## Approximate Data Structures with Applications (Extended Abstract) (1994)

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Citations: | 12 - 7 self |

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@MISC{Matias94approximatedata,

author = {Yossi Matias and Jeffrey Scott Vitter and Neal E. Young},

title = {Approximate Data Structures with Applications (Extended Abstract)},

year = {1994}

}

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### Abstract

In this paper we introduce the notion of approximate data structures, in which a small amount of error is tolerated in the output. Approximate data structures trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure [28, 20], except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is 1/polylog(n), and in O(loglog n) time provided the error is 1/polynomial(n), for n elements in the data structure. We consider

### Citations

2542 |
The Design and Analysis of Computers Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...factor of some 1 + ǫ. For the additively approximate variant, the function f preserves the order of any two elements differing additively by at least some ∆. Let the elements be taken from a universe =-=[1, U]-=-. On an arithmetic ram with b-bit words, the times required per operation in our approximate data structures are as follows: time multiplicative approx. (1 + ǫ) ( ) log U O log logb ǫ additive approx.... |

1594 | A note on two problems in connexion with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ...ion 4. The proofs are simple and are given in the full paper. 2.1 Minimum spanning trees. For the minimum spanning tree problem, we show the following result about the performance of Prim’s algorithm =-=[16, 25, 7]-=- when our approximate veb data structure is used to implement the priority queue: Theorem 2.1. Given a graph with edge weights in {0, .., U}, Prim’s algorithm, when implemented with our approximate ve... |

618 | Data Structures and Network Algorithms - Tarjan - 1996 |

573 |
Shortest connection networks and some generalizations, Bell System Technical Journal 36
- Prim
- 1957
(Show Context)
Citation Context ...ion 4. The proofs are simple and are given in the full paper. 2.1 Minimum spanning trees. For the minimum spanning tree problem, we show the following result about the performance of Prim’s algorithm =-=[16, 25, 7]-=- when our approximate veb data structure is used to implement the priority queue: Theorem 2.1. Given a graph with edge weights in {0, .., U}, Prim’s algorithm, when implemented with our approximate ve... |

260 |
An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set
- Graham
- 1972
(Show Context)
Citation Context ...ained within the true convex hull such that the distance of any point on the true hull to the approximate hull is O(∆) times the diameter. We show the following result about the Graham scan algorithm =-=[12]-=- when run using our approximate veb data structure: Theorem 2.3. The on-line (1+∆)-approximate convex hull can be computed by a Graham scan in constant amortized time per update if ∆ ≥ log −c n for an... |

200 |
Design and implementation of an efficient priority queue
- BOAS, KAAS, et al.
- 1977
(Show Context)
Citation Context ...cal speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure =-=[28, 20]-=-, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is 1/polylog(n), and in O(log log n) time provided the error ... |

161 |
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
- Fredman, Willard
- 1994
(Show Context)
Citation Context ... size O(log U). In [10], they presented the fusion tree data structure. Briefly, fusion trees implement the veb data type in time O(log n/ loglog n). They also presented an atomic heap data structure =-=[11]-=- based on their fusion tree and used it to obtain a lineartime minimum spanning tree algorithm and an O(m + n log n/ loglog n)-time single-source shortest paths algorithm. Willard [29] also considered... |

100 | Epsilon geometry: Building robust algorithms from imprecise computations
- Guibas, Salesin, et al.
- 1989
(Show Context)
Citation Context ...es is related in spirit to the challenging problems that arise from various types of error in numeric computations. Such errors has been studied, for example, in the context of computational geometry =-=[8, 9, 13, 14, 21, 22, 23]-=-. We discuss this further in Section 6. Approximate sorting. Bern, Karloff, Raghavan, and Schieber [3] introduced approximate sorting and applied it to several geometric problems. Their results includ... |

80 |
Finiteresolution computational geometry
- Greene, Yao
- 1986
(Show Context)
Citation Context ...es is related in spirit to the challenging problems that arise from various types of error in numeric computations. Such errors has been studied, for example, in the context of computational geometry =-=[8, 9, 13, 14, 21, 22, 23]-=-. We discuss this further in Section 6. Approximate sorting. Bern, Karloff, Raghavan, and Schieber [3] introduced approximate sorting and applied it to several geometric problems. Their results includ... |

