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## Proximity Problems on Moving Points (1997)

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- [graphics.stanford.edu]
- [theory.stanford.edu]
- [graphics.stanford.edu]
- [www.basch.org]
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### Other Repositories/Bibliography

Venue: | In Proc. 13th Annu. ACM Sympos. Comput. Geom |

Citations: | 49 - 15 self |

### Citations

343 | A data structure for dynamic trees
- Sleator, Tarjan
- 1983
(Show Context)
Citation Context ...e, and (3) e 1 is on the MST path connecting the two endpoints of e 2 . This last condition can be detected and maintained within O(log n) time per operation using the link-cut tree data structure of =-=[ST83]-=-. Thus, we have a kinetic data structure to maintain the #-MST where # is a polyhedral metrics induced by a set of dense vectors. Let m = max #v#V |C(#v)|. The number of cones M is then bounded by |V ... |

298 | Computational Geometry: An Introduction Through Randomized Algorithms. - Mulmuley - 1994 |

256 | Data Structures for Mobile Data”,
- Basch, Guibas
- 1997
(Show Context)
Citation Context ...ence in the motion of the points to gain e#ciency. These new algorithms are kinetic --- they are instances of kinetic data structures (KDSs for short), as introduced by Basch, Guibas, and Hershberger =-=[BGH97]-=-. In the kinetic setting, a set of objects is assumed to be continuously changing, or moving. Each object follows a posted flight plan, but a plan which can change at any moment through a flight plan ... |

222 | On constructing minimum spanning trees in K-dimensional spaces and related problems’,
- Yao
- 1982
(Show Context)
Citation Context ...nd then use ordinary graph algorithm on this subgraph to compute the MST. Delaunay triangulation, lunar graphs etc. are such graphs. Here, we are going to use Yao's geographic nearest neighbor graphs =-=[Yao82]-=-. Consider a set of cones C with common apex at the origin and forming a partitioning of the space. For a point p # S and C # C, let the geographic nearest neighbor to p in C be the point(other than p... |

151 | Randomized Search Trees,
- Aragon, Seidel
- 1989
(Show Context)
Citation Context ...ay to dynamize T is to do some local and global rebuilding after every few operations, which gives an amortized bound of O(t f log k n) per operation [Meh84, Ove83], or to use randomized search trees =-=[AS89]-=- and obtain the same bounds in expectation. 4 2.2 Kinetizing the tree T We now assume that we are given a set S of points whose positions and weights are continuous functions of time; we wish to maint... |

137 |
Data structures and algorithms 3: Multidimensional searching and computational geometry
- Mehlhorn
- 1984
(Show Context)
Citation Context ...the reader is referred to [BGH97]. The results of this paper are based on kinetizing (i.e., maintaining under continuous point motion) a certain type of multidimensional range search tree (MDRS tree) =-=[Meh84]-=- used to query the value of some specified function on subsets of the points [BS80]. We apply this MDRS tree to obtain kinetic data structures for the closest pair among n moving points in IR d , the ... |

134 | A randomized linear-time algorithm for finding minimum spanning trees - Karger, Klein, et al. - 1995 |

114 |
Efficient algorithms for finding minimum spanning trees in undirected and directed graphs.
- Gabow, Galil, et al.
- 1986
(Show Context)
Citation Context ...the total number of cones in our construction. 2 Hence, to construct the ffi ffl -MST, we first build the LNNG, then find the MST of the LNNG by either a slightly superlinear deterministic algorithm (=-=[GGST86]-=-) or a linear randomizedsalgorithm([KKT95]). Once again, the LNNG can be found by the multidimensional range search technique. In order to perform a d dimensional range search, we triangulate each Vor... |

113 | Decomposable searching problems I: Static-to-dynamic transformation - Bentley, Saxe - 1980 |

104 | Improved bounds on planar k-sets and related problems.
- Dey
- 1998
(Show Context)
Citation Context ... times for constant degree algebraic point motions. Thus there are at most O((Mn) 2 ) events of third type in each stage. The total is O(M 2 n 3 ). 13 According to the recent result of Dey on k-level =-=[Dey98]-=-, combined with the result in [KTI95], the number of changes of the MSTs under polyhedral metric for linearly moving points is bounded by O(M 4/3 n 7/3 ). Thus, the above algorithm is not e#cient for ... |

67 | Euclidean minimum spanning trees and bichromatic closest pairs, Discrete Comput - Agarwal, Edelsbrunner, et al. - 1991 |

65 | Transitions in geometric minimum spanning trees’, in - Monma, Suri - 1991 |

55 |
Adding range restriction capability to dynamic data structures
- Willard, Lueker
- 1985
(Show Context)
Citation Context ... in the recursive definition of T to a balanced partitioning and using a careful rotation scheme, the above structure can be maintained and queried within O(t f log k n) worst case time per operation =-=[WL85]-=-. Another way to dynamize T is to do some local and global rebuilding after every few operations, which gives an amortized bound of O(t f log k n) per operation [Meh84, Ove83], or to use randomized se... |

