Matthew T. Dickerson, R. L. Scot Drysdale, and Jorg-Rudiger Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. International Journal of Computational Geometry and Applications, 2:221--239, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....finding nearest neighbors An efficient technique for finding neighbors of (d 2) flats might also lead to a faster minimum boundary measure algorithm. Finally, is it possible to find rectilinear neighbors of points in higher dimensions in o(mn log n) time 12 Recent Results Dickerson et al. [13] describe an algorithm for finding k nearest neighbors in the plane, under any metric, in time O(n log n kn log k) and space O(n) Except for L1 diameter, we can replace our neighbor algorithm with theirs in each of the algorithms in Section 5, without decreasing the time. The resulting ....

M. T. Dickerson and R. L. Drysdale and J. R. Sack. Simple algorithm for enumerating interpoint distances and finding k nearest neighbors. Internat. J. Comput. Geom. Appl. 2:221--239, 1993.

....and deletions are allowed, Smid [22] described a data structure which uses O(n log D n) space and runs in O(log D n log log n) amortized time per update. Another data structure due to Smid [21] with improvements stemming from results of Salowe [17] and Dickerson, Drysdale, and Sack [7], uses O(n) space and requires O( # n log n) time for updates. Very recently, after a preliminary version of this paper was presented, Kapoor and Smid [13] devised a deterministic data structure of linear size which achieves polylogarithmic amortized update time, namely, O(log D 1 n log log n) ....

M. T. Dickerson, R. L. Drysdale, and J. R. Sack, Simple algorithms for enumerating interpoint distances and finding k nearest neighbors, Internat. J. Comput. Geom. Appl., 2 (1992), pp. 221--239.

....3 3.1. The main loop. input two point sets P and Q and the parameters k, ae and t. Step 1. Precomputations a) For each point in P find the k nearest neighbors (in order of closeness) and store in nearest neighbor list. Do the same for Q. This can be done using the algorithms described in [5] or [4]. b) Divide the smallest square into which Q fits into a two dimensional array of squares of side length r= p n. Let each entry in the array contain a list of the points in Q that lie in that square. This will allow us to quickly check whether there is a point in Q at a given coordinate. See ....

M. T. Dickerson,R. L. Drysdale, and J-R Sack. Simple algorithms for enumerating inter-point distances and finding k nearest neighbors. Internat. J. Comput. Geom. Appl., 1992 2(3):221--239. A FAST ALGORITHM FOR THE POINT PATTERN MATCHING PROBLEM 11

....to the maximum value of . Lemma 11. The algorithm has a performance guarantee of = 3:6. Note that for each point a, the points of S that lie in C 0 a must be determined. Since we have N(C 0 a ) 5, these points can be computed in O(n log n) time with the algorithm of Dickerson et al. [4], after which the algorithm takes only linear time to compute P and D regions for all the points; solving each 2SAT instance takes only linear time; and the while loop iterates O(log R ) times. Theorem 1. In O(n log n n log(R ) time every point can be labeled with circles of size 5R ....

M.T. Dickerson, R.L. Drysdale, and J.R. Sack, \Simple algorithm for enumerating interpoint distances and nding k nearest neighbors," Int'l J. Comput. Geom. and Appl. 2, 2 (1992), 221-239.

....computed in O(n log n) time. 7. The minimum spanning tree is a subgraph of the Delaunay triangulation, and in fact, a single linkage clustering (or dendrogram) can be found in O(n log n) time from D(S) 8. The u nearest neighbors of a point s i can be found in O(u log u) expected time from D(S) [19]. The algorithm is simple and practical. Place the Delaunay neighbors of s i 2 S in a priority queue with the Euclidean distance to s i as its key. Repeatedly extract the item with smallest key and place its Delaunay neighbors not already examined in the priority queue (again, the key is the ....

M. T. Dickerson, R. L. S. Drysdale, and J.-R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbours. International Journal of Computational Geometry & Applications, 2(3):221--239, 1992. Robust distance-based clustering 27

....is still attainable when the points have been pre sorted along each of the d coordinates. The more general problem of enumerating the k closest pairs (or enumerating the first k smallest distances) has also received much attention. One of the earliest reported algorithms is by Dickerson et al. [12], who used the Delaunay triangulation to enumerate the k closest pairs in O( n k) log n) time in two dimensions. Their algorithm actually enumerates the distances in sorted order. Salowe [33] was the first to give an O(n log n k) algorithm for any fixed dimension. His algorithm employs the A ....

M. T. Dickerson, R. L. S. Drysdale, and J.-R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. Int. J. Comput. Geom. Appl., 2:221--239, 1992.

