## Kinetic data structures (2004)

Venue: | Handbook of Data Structures and Applications |

Citations: | 20 - 1 self |

### BibTeX

@INPROCEEDINGS{Guibas04kineticdata,

author = {Leonidas Guibas},

title = {Kinetic data structures},

booktitle = {Handbook of Data Structures and Applications},

year = {2004},

publisher = {CRC Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

972 |
O.: Computational Geometry: Algorithms and Applications
- Berg, Kreveld, et al.
- 1997
(Show Context)
Citation Context ... convex hull of the primal points. Such dualities represent a powerful toolkit in computational geometry; readers are referred to any standard computational geometry textbook for details, for example =-=[dBvKOS00]-=-. For simplicity we focus only on the upper envelope of the moving lines from now on; the lower envelope case is entirely symmetric. Using a standard divide-and-conquer approach, we partition our line... |

657 | OBBTree : A hierarchical structure for rapid interference detection
- Gottschalk, Lin, et al.
- 1996
(Show Context)
Citation Context ...separating subdivision of the objects, and thus sensitive to how intertwined the objects are. Deformable objects are more challenging to handle. Classical methods, such as bounding volume hierarchies =-=[GLM96]-=-, become expensive, as the fixed object hierarchies have to be rebuilt frequently. One possibility for mitigating this cost is to let the hierarchies themselves deform continuously, by having the boun... |

231 | Data Structures for Mobile Data
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ... theoretical question for several geometric attribute functions.) 1.4.3 The convex hull, revisited We now briefly describe a KDS for maintaining the convex hull of n points moving around in the plane =-=[BGH99]-=-. The key goal in designing a KDS is to produce a repairable certification of the geometric object we want to track. In the convex hull case it turns out that it is a bit more intuitive to look at the... |

154 | Geometric spanners for routing in mobile networks - Gao, Guibas, et al. |

107 |
The visibility complex
- Pocchiola, Vegter
- 1996
(Show Context)
Citation Context ...parallel to a given direction). It turns out that in practice these generate “sliver-like” BSP tiles that lead to robustness issues [Com99]. As the pioneering work on the visibility complex has shown =-=[PV96]-=-, another structure that is well suited to visibility queries in R 2 is an appropriate pseudotriangulation. Given a moving observer and convex moving obstacles, a full radial decomposition of the free... |

96 | Discrete mobile centers - Gao, Guibas, et al. |

93 | Realistic input models for geometric algorithms
- Berg, Katz, et al.
(Show Context)
Citation Context ...ed at unrealistic worst-case behavior, without capturing solutions that exploit the special structure of the motion data that actually arise in practice — as already discussed in a related setting in =-=[dBKvdSV97]-=-. Thus it is important to develop a class of kinetic motion-sensitive algorithms, whose performance can be expressed a function of how coherent the motions of the underlying objects are. Non-canonical... |

90 | Kinetic data structures: a state of the art report
- Guibas
- 1998
(Show Context)
Citation Context ...d-trees and R-trees have recently been investigated for moving objects [AHPP02, AGG02]. 1.7 Sources and Related Materials Results not given an explicit reference above may be traced in these surveys. =-=[Gui98]-=-: An early, and by now somewhat dated, survey of KDS work. [AG + 03]: A report based on an NSF-ARO workshop, addressing several issues on modeling motion from the perspective of a variety of disciplin... |

84 |
diagrams: properties, algorithms and applications
- Power
- 1987
(Show Context)
Citation Context ... [AGMR98]. This principle of a set of easily checked local conditions that implies a global property has been used in kinetizing other proximity structures as well. For instance, in the power diagram =-=[Aur87]-=- of a set of disjoint balls, the two closest balls must be neighbors [GZ98]— and this diagram can be kinetized by a similar approach. Voronoi diagrams of more general objects, such as convex polytopes... |

84 |
Almost tight upper bounds for lower envelopes in higher dimensions
- Sharir
- 1994
(Show Context)
Citation Context ... pseudo-algebraic. A proof of efficiency can be developed by extruding the moving lines into space-time surfaces. Using certain well-known theorems about the complexity of upper envelopes of surfaces =-=[Sha94]-=- and the overlays of such envelopes [ASS96] it can be shown that in the worst case the number of events processed by this algorithm is near quadratic (O(n 2+ɛ )). Since the convex hull of even linearl... |