38 | Dynamic Euclidean minimum spanning trees and extrema of binary functions - Eppstein - 1995 |

35 |
Fast collision detection among multiple moving spheres
- Kim, Guibas, et al.
- 1997
(Show Context)
Citation Context ... an e#cient, compact, responsive and local kinetic data structure is given for the closest pair in the plane. Another event-based algorithm for detecting the collision of moving balls was proposed in =-=[KGS97]-=-. For arbitrary dimensions the performance of the algorithms given in this paper is nearly optimal as a function of n (the number of points or objects), but not of d (the dimension --- the `hidden' co... |

32 | M.R.: Parametric and kinetic minimum spanning trees
- Agarwal, Eppstein, et al.
- 1998
(Show Context)
Citation Context ...nges of the MST itself for any type of metric. Even for a fixed (and even planar) graph with variable edge lengths, the problem of kinetically maintaining an MST e#ciently seems quite challenging. In =-=[AEGH98]-=-, a sub-quadratic algorithm is presented for planar graphs with linearly changing weights. Finally, a kinetic data structure that can maintain the exact EMST in a responsive, compact, and local manner... |

32 |
Farthest neighbors, maximum spanning trees and related problems
- Agarwal, Matousek, et al.
- 1992
(Show Context)
Citation Context ...sed by Vaidya [Vai84], is the basis of our method for approximating the Euclidean distance by a certain polyhedral metric and thus the EMST by a polyhedral MST. Similar constructions were employed in =-=[AMS92]-=- for the maximum spanning tree problem. However, the technical development given here is di#erent. In the previous section, we defined #-dense vector sets. It is easy to see that a metric induced by a... |

28 | The Design of Dynamic Data Structures, volume 156 - Overmars |

18 | Geometric Lower Bounds for Parametric Matroid Optimization - Eppstein - 1995 |

18 | An O(|E| log log |V |) algorithm for finding minimum spanning trees - Yao - 1975 |

17 | Minimum spanning trees in k-dimensional space - Vaidya - 1988 |

14 | Dynamic Half-Space Reporting, Geometric Optimization, and Minimum Spanning Trees". - Agarwal, Eppstein, et al. - 1992 |

13 | A faster deterministic algorithm for minimum spanning trees - Chazelle - 1997 |

13 |
On minimum and maximum spanning trees of linearly moving points, Discrete Comput
- Katoh, Tokuyama, et al.
- 1995
(Show Context)
Citation Context ...or the closest pair maintenance. It is proved in [Yao82] that there exist a set C with constant number (dependent on d) of cones so that the EMST is a subgraph of GNNG. The same graph is also used in =-=[KTI95]-=- to bound the number of combinatorial changes of the MST of linearly moving points under L 1 , L 2 , L# metrics. On the other hand, the cone structure also appears when we compute the distance under a... |

11 |
Maintaining the minimal distance of a point set in polylogarithmic time
- Smid
- 1992
(Show Context)
Citation Context ...e "diagonal" of this orthant (the line through the origin in the orthant making equal angles with the axes). Such "quadrant-tilings" have also been previously used for dynamic clos=-=est-pair algorithms [Smi92]-=-. What is surprising here is that our rank functions, which are all based on ordering the points along certain fixed directions, can be used to find an exact (not approximate) L 2 closest pair. As usu... |

9 | Bounds for the Parametric Spanning Tree Problem - Gusfield - 1979 |

6 | A randomized linear-time algorithm to ¯nd minimum spanning trees - Karger, Klein, et al. - 1995 |

5 | Probabilistic analysis for combinatorial functions of moving points. This volume
- Basch, Devarajan, et al.
(Show Context)
Citation Context ...nce is crucial in designing the KDS for the closest pair. However, such packing property does not seem to help us to maintain the nearest neighbor graph --- another important proximity structures. In =-=[BDIZ97]-=-, an alternate probabilistic setting 16 for judging the e#ciency of kinetic data structures was proposed. In their setting, one computes the expected number of events processed when a kinetic data str... |

4 | E cient algorithms for nding minimum spanning trees in undirected and directed graphs, Combinatorica 6:2 - Gabow, Galil, et al. - 1986 |

2 |
A fast approximation for minimum spanning trees in k-dimensional space
- Vaidya
- 1984
(Show Context)
Citation Context ...tors to approximate the Euclidean distance. Or equivalently, we use a polyhedral metric induced by a set of dense vectors to approximate the Euclidean metric. This idea, which was also used by Vaidya =-=[Vai84]-=-, is the basis of our method for approximating the Euclidean distance by a certain polyhedral metric and thus the EMST by a polyhedral MST. Similar constructions were employed in [AMS92] for the maxim... |