73 |
Verifiable implementations of geometric algorithms using finite precision arithmetic
- Milenkovic
- 1988
(Show Context)
Citation Context ...es is related in spirit to the challenging problems that arise from various types of error in numeric computations. Such errors has been studied, for example, in the context of computational geometry =-=[8, 9, 13, 14, 21, 22, 23]-=-. We discuss this further in Section 6. Approximate sorting. Bern, Karloff, Raghavan, and Schieber [3] introduced approximate sorting and applied it to several geometric problems. Their results includ... |

48 | Data Structures and Algorithms
- Mehlhorn
- 1984
(Show Context)
Citation Context ...cal speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure =-=[28, 20]-=-, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is 1/polylog(n), and in O(log log n) time provided the error ... |

46 |
Stable maintenance of point set triangulations in two dimensions
- Fortune
- 1989
(Show Context)
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40 |
auf der Heide, A new universal class of hash functions and dynamic hashing in real time
- Dietzfelbinger, Meyer
- 1990
(Show Context)
Citation Context ...S)| is the number of distinct elements under the mapping), and t is the time required per operation. The overhead incurred by using dynamic hashing is constant per memory access with high probability =-=[6, 5]-=-. Thus, if the data structures are implemented to use nearly linear space, the times given per operation hold only with high probability. 1.1 Description of the data structure. The approach is simple ... |

27 |
Polynomial hash functions are reliable
- Dietzfelbinger, Gil, et al.
- 1992
(Show Context)
Citation Context ...S)| is the number of distinct elements under the mapping), and t is the time required per operation. The overhead incurred by using dynamic hashing is constant per memory access with high probability =-=[6, 5]-=-. Thus, if the data structures are implemented to use nearly linear space, the times given per operation hold only with high probability. 1.1 Description of the data structure. The approach is simple ... |

27 | Numerical stability of algorithms for line arrangements
- Fortune, Milenkovic
- 1991
(Show Context)
Citation Context |

27 | Double Precision Geometry: a General Technique for Calculating Line and Segment Intersections using Rounded Arithmetic
- Milenkovic
- 1989
(Show Context)
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21 |
Upper bounds for sorting integers on random access machines, Theoret
- Kirkpatrick, Reisch
(Show Context)
Citation Context ... as a combination of the algorithms by Shamos and by Bentley et al., with the replacement of an exact veb data structure by an approximate variant. Computation with large words. Kirkpatrick and Reich =-=[17]-=- considered exact sorting with large words, giving upper and lower bounds. Their interest was theoretical, but Lemma 5.1, which in some sense says that maintaining an approximate veb data structure is... |

21 |
A randomized lineartime algorithm to minimum spanning trees
- Karger, Klein, et al.
- 1995
(Show Context)
Citation Context ...they are more complicated and involve larger constants. Subsequent to our work Klein and Tarjan recently announced a randomized minimum spanning tree algorithm that requires only expected linear time =-=[18]-=-. Arguably, our algorithm is simpler and more practical. 4 Model of computation The model of computation assumed in this paper is a modernized version of the random access machine (ram). Many ram mode... |

19 |
On maintaining the width and diameter of a planar point-set online
- JANARDAN
- 1993
(Show Context)
Citation Context ...ep. Bentley, Faust, and Preparata [2] give an O(n + 1/∆)-time algorithm that finds a (1 + ∆)approximate convex hull. Their result was superseded by the result of Bern et al. mentioned above. Janardan =-=[15]-=- gave an algorithm maintaining a fully dynamic (1 + ∆)-approximate convex hull (allowing deletion of points) in O(log(n)/∆) time per request. Our on-line approximation algorithm is based on Graham’s s... |

17 | Calculating approximate curve arrangements using rounded arithmetic
- Milenkovic
- 1972
(Show Context)
Citation Context |

17 |
Applications of the fusion tree method to computational geometry and searching
- Willard
- 1992
(Show Context)
Citation Context ... data structure [11] based on their fusion tree and used it to obtain a lineartime minimum spanning tree algorithm and an O(m + n log n/ loglog n)-time single-source shortest paths algorithm. Willard =-=[29]-=- also considered similar applications to related geometric and searching problems. Generally, these works assume a machine model similar to ours and demonstrate remarkable theoretical consequences of ... |

