## Efficient Regular Data Structures and Algorithms for Dilation, Location and Proximity Problems

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Citations: | 12 - 0 self |

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@MISC{Amir_efficientregular,

author = {Arnon Amir and Alon Efrat and Piotr Indyk and Hanan Samet},

title = {Efficient Regular Data Structures and Algorithms for Dilation, Location and Proximity Problems},

year = {}

}

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### Abstract

In this paper we investigate data-structures obtained by a recursive partitioning of the input domain into regions of equal size. One of the most well known examples of such a structure is the quadtree, used here as a basis for more complex data structures; we also provide multidimensional versions of the stratified tree by van Emde Boas [20]. We show that under the assumption that the input points have limited precision (i.e. are drawn from the integer grid of size u) these data structures yield efficient solutions to many important problems. In particular, they allow us to achieve O(log log u) time per operation for dynamic approximate nearest neighbor (under insertions and deletions) and exact on-line closest pair (under insertions only) in any constant dimension. They allow O(log log u) point location in a given planar shape or in its expansion (dilation by a ball of a given radius). Finally, we provide a linear time (optimal) algorithm for computing the expansion of a shape...

### Citations

1174 |
The Design and Analysis of Spatial Data Structures
- Samet
- 1990
(Show Context)
Citation Context ... bounded region into blocks, where the blocks of each partition are of equal size; we call such structures regular. One of the most popular examples of such structures is the quadtree (e.g., [S1] and =-=[S2]-=-), which is based on a recursive decomposition of a square into four quadrants; another example is the stratified tree structure of van Emde Boas [vEB]. The quadtree data structure and its numerous va... |

974 |
Computational geometry: algorithms and applications
- Berg, Cheong, et al.
- 2008
(Show Context)
Citation Context ... of n nodes, and let r be a given radius. We can construct a segment quadtree ˜T that stores D(S) in time and space O(n log 2 u). PROOF. The constructive proof requires the following definitions (see =-=[dBvKOS]-=-): A balanced quadtree is a quadtree with the additional property that if v1 and v2 are nodes in the tree such that Rv1 and Rv2 share an edge or a portion of an edge, then the depth of v1 differs by a... |

775 | An optimal algorithm for approximate nearest neighbor searching in fixed dimensions - Arya, Mount, et al. |

713 | Approximate nearest neighbors: Towards removing the curse of dimensionality
- Indyk, Motwani
- 1998
(Show Context)
Citation Context ... is a data structure which finds a d 2 -approximate nearest neighbor in the l∞ norm. Having a rough approximation of the nearest neighbor distance (call it R), we refine it by using the techniques of =-=[IM]-=- to obtain a (1 + ɛ)approximation in the following manner. During the preprocessing, for any r = 1,(1 + ɛ),(1 + ɛ) 2 ,...(i.e., for O((log u)/ɛ) different values of r) and for each database point p we... |

418 | Davenport-Schinzel sequences and their geometric applications
- Agarwal, Sharir
- 2000
(Show Context)
Citation Context ...ity of each of them is a constant, and the boundaries of each pair of objects intersect ≤ s times for a constant s. Let λs(n) denote the maximal length of the (n, s)-Davenport–Schinzel sequences (see =-=[SA]-=-). It is known that λs(n) is almost linear in n for any constant s. It is shown in [E] (see also [ES]) that the number N of vertices of ∂ ∪C is only O(λs+2(n) log 2 n log log n). THEOREM 5.1. Let T be... |

140 |
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete and Computational Geometry 1
- Kedem, Livne, et al.
- 1986
(Show Context)
Citation Context ...xity of the deleted portion qβ. Since each part in the region is created only once, and can be removed only once, and since the number of elements in opd(α) is proportional to the complexity of C (by =-=[KLPS]-=- cited above) the execution time over the course of the procedure is linear in the complexity of C. The Merging (Zipping) Process. To explain the dilation merging procedure, we need the following defi... |

112 |
Emde Boas. Preserving order in a forest in less than logarithmic time and linear space
- van
- 1978
(Show Context)
Citation Context ...best known examples of such a structure is the quadtree. It is used here as a basis for more complex data structures. We also provide multidimensional versions of the stratified tree by van Emde Boas =-=[vEB]-=-. We show that under the assumption that the input points have limited precision (i.e., are drawn from the integer grid of size u) these data structures yield efficient solutions to many important pro... |

88 | Preserving order in a forest in less than logarithmic time - Boas, P - 1990 |

85 |
Log-logarithmic worst-case range queries are possible
- Willard
- 1983
(Show Context)
Citation Context ...h lies on the path leading to the leaf including q). We can find the leaf containing q, or determine that no such block exists, in expected time O(log log u). This idea has appeared in the literature =-=[W]-=-. When T is a compressed tree, the analysis is a bit more complicated. This is due to the fact that simple replication of the previous approach would seemingly require hashing all the paths in the tre... |

62 | Optimal bounds for the predecessor problem
- Beame, Fich
- 1999
(Show Context)
Citation Context ...by van Emde Boas [vEB], yields a dynamic data structure for nearest-neighbor queries. The running time which it guarantees, O(log log u), has been recently improved to O(log log u/log log log u) (see =-=[BF]-=-). We are not aware, however, of any prior work in which its multidimensional version has been used. The first multidimensional stratified tree allows O(log log u) time per operation for dynamic appro... |

44 |
A priority queue in which initialization and queue operations take O(log log D) time
- Johnson
- 1982
(Show Context)
Citation Context ...ratified tree was proposed by van Emde Boas [vEB]. It supports the performance of a nearest neighbor query in the integer interval [u] inO(log log u) time. The data structure can be made dynamic (see =-=[J]-=-). Each addition or deletion of a point takes O(log log u) time. The tree requires O(n) space, where n denotes the maximum number of points present in the tree at any time. 9 It is assumed that standa... |

