M. T. Dickerson and D. Eppstein, "Algorithms for proximity problems in higher dimensions," Comput. Geom. Theory and Appl., vol. 5, no. 5, pp. 277--291, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for randomness [18] The k NNG problem consists of finding the set N k,i of k nearest neighbors of each point X i in the set X . This problem has exact solutions which run in linear loglinear time and the weight is L k NNG e#Nk,i . The k NNG arises in computational geometry [19], clustering and pattern recognition [20] spatial statistics [21] and adaptive vector quantization [22] The following technical conditions on a Euclidean functional L # were defined in [3, 2] Null condition: L # (#) 0, where # is the null set. Subadditivity: Let i=1 be a ....

M. T. Dickerson and D. Eppstein, "Algorithms for proximity problems in higher dimensions," Comput. Geom. Theory and Appl., vol. 5, no. 5, pp. 277--291, 1996.

....first experimental evidence of the validity of algebraic degree as a complexity measure. This problem of finding intersecting pairs in a set of unit balls is known as the Fixed Radius Near Neighbors Search, Dickerson and Eppstein solved it in O(n log n k) time in fixed arbitrary dimension [7]. Similarly, our algorithm is optimal only when the dimension d is constant, but our experimental results are very good and it extends to the following more general setting. Let E be a set of n objects in fixed dimension d. They are supposed to be thick in the sense that each one of them has ....

....slower. Of course, when E is sparse our algorithm is far more efficient. Then I tried it against a Voronoi diagram computation. I believe it to be an interesting comparison for two reasons. First, no experimental result has been published for the previous fixed radius near neighbors algorithms [2, 7]. Second, Dickerson and Eppstein algorithm [7] starts by computing a Delaunay triangulation, which is equivalent to computing a Voronoi diagram. When k n, my program is about 100 times faster than an exact Voronoi diagram computation with CGAL [9] 4 Conclusion The main weakness of our ....

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M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comput. Geom. Theory Appl., 5:277--291, 1996.

....n) If = 2, then the expected preprocessing time is roughly n 3=2 . For = 3, it is roughly n 4=3 . For larger values of , the time bound remains roughly n 4=3 , but then the approximation ratio increases. Assume that d 3. Agarwal, in a personal communication to Dickerson and Eppstein [10], has shown that T sel (n) O(n 2(1 1= d 1) 7) 21 where is an arbitrarily small positive real constant. The constant in the Big Oh bound depends on . Agarwal and Matou sek [1] and Matou sek and Schwarzkopf [12] have given a static nearest neighbor data structure for which nQ s ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comput. Geom. Theory Appl., 5:277-291, 1996.

....# = 2, then the expected preprocessing 20 time is roughly n 3 2 . For # = 3, it is roughly n 4 3 . For larger values of #, the time bound remains roughly n 4 3 , but then the approximation ratio increases. Assume that d # 3. Agarwal, in a personal communication to Dickerson and Eppstein [10], has shown that T sel (n) O(n 2(1 1 (d 1) # ) 7) where # is an arbitrarily small positive real constant. The constant in the Big Oh bound depends on #. Agarwal and Matousek [1] and Matousek and Schwarzkopf [12] have given a static nearest neighbor data structure for which nQ s NN ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comput. Geom. Theory Appl., 5:277--291, 1996.

....of comparison proves the superiority of AUTOCLUST in terms of eciency and e ectiveness. We expect that our methods will extended in two directions. First, the approach can be extended to 3 dimensions by computing a spanner graph of linear size and obtaining local and global information eciently [6] from this graph. Second, the approach can naturally be extended to other metrics with well de ned Delaunay Diagrams like Manhattan distance or L1 distances. Further research will explore these directions. ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Computational Geometry - Theory and Applications, 5:277-291, 1996.

....Vaidya s all nearest neighbor method, making implementation difficult. Lenhof and Smid [24] subsequently pointed out a much simplified algorithm the only geometric structure needed is a grid. Alternatives that apply more advanced geometric structures have been proposed by Dickerson and Eppstein [13] (higher dimensional Delaunay triangulations) and Arya and Smid [4] spanners) In this paper, we add two more distance enumeration algorithms to the list: ffl A further simplification of Lenhof and Smid s algorithm, running in O(n log n k) time. In their algorithm, parametric search is avoided ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comput. Geom. Theory Appl., 5:277--291, 1996.

....as parametric searching [28] or Vaidya s all nearest neighbor method. Lenhof and Smid [23] subsequently pointed out a much simplified algorithm; the only geometric structure needed is a grid. Alternatives that apply more advanced geometric structures have been proposed by Dickerson and Eppstein [14] (higher dimensional Delaunay triangulations) and Arya and Smid [4] spanners) In this paper, we add two more distance enumeration algorithms to the list: ffl A further simplification of Lenhof and Smid s algorithm, running in O(n log n k) time. In their algorithm, parametric search is avoided ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comput. Geom. Theory Appl., 5:277--291, 1996.