77 | L.: Deformable free space tilings for kinetic collision detection - Agarwal, Basch, et al. - 2002 |

65 | B.: Kinetic collision detection for simple polygons - Kirkpatrick, Snoeyink, et al. - 2002 |

49 | L.: Proximity problems on moving points
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ...ectional neighbors here we mean that we measure distance only alongsKinetic Data Structures 1-9 a given direction in that cone. This geometric fact follows from a packing argument and is exploited in =-=[BGZ97]-=- to give a different method for maintaining the closest pair of points in R d . The advantage of this method is that the kinetic events are changes of the sorted order of the points along a set of dir... |

48 |
Some dynamic computational geometry problems
- Atallah
- 1985
(Show Context)
Citation Context ..., the steady-state form of the hull (after all changes have occurred), or get an upper bound on how many times the convex hull changes during this motion. Such problems were introduced and studied in =-=[Ata85]-=-. A number of other authors have dealt with geometric problems arising from motion, such as collision detection or minimum separation determination [GJS96, ST95, ST96]. For instance, [ST96] shows how ... |

47 | Dynamic skin triangulation
- Cheng, Dey, et al.
(Show Context)
Citation Context ... avoid undesirably “thin” triangles. In certain cases the number of nodes (points) may also have to change in order to stay sufficiently faithful to the underlying physical process; see, for example, =-=[CDES01]-=-. Because in general a triangulation meeting certain criteria is not unique or canonical, it becomes more difficult to assess the efficiency of kinetic algorithms for solving such problems. The lower-... |

46 | Voronoi diagrams of moving points
- Albers, Guibas, et al.
- 1998
(Show Context)
Citation Context ..., an exact upper bound for the total number of such events in the worst-case is still elusive even in R 2 , where the best known upper bound is nearly cubic, while the best lower bound only quadratic =-=[AGMR98]-=-. This principle of a set of easily checked local conditions that implies a global property has been used in kinetizing other proximity structures as well. For instance, in the power diagram [Aur87] o... |

46 | Efficient collision detection for moving polyhedra - Schömer, Thiel - 1995 |

35 | Collision detection for deforming necklaces
- Agarwal, Guibas, et al.
(Show Context)
Citation Context ...olumes defined implicitly in terms of object features. Such an approach was developed for flexible linear objects (such as rope or macromolecules), using combinatorially defined sphere hierarchies in =-=[GNRZ02]-=-. In that work a bounding sphere is defined not in the usual way, via its center and radius, but in an implicit combinatorial way, in terms of four feature points of the enclosed object geometry. As t... |

29 | Parametric and kinetic minimum spanning trees
- Agarwal, Eppstein, et al.
- 1998
(Show Context)
Citation Context ...dratic in the graph size, or O(n 4 ) in our case). This was improved when the weights are linear functions of time to nearly O(n 11/6 ) (subquadratic) for planar graphs or other minor-closed families =-=[AEGH98]-=-.s1-12 When the weights are the Euclidean distances between moving points, only approximation algorithms are known and the best event bounds are nearly cubic [BGZ97]. For many other optimization probl... |

29 | Cylindrical static and kinetic binary space partitions
- Agarwal, Guibas, et al.
- 1997
(Show Context)
Citation Context ...d tree rearrangements — most of the classical BSP construction algorithms do not have this property. In R 2 , a randomized algorithm for maintaining a BSP of moving disjoint line segments is given in =-=[AGMV00]-=-. The algorithm processes O(n 2 )events, the expected cost per tree update is O(log n), and the expected tree size is O(n log n). The maintenance cost increases to O(nλs+2(n)log 2 n) [AEG98] for disjo... |

27 | Kinetic collision detection between two simple polygons - Basch, Erickson, et al. - 1999 |

25 |
The overlay of lower envelopes and its applications. Discrete Comput. Geom
- Agarwal, Schwarzkopf, et al.
- 1996
(Show Context)
Citation Context ...n be developed by extruding the moving lines into space-time surfaces. Using certain well-known theorems about the complexity of upper envelopes of surfaces [Sha94] and the overlays of such envelopes =-=[ASS96]-=- it can be shown that in the worst case the number of events processed by this algorithm is near quadratic (O(n 2+ɛ )). Since the convex hull of even linearly moving points can change Ω(n 2 ) times [A... |