40 | The complexity and construction of many faces in arrangements of lines and of segments - Edelsbrunner, Guibas, et al. - 1990 |

36 | A general approach to connected-component labeling for arbitrary image representations
- Dillencourt, Samet, et al.
- 1992
(Show Context)
Citation Context ...nt Regular Data Structures 165 • Versatility: many operations on such data structures can be performed very efficiently (for example, computing the union/intersection, or connected component labeling =-=[DST]-=-, [S2]). Despite their usefulness, however, regular data structures for geometric problems have not been investigated much from the theoretical point of view. One of the main reasons is that in the wi... |

35 | Improvements on bottleneck matching and related problems using geometry
- Efrat, Itai
- 1996
(Show Context)
Citation Context ...the down edge of a block and C is the down envelope). We need to compute op(C, down), the region of D(C) which is below ℓ (see Figure 8). To perform this task, we need the following lemma, taken from =-=[EI]-=-; LEMMA 4.3. Assume C consists of Cl, a left part, and Cr, its right part, where Cl is completely to the left of Cr. Then op(Cl, down) and op(Cr, down) intersect at most once. Fig. 8. Computing the ou... |

25 | Dynamic data structures for fat objects and their applications
- Efrat, Katz, et al.
(Show Context)
Citation Context ...whose center coincides with the center of Rv, and whose edge-length is larger by some factor κ ′ > 1 than the edge-length of R, where κ ′ > 1 depends on α. Using arguments similar to the ones used in =-=[EKNS]-=-, we can show that at least one of the following cases must occur: • There is an object c ∈ C such that ¯R contains z, a rightmost, a leftmost, a highest or a lowest point of ∂c. • ¯R contains a verte... |

23 |
The complexity of the union of (α, β)-covered objects
- Efrat
(Show Context)
Citation Context ...bjects in [u] 2 , then S can be represented as a segment quadtree with N = O(|∂ S| log u) leaves, where |∂ S| is the complexity of the boundary of S, which in turn is known to be close to linear in n =-=[E]-=-. After we show that a segment quadtree can be an efficient shape representation, in Section 5.2 we give an efficient algorithm to construct it. Given a decomposition of S into n (not necessarily fat)... |

20 | Two-dimensional and three-dimensional point location in rectangular subdivisions - Berg, Kreveld, et al. - 1995 |

15 |
Blasting through the information theoretic barrier with fusion trees
- Fredman, Willard
- 1990
(Show Context)
Citation Context ...uences, each consisting of a 1 followed by v 2 − 1 zeros. Step (b) uses just one Boolean operation. The last step can be implemented using a constant number of Boolean and arithmetic operations as in =-=[FW]-=-. CLAIM 2.3. For any i, j searching in Eij takes time O((C(ɛ, u) + 1/ɛ) · 1/ɛ 3 ). PROOF. Recall that the data structure Eij does not contain other E-type structures. Therefore, it contains only H-typ... |

15 | An algorithm to compute the minkowski sum outer-face of two simple polygons - Ramkumar - 1996 |

9 | A general approach to connected component labeling for arbitrary image representations - Dillecourt, Samet, et al. - 1992 |

8 |
On the complexity of the union of fat objects in the plane
- Efrat, Sharir
(Show Context)
Citation Context ... a constant s. Let λs(n) denote the maximal length of the (n, s)-Davenport–Schinzel sequences (see [SA]). It is known that λs(n) is almost linear in n for any constant s. It is shown in [E] (see also =-=[ES]-=-) that the number N of vertices of ∂ ∪C is only O(λs+2(n) log 2 n log log n). THEOREM 5.1. Let T be a segment quadtree, constructed for the union of C. Then the number of leaves in T is O(N log u), pr... |

7 | The complexity of a single face of a Minkowski sum - Har-Peled, Chan, et al. - 1995 |

6 |
Rasterized point location
- Müller
- 1985
(Show Context)
Citation Context ...size is large. There have been a number of papers discussing computational geometry problems on a grid. Examples include nearest neighbor searching using the L1 norm [RGK], the point location problem =-=[M]-=-, and orthogonal range searching [O]. The solutions given in these papers provide static data structures for two-dimensional data with query time O(log log u). A number of off-line problems have been ... |

6 |
Proximity on a Grid
- Karlsson, Munro
- 1985
(Show Context)
Citation Context ...lly for scenarios where the input size is large. There have been a number of papers discussing computational geometry problems on a grid. Examples include nearest neighbor searching using the L1 norm =-=[RGK]-=-, the point location problem [M], and orthogonal range searching [O]. The solutions given in these papers provide static data structures for two-dimensional data with query time O(log log u). A number... |

5 | A new region expansion for quadtrees - Ang, Samet, et al. - 1990 |

4 | The complexity of the union of (#, #)-covered objects - Efrat - 1999 |

1 |
Range searching on a grid
- Overmars
- 1988
(Show Context)
Citation Context ...ber of papers discussing computational geometry problems on a grid. Examples include nearest neighbor searching using the L1 norm [RGK], the point location problem [M], and orthogonal range searching =-=[O]-=-. The solutions given in these papers provide static data structures for two-dimensional data with query time O(log log u). A number of off-line problems have been also considered (see [O] for more de... |

1 | The complexity of the union of (ff; fi)- covered objects - Efrat - 1998 |

1 | On the complexity of the union of fat objects in the pl ane - Efrat, Sharir - 1997 |

1 | Optimal decomposition of morphological structuring elements - Yang, Lee - 1996 |