....if E opt = f g opt (e) j e 2 R g denotes the set of matched nodes of E in the optimal matching g opt , each of the arcs between R and E opt is visited at most O(n) times. The only operation which we have not discussed is the generation of all edges in E according to length. Dickerson et al. [DDS] have shown that the m shortest edges in an n point set P may be generated in O( m n) log n) time and O(m n) space. By trying successively the values m = n; 2n; 4n; 8n; we can make sure that we never generate more than twice as many edges as we actually need. The O(log n) overhead for ....

M. Dickerson, R. L. Drysdale, and J.-R. Sack, Simple algorithms for enumerating interpoint distances and finding k nearest neighbors, Int. J. Computational Geometry & Appl. 3 (1992), 221-239.

....is still attainable when the points have been pre sorted along each of the d coordinates. The more general problem of enumerating the k closest pairs (or enumerating the first k smallest distances) has also received much attention. One of the earliest reported algorithms is by Dickerson, et al. [13], who used the Delaunay triangulation to enumerate the k Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124 4250, USA, E mail: tchan cs.miami.edu Copyright c fl 1998 by the Association for Computing Machinery, Inc. Permission to make digital or hard ....

M. T. Dickerson, R. L. S. Drysdale, and J.-R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. Int. J. Comput. Geom. Appl., 2:221--239, 1992.

....k whose distance is smallest. This gives a set of at most 2k points that contains the k closest pairs of the set S. We apply the above algorithms on this small set. The result is an algorithm for solving the k closest pairs problem in O(n log n k p k log k) time. Dickerson, Drysdale and Sack [52] give a simple algorithm for the planar case that uses the Delaunay triangulation. Here is a description of their algorithm. Given a set S of n points in the plane, compute the Delaunay triangulation DT of S. For each point p of S, sort the edges of DT that are incident to p in increasing order ....

....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 algebraic computation tree model. We remark that Arya and Smid [15] have shown that the algorithm of [52] also works if we replace the Delaunay triangulation by any bounded degree spanner. See Section 6.1. The first optimal algorithm for the k closest pairs problem is due to Salowe [105] He first combines a variant of Vaidya s algorithm [125] with the parametric search technique to compute the ....

[Article contains additional citation context not shown here]

M.T. Dickerson, R.L. Scot Drysdale and J.R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. International Journal of Computational Geometry & Applications 2 (1993), pp. 221--239.

.... a given set of points in d space the nearest neighbour of each (the all nearest neighbours problem) For the latest and most practical computational geometric results concerning nearest neighbour search in all dimensions and for pointers to many other key recent results the reader is referred to [DSS92], Yi93] and [AM93] 6. ....

Dickerson, M. T., Scott Drysdale, R. L. and Sack, J.-R., "Simple algorithms for enumerating interpoint distances and finding k nearest neighbors," International Journal of Computational Geometry and Applications, vol. 2, No. 3, 1992, pp. 221-239.

....nearest neighbors An efficient technique for finding neighbors of (d Gamma 2) flats might also lead to a faster minimum boundary measure algorithm. Finally, is it possible to find rectilinear neighbors of points in higher dimensions in o(mn log n) time 12 Recent Results Dickerson et al. [13] describe an algorithm for finding k nearest neighbors in the plane, under any metric, in time O(n log n kn log k) and space O(n) Except for L1 diameter, we can replace our neighbor algorithm with theirs in each of the algorithms in Section 5, without decreasing the time. The resulting ....

M. T. Dickerson and R. L. Drysdale and J. R. Sack. Simple algorithm for enumerating interpoint distances and finding k nearest neighbors. Internat. J. Comput. Geom. Appl. 2:221--239, 1993.

....if E opt = f g opt (e) j e 2 R g denotes the set of matched nodes of E in the optimal matching g opt , each of the arcs between R and E opt is visited at most O(n) times. The only operation which we have not discussed is the generation of all segments in E according to length. Dickerson et al. [DDS] have shown that the c shortest edges in an n point set P may be generated in O( c n) log n) time and O(c n) space. By trying successively the values c = n; 2n; 4n; 8n; we can make sure that we never generate more than twice as many edges as we actually need. The O(log n) overhead for ....

M. Dickerson, R. L. Drysdale, and J.-R. Sack, Simple algorithms for enumerating interpoint distances and finding k nearest neighbors, Int. J. Computational Geometry & Appl. 3 (1992), 221-239.

....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 ....

....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 log n kn) and O(n log n) space. In the algebraic Pbm Previous ....

[Article contains additional citation context not shown here]

M.T. Dickerson, R.L.S Drysdale, J-R. Sack, "Simple algorithms for enumerating interpoint distances and finding k nearest neighbors", Internat. J. Comput. Geom. Appl., 2(3):221--239, 1992.