15 |
Blasting through the information theoretic barrier with fusion trees
- Fredman, Willard
- 1990
(Show Context)
Citation Context ...g and data structures. Exploiting the power of RAM. Fredman and Willard have considered a number of data structures taking advantage of arithmetic and bitwise operations on words of size O(log U). In =-=[10]-=-, they presented the fusion tree data structure. Briefly, fusion trees implement the veb data type in time O(log n/ loglog n). They also presented an atomic heap data structure [11] based on their fus... |

15 |
An optimal real-time algorithm for planar convex hulls
- Preparata
- 1979
(Show Context)
Citation Context ...re several relevant works for the on-line convex hull problem. Shamos (see, e.g., [26]) gave an on-line algorithm for (exact) convex hull that takes O(log n) amortized time per update step. Preparata =-=[24]-=- gave a real-time on-line (exact) convex hull algorithm with O(log n)-time worst-case time per update step. Bentley, Faust, and Preparata [2] give an O(n + 1/∆)-time algorithm that finds a (1 + ∆)appr... |

14 |
Approximation algorithms for convex hulls
- Bentley, Faust, et al.
- 1982
(Show Context)
Citation Context ... takes O(log n) amortized time per update step. Preparata [24] gave a real-time on-line (exact) convex hull algorithm with O(log n)-time worst-case time per update step. Bentley, Faust, and Preparata =-=[2]-=- give an O(n + 1/∆)-time algorithm that finds a (1 + ∆)approximate convex hull. Their result was superseded by the result of Bern et al. mentioned above. Janardan [15] gave an algorithm maintaining a ... |

14 |
Fast geometric approximation techniques and geometric embedding problems
- Bern, Karloff, et al.
- 1990
(Show Context)
Citation Context ...rs has been studied, for example, in the context of computational geometry [8, 9, 13, 14, 21, 22, 23]. We discuss this further in Section 6. Approximate sorting. Bern, Karloff, Raghavan, and Schieber =-=[3]-=- introduced approximate sorting and applied it to several geometric problems. Their results include an O((n log log n)/ǫ)-time algorithm that finds a (1+ǫ)approximate Euclidean minimum spanning tree. ... |

11 | Sorting helps for Voronoi diagrams
- Chew, Fortune
- 1997
(Show Context)
Citation Context ...nd Milenkovic [9], the priority queue can be replaced by an approximate priority queue with minor adjustments, to obtain an output with similar accuracy. If the sweeping algorithm of Chew and Fortune =-=[4]-=- can be shown to be appropriately robust then the use of the van Emde Boas priority queue there can be replaced by an approximate variant; an improved running time may imply better performance for alg... |

10 | An e cient algorithm for determining the convex hull of a nite planar set - Graham - 1972 |

8 | Design and implementation of an e cient priority queue - Boas, Kaas, et al. - 1977 |

1 | ik. O jist'em probl'emu minim'almn ' im. Pr'aca Moravsk'e Pr ' irodovedeck'e Spolecnosti, 6:57--63 - Jarn - 1930 |

1 |
Dynamic generation of random variates
- Matias, Vitter, et al.
- 1993
(Show Context)
Citation Context ...point is constant. 3 Related work Our work was inspired by and improves upon data structures developed for use in dynamic random variate4 Matias, Vitter, & Young generation by Matias, Vitter, and Ni =-=[19]-=-. Approximation techniques such as rounding and bucketing have been widely used in algorithm design. This is the first work we know of that gives a generalpurpose approximate data structure. Finite pr... |

1 | A note on two problems in connexion vith graphs - Dijkstra - 1959 |

1 | O jistm problmu minimlmnm - Jarnk - 1930 |

1 | Verifiable hnplementations of Geometric Algorithms using Finite Precision Arithmetic - Milenkovic - 1988 |

1 | Shortest connection netvorks and some generalizations - Prim - 1957 |

1 | Data Structures and Network Algorithms - Trojan - 1983 |

1 |
O jistém problému minimálmním. Práca Moravské Prírodovedecké Spolecnosti, 6:57–63
- Jarník
- 1930
(Show Context)
Citation Context ...ion 4. The proofs are simple and are given in the full paper. 2.1 Minimum spanning trees. For the minimum spanning tree problem, we show the following result about the performance of Prim’s algorithm =-=[16, 25, 7]-=- when our approximate veb data structure is used to implement the priority queue: Theorem 2.1. Given a graph with edge weights in {0, .., U}, Prim’s algorithm, when implemented with our approximate ve... |

1 | Veri able Implementations ofGeometric Algorithms using Finite Precision Arithmetic - Milenkovic - 1988 |