25 | tracking, and reasoning with relations - Sensing - 2002 |

25 |
A model for data in motion
- Kahan
- 1991
(Show Context)
Citation Context ...s typically given at once. In real-world settings, the motion of objects may be imperfectly known and better information may only be obtainable at considerable expense. The model of data in motion of =-=[Kah91]-=- assumes that upper bounds on the rates of change are known, and focuses on being selective in using sensing to obtain additional information about the objects, in order to answer a series of queries.... |

24 |
Sur la sphère vide. A la memoire de Georges Voronoi
- Delaunay
- 1934
(Show Context)
Citation Context ...ell as a wealth of other proximity information among the points. From the kinetic point of view, these are nice structures, because they admit completely local certifications. Delaunay’s 1934 theorem =-=[Del34]-=- states that if a local empty sphere condition is valid for each (d−1)-simplex in a triangulation of points in R d , then that triangulation must be Delaunay. This makes it simple to maintain a Delaun... |

24 | Separation-sensitive collision detection for convex objects
- Erickson, Guibas, et al.
- 1999
(Show Context)
Citation Context ...ling and undersampling the system. For two moving convex polygons in the plane, a kinetic algorithm where the number of events is a function of the relative separation of the two polygons is given in =-=[EGSZ99]-=-. The algorithm is based on constructing certain outer hierarchies on the two polygons. Analogous methods for 3D polytopes were presented in [GXZ01], together with implementation data. A tiling of the... |

24 | Kinetic connectivity for unit disks
- GUIBAS, HERSHBERGER, et al.
(Show Context)
Citation Context ... respective regions overlap. There has been work on kinetically maintaining the connected components of the union of a set of moving geometric regions for the case of rectangles [HS99] and unit disks =-=[GHSZ00]-=-. Clustering mobile nodes is an essential step in many algorithms for establishing communication hierarchies, or otherwise structuring ad hoc networks. Nodes in close proximity can communicate directl... |

22 | Kinetic Medians and kd-Trees - Agarwal, Gao, et al. - 2002 |

19 | Maintaining the extent of a moving point set
- Agarwal, Guibas, et al.
- 1997
(Show Context)
Citation Context ...6] it can be shown that in the worst case the number of events processed by this algorithm is near quadratic (O(n 2+ɛ )). Since the convex hull of even linearly moving points can change Ω(n 2 ) times =-=[AGHV01]-=-, the efficiency result follows. No comparable structure is known for the convex hull of points in dimensions d ≥ 3.sKinetic Data Structures 1-7 a a a d d b e b d e b e c c ab<xde de<xbc ab<yd de<yb d... |

17 | M.: Fast algorithms for collision and proximity problems involving moving geometric objects - Gupta, Janardan, et al. - 1996 |

16 | bounds for kinetic planar subdivisions - Agarwal, Basch, et al. - 1999 |

15 |
Kinetic collision detection: Algorithms and experiments
- Guibas, Xie, et al.
(Show Context)
Citation Context ...elative separation of the two polygons is given in [EGSZ99]. The algorithm is based on constructing certain outer hierarchies on the two polygons. Analogous methods for 3D polytopes were presented in =-=[GXZ01]-=-, together with implementation data. A tiling of the free space around objects can serve as a proof of non-intersection of the objects. If such a tiling can be efficiently maintained under object moti... |

15 | Kinetic connectivity of rectangles
- Hershberger, Suri
- 1999
(Show Context)
Citation Context ...k if and only if their respective regions overlap. There has been work on kinetically maintaining the connected components of the union of a set of moving geometric regions for the case of rectangles =-=[HS99]-=- and unit disks [GHSZ00]. Clustering mobile nodes is an essential step in many algorithms for establishing communication hierarchies, or otherwise structuring ad hoc networks. Nodes in close proximity... |