....problem leads to a complexity of O(n 4 ) time and O(n 3 ) space. However, we can follow the greedy approach outlined in Section 4.4 to obtain a faster algorithm that uses less space. First, we generate the edges in E in increasing order of length until jRj of them are matched. Dickerson et al. [DDS92] have shown that the m shortest edges in an n point set P may be generated in O( m n) log n) time and O(m n) space. By trying successively the values m = n; 2n; 4n; 8n; we can make sure that we never generate more than twice as many edges as we actually need. In the iterative step we ....

M. Dickerson, R. L. Drysdale, and J.-R. Sack, Simple algorithms for enumerating interpoint distances and finding k nearest neighbors, Int. J. Computational Geometry & Appl. 3 (1992), 221--239.

....of enumerating the k smallest distances, the following was known. Salowe [13] and Lenhof and Smid [9] achieve O(n log n k) time for any dimension, but in both algorithms, the value of k must be known in advance and the distances are not enumerated in sorted order. In the plane, Dickerson et al.[8] show that given the Delaunay triangulation, the k smallest distances can be enumerated in O(n k log k) time. In this algorithm, the value of k need not be known in advance and the distances are enumerated in sorted order. Hence our spanner can be regarded as an efficient data structure that ....

....Let p and q be two points of S. The weight of this pair is defined as the Euclidean distance between p and q, and its pseudoweight is defined as the Euclidean length of a shortest path in G between p and q. The algorithm for approximate distance enumeration is similar to that of Dickerson et al.[8]. We initialize a priority queue with all pairs of points corresponding to the edges of G, with priority given by the pseudo weight of the pair. In each iteration, we extract the pair p; q with smallest priority and report it together with its weight. For each edge (q; r) of G, we compute the ....

M.T. Dickerson, R.L. Drysdale and J.R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. Internat. J. Comput. Geom. Appl. 2 (1992), pp. 221--239.

....If not, then perhaps some sort of sampling or bucketing via quad trees or k d trees would allow the points in the wedges to be found quickly on the average. It is also possible that some variation of searching on the Delaunay triangulation, as is done in papers by Dickerson, Drysdale, and Sack [6, 8], could prove useful. It seems likely that this sort of algorithm could work quite well for relatively smooth distributions. Some obvious questions arise from this work. Problem 1. What are reasonable ways to adapt this algorithm to other distributions Ideally the algorithm would adapt to any ....

M. Dickerson, R. L. Drysdale, and J.-R. Sack, "Simple Algorithms for enumerating interpoint distances and finding k nearest neighbors." International Journal of Computational Geometry and Applications 3 (1992) 221--239.

....that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S; and an O(n log n kn) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the method of Bern, Eppstein, and Gilbert [3] for augmenting a point set to have a linear size bounded degree Delaunay triangulation. Thus, in addition to providing new solutions to these problems, the paper also shows how the Delaunay triangulation can be used as the underlying data ....

....are not necessarily enumerated in order, and also an easier version of Problem 2 where we do not require the neighbors to be enumerated in order by distance. The algorithms we present in this paper extend the recent planar results of Dickerson and Drysdale [9] and Dickerson, Drysdale, and Sack [11] to higher dimensions by making use of the results of Bern, Eppstein, and Gilbert [3] on provably good mesh generation. Bern, et al. showed how for a set S of points in arbitrary dimension, a superset S # of S could be found in O(n log n) time so that the Delaunay triangulation of S # has linear ....

[Article contains additional citation context not shown here]

M. Dickerson, R. L. Drysdale and J.-R. Sack "Simple algorithms for enumerating interpoint distances and finding k nearest neighbors", International Journal of Computational Geometry and Applications 2:3 (1992) 221--239.

....is less sensitive to the exact distribution. It will dynamically decide when to switch from one phase to the next in an attempt to balance the amount of work done in each phase. In the first phase it generates possible edges in increasing order using an algorithm of Dickerson, Drysdale, and Sack [8]. Algorithms to enumerate the k closest interpoint pairs have been invented by Salowe and by Lenhof and Smid, but because they need to know k in advance they are less appropriate in this context [32, 17] When the number of pairs of not closed points is proportional to the number of pairs ....

M. Dickerson, R.L. Drysdale, and J-R Sack, "Simple Algorithms for enumerating interpoint distances and finding k nearest neighbors." International Journal of Computational Geometry and Applications 3 (1992) 221--239.

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Matthew T. Dickerson, R. L. Scot Drysdale, and Jorg-Rudiger Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. International Journal of Computational Geometry and Applications, 2:221--239, 1992.

No context found.

M. Dickerson, R. Drysdale, and J.-R. Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. International J. Computational Geometry and Applications, 2(3):221--239, 1992.

No context found.

Matthew T. Dickerson, R. L. Scot Drysdale, and Jorg-Rudiger Sack. Simple algorithms for enumerating interpoint distances and finding k nearest neighbors. International Journal of Computational Geometry and Applications, 2:221--239, 1992.

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