloud data in the leaf nodes.  ...  verify the effectiveness and practicability of the proposed approach.  ...  Since the time complexity and space complexity of the Ball-Pivoting algorithm are also linear [5] , and we cannot determine the complexity of the RVCP algorithm, the memory usage and time consumption  ... 
doi:10.1155/2020/8670151 fatcat:q77c225mvvcldosuaka63zi4cy

MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1995 Lecture Notes Series on Computing  
We discuss algorithms both for the optimization of triangulations on a xed set of vertices and for the placement of new vertices (Steiner points).  ...  We brie y survey the heuristic algorithms used in some practical mesh generators. remove the rst edge e from the queue. If Q e is not reversed, we simply continue to the next edge.  ...  This material spans a spectrum from purely theoretical results (for example, Chazelle's linear-time triangulation algorithm), through a middle ground (our own work on Steiner triangulations), to practical  ... 
doi:10.1142/9789812831699_0003 fatcat:ec6436f5i5bszheuiherz452ae

MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1992 Lecture Notes Series on Computing  
We discuss algorithms both for the optimization of triangulations on a xed set of vertices and for the placement of new vertices (Steiner points).  ...  We brie y survey the heuristic algorithms used in some practical mesh generators. remove the rst edge e from the queue. If Q e is not reversed, we simply continue to the next edge.  ...  This material spans a spectrum from purely theoretical results (for example, Chazelle's linear-time triangulation algorithm), through a middle ground (our own work on Steiner triangulations), to practical  ... 
doi:10.1142/9789814355858_0002 fatcat:zjlfgakee5df7nihosmpeuhco4
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