14 |
Kinetic BSPs for intersecting segments and disjoint triangles
- Agarwal, Erickson, et al.
- 1998
(Show Context)
Citation Context ...given in [AGMV00]. The algorithm processes O(n 2 )events, the expected cost per tree update is O(log n), and the expected tree size is O(n log n). The maintenance cost increases to O(nλs+2(n)log 2 n) =-=[AEG98]-=- for disjoint moving triangles in R 3 (s is a constant depending on the triangle motion). Both of these algorithms are based on variants on vertical decompositions (many of the cuts are parallel to a ... |

13 | Euclidean proximity and power diagrams
- Guibas, Zhang
- 1998
(Show Context)
Citation Context ...mplies a global property has been used in kinetizing other proximity structures as well. For instance, in the power diagram [Aur87] of a set of disjoint balls, the two closest balls must be neighbors =-=[GZ98]-=-— and this diagram can be kinetized by a similar approach. Voronoi diagrams of more general objects, such as convex polytopes, have also been investigated. For example, in R 2 [GSZ00] shows how to mai... |

11 | Star-tree: An efficent self-adjusting index for moving points - Procopiuc, Agarwal, et al. - 2002 |

10 | C.: Subquadratic algorithms for the general collision detection problem
- Schomer, Theil
- 1996
(Show Context)
Citation Context ...tudied in [Ata85]. A number of other authors have dealt with geometric problems arising from motion, such as collision detection or minimum separation determination [GJS96, ST95, ST96]. For instance, =-=[ST96]-=- shows how to check in subquadratic time whether two collections of simple geometric objects (spheres, triangles) collide with each other under specified polynomial motions. 1.3 Motion Models An issue... |

9 |
Kinetic Visibility
- Hall-Holt
- 2002
(Show Context)
Citation Context ...bserver. Thus one can have a structure that compactly encodes the changing visibility polygon around the observer, while being quite stable in regions of the free space welloccluded from the observer =-=[OH02]-=-. 1.5.7 Result summary We summarize in Table 1.9 the efficiency bounds on the main KDSs discussed above. 1.5.8 Open problems As mentioned above, we still lack efficient kinetic data structures for man... |

7 | Using sparsification for parametric minimum spanning tree problems
- Fernández-Baca, Slutzki, et al.
- 1996
(Show Context)
Citation Context ...cation graph. Another fundamental kinetic question is the maintenance of a minimum spanning tree (MST) among n mobile points in the plane, closely related to earlier work on parametric spanning trees =-=[FBSE96]-=- in a graph whose edge weights are functions of a parameter λ (λ is time in the kinetic setting). Since the MST is determined by the sorted order of the edge weights in the graph, a simple algorithm c... |

6 | Compact voronoi diagrams for moving convex polygons
- Guibas, Snoeyink, et al.
- 2000
(Show Context)
Citation Context ...must be neighbors [GZ98]— and this diagram can be kinetized by a similar approach. Voronoi diagrams of more general objects, such as convex polytopes, have also been investigated. For example, in R 2 =-=[GSZ00]-=- shows how to maintain a compact Voronoi-like diagram among moving disjoint convex polygons; again, a set of local conditions is derived which implies the global correctness of this diagram. As the po... |

5 | et al. Algorithmic issues in modeling motion - Agarwal, Guibas - 2002 |

4 |
Agarwal and Sariel Har-Peled. Maintaining approximate extent measures of moving points
- Pankaj
- 2001
(Show Context)
Citation Context ...proximations to the measures, then an approximate smallest orthogonal rectangle, diameter, and smallest enclosing disk can be maintained with a number of events that is a function ɛ only and not of n =-=[AHP01]-=-. For example, the bound of the number of approximate diameter updates in R 2 under linear motion of the points is O(1/ɛ). 1.5.2 Proximity problems The fundamental proximity structures in computationa... |

4 |
Kinetic Vertical Decomposition Trees
- Comba
- 2000
(Show Context)
Citation Context ... based on variants on vertical decompositions (many of the cuts are parallel to a given direction). It turns out that in practice these generate “sliver-like” BSP tiles that lead to robustness issues =-=[Com99]-=-. As the pioneering work on the visibility complex has shown [PV96], another structure that is well suited to visibility queries in R 2 is an appropriate pseudotriangulation. Given a moving observer a... |