....They also show that the running time of the algorithm is bounded by O(n log n k log k) The efficiency of the algorithm of Dickerson et al. heavily depends on the fact that the Delaunay triangulation has linear size. This does not hold for dimensions greater than two. Dickerson and Eppstein [53], however, circumvent this by using the following result of Bern, Eppstein and Gilbert [26] Given a set S of n points in IR D , there is a superset S 0 of S having size O(n) such that the Delaunay triangulation DT 0 of S 0 has size O(n) and bounded degree. Such a superset can be computed ....

....and bounded degree. Such a superset can be computed in O(n log n) time. Applying the algorithm given above to DT 0 gives the k closest pairs of S. Clearly, the correctness proof is more complicated than in the planar case, because we use the Delaunay triangulation of S 0 instead of S. See [53]. The entire algorithm solves the k closest pairs problem in O(n log n k log k) time. Note that this algorithm reports the k closest pairs in sorted order. In [53] a variant is given that reports the k closest pairs in no particular order in O(n log n k) time. This is optimal in the ....

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M.T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Computational Geometry, Theory and Applications 5 (1996), pp. 277-291.

....the smallest constrained circular ring (or a sector of a constrained ring) that contains k (k n 2 ) points of S. 6. Given a number k n 2 , decide whether a query rectangle contains k points or less. 1. 2 Background Most of the problems mentioned above have been considered in previous papers [7, 8, 9, 11, 18]. We summarize our and previous results in Table 1. Dickerson et al. 7] present an algorithm for the first problem which runs in time O(n log n nk log k) and works for any convex distance function. Eppstein and Erickson [11] solve the first problem on a random access machine model in time O(n ....

....problem which runs in time O(n log n nk log k) and works for any convex distance function. Eppstein and Erickson [11] solve the first problem on a random access machine model in time O(n log n kn) and O(n log n) space. In the algebraic Pbm Previous results Our results 1 O(n log n kn) [9] O(n log n (n Gamma k)n) 2 O(n log n k log k log n log log n ) expected) 8] O(n k log n) 3 O(n log n k) 9] O(n log n k) 4 open, constrained open, non constrained O(n(n Gamma k) log (n Gamma k) O(n(n Gamma k) 4 log n) 5 open O(n 2 n(n Gamma k) log n) 6 Preprocess: ....

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M.T. Dickerson, D. Eppstein, "Algorithms for proximity problems in higher dimensions", Computational Geometry: Theory and Applications 5, 277--291, 1996.

....for this problem [21, 22, 25] reduce it to finding the k minimum weight nodes in a heap ordered tree, defined using the best swap in a sequence of graphs. Heap ordered tree selection has also been used to find the smallest interpoint distances or the nearest neighbors in geometric point sets [16]. We apply a similar tree selection technique to the k shortest path problem, however the reduction of k shortest paths to heap ordered trees is very different from the constructions in these other problems. 2 The Basic Algorithm Finding the k shortest paths between two terminals s and t has ....

M.T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Computational Geometry Theory and Applications 5:277--291, 1996.

....depending exponentially on the dimension [6, 16] We can generalize NNG(V ) to k NNG(V ) the k nearest neighbors graph of V , by introducing k edges from a vertex to its k nearest neighbors. In any constant dimension #, one can compute k NNG(V ) in time O(kn log n) 16] or even O(kn n log n) [4, 5, 9]. The k nearest neighbors graphs are useful for certain clustering problems [12] However at present they have not been studied extensively, and few of their combinatorial properties are known. 3 Monotone Logical Grid Boris [3] proposed a data structure, called the Monotone Logical Grid (MLG) as ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Computational Geometry Theory & Applications 5 (1996) 277--291.

....for this problem [11, 13] reduce it to finding the k minimum weight nodes in a heap ordered tree, defined using the best swap in a sequence of graphs. Heap ordered tree selection has also been used to find the smallest interpoint distances or the nearest neighbors in geometric point sets [8]. We apply a similar tree selection technique to the k shortest path problem, however the reduction of k shortest paths to heap ordered trees is very di#erent from the constructions in these other problems. 2 Preliminaries We assume throughout that our input graph G has n vertices and m edges. ....

M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Comp. Geom. Theory & Appl. To appear.

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M. T. Dickerson and D. Eppstein, "Algorithms for proximity problems in higher dimensions," Comput. Geom. Theory and Appl., vol. 5, no. 5, pp. 277--291, 1996.

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