M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Rev., 2 (1991), pp. 33--44.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....version of the same problem also for This work has been partially supported by the National Science Foundation under grant IRI 9610240. d 2. Salowe [21] shows how to compute the k th closest pair in O(n log d n) time under L1 . The first result for enumerating more than one pair was by Smid [25] who solved CP n 2=3 in O(n log n) time. Lenhof and Smid [19] followed with a solution to the general CP k problem that runs in O(n log n k) time. Unfortunately, when d is not viewed as a constant, all of the algorithms reported above suffer from the so called dimensionality curse the ....

M. Smid. Maintaining the minimal distance of a point set in less than linear time. Algorithms Rev., 2:33--44, 1991.

....an optimal data structure for a special case of 2D range reporting in which the query ranges are translates of a polygon. 34 Pankaj Agarwal and Jeff Erickson Problem Size Query Time Update Time Source Counting n log 2 n log 2 n [55] n k log 2 (2n=k) log 2 n [55] n n k log 2 n [234] Reporting n log n log n log log n k log n log log n [192] n log n log log n log 2 n log log n k log 2 n log log n [234] Semigroup n log 4 n log 4 n [55] Table 6. Asymptotic upper bounds for dynamic 2D orthogonal range searching. Although Matousek s O(n log n) size data ....

.... and Jeff Erickson Problem Size Query Time Update Time Source Counting n log 2 n log 2 n [55] n k log 2 (2n=k) log 2 n [55] n n k log 2 n [234] Reporting n log n log n log log n k log n log log n [192] n log n log log n log 2 n log log n k log 2 n log log n [234] Semigroup n log 4 n log 4 n [55] Table 6. Asymptotic upper bounds for dynamic 2D orthogonal range searching. Although Matousek s O(n log n) size data structure for d dimensional halfspace range reporting [179] can be dynamized, the logarithmic query time data structure is not easy to ....

M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Rev., 2 (1991), 33--44.

....We describe our methods with respect to the Euclidean L 2 metric, but we do not use this metric in any essential way; similar techniques and results apply to other metrics on IR d , and in particular to the L p metrics. 1. 1 Background and Previous Results Problem 2 was posed by Smid [24]. He presented an O(n log n) time O(n) space algorithm for enumerating the O(n 2 3 ) smallest distances for a set of n points in d space for any L p metric, and posed as an open problem enumerating the #(n) smallest distances in O(n log n) time and O(n) space. He used this as a subroutine in ....

M. Smid, "Maintaining the minimal distance of a point set in less than linear time", Algorithms Review 2 (1991) 33--44.

.... 2pdxd Gamma (p 2 1 : p 2 d ) Then p is a closest neighbor of a query point q = q1 ; q2 ; q d ) if and only if fp(q1 ; q2 ; q d ) maxr2Sfr (q1 ; q2 : qd ) is amazing that most of the practical clustering software use the brute force to maintain the closest pair[17] Smid[35] gives two fully dynamic techniques for maintaining the minimal L r distance of a point set in d dimensional spaces. The data structure uses O(n) space and supports updates in O(n 2=3 log n) time, by giving a method to compute the O(n 2=3 ) smallest distances defined by a set of n points in ....

M.Smid. Maintaining the Minimal Distance of Point Set in Less than Linear Times. Algorithms Review 2(1),1991,33-44.

....is not easy to dynamize because some of the points may be stored at Omega Gamma n bd=2c ) nodes of the tree. Agarwal 38 Pankaj K. Agarwal and Jeff Erickson Problem Size Query Time Update Time Source Counting n log 2 n log 2 n [58] n k log 2 (2n=k) log 2 n [58] n n k log 2 n [264] Reporting n log n log n log log n k log n log log n [212] n log n log log n log 2 n log log n k log 2 n log log n [264] Semigroup n log 4 n log 4 n [58] Table 6. Asymptotic upper bounds for dynamic 2D orthogonal range searching. and Matousek [9] developed a rather ....

.... and Jeff Erickson Problem Size Query Time Update Time Source Counting n log 2 n log 2 n [58] n k log 2 (2n=k) log 2 n [58] n n k log 2 n [264] Reporting n log n log n log log n k log n log log n [212] n log n log log n log 2 n log log n k log 2 n log log n [264] Semigroup n log 4 n log 4 n [58] Table 6. Asymptotic upper bounds for dynamic 2D orthogonal range searching. and Matousek [9] developed a rather sophisticated data structure that can insert or delete a point in time O(n bd=2c Gamma1 ) time and can answer a query in O(log n k) time. As in ....

M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Rev., 2 (1991), 33--44.

....described an optimal data structure for a special case of 2D range reporting where the query ranges are translates of a polygon. TABLE 7 Dynamic 2D orthogonal range searching Mode S(n) Q(n) U(n) Source Counting n log 2 n log 2 n [27] n k log 2 (2n=k) log 2 n [27] n n k log 2 n [104] Reporting n log n log n log log n k log n log log n [89] n log n log log n log 2 n log log n k log 2 n log log n [104] Semigroup n log 4 n log 4 n [27] Range Searching 15 Since an arbitrary sequence of deletions is difficult to handle in general, researchers have examined ....

.... 7 Dynamic 2D orthogonal range searching Mode S(n) Q(n) U(n) Source Counting n log 2 n log 2 n [27] n k log 2 (2n=k) log 2 n [27] n n k log 2 n [104] Reporting n log n log n log log n k log n log log n [89] n log n log log n log 2 n log log n k log 2 n log log n [104] Semigroup n log 4 n log 4 n [27] Range Searching 15 Since an arbitrary sequence of deletions is difficult to handle in general, researchers have examined whether a random sequence of insertions and deletions can be handled efficiently; see [92, 93, 102] Mulmuley [92] has shown that there ....

M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Rev., 2 (1991), 33--44.

....of size O(n) that supports insertions in O(log k Gamma1 n) amortized time. Schwarz, Smid and Snoeyink [26] presented a data structure of size O(n) that maintains the closest pair in O(log n) amortized time per insertion. Several algorithms are obtained for the dynamic closest pair problem [19, 20, 22, 24, 29 31]. In [20, 22, 29] the problem is solved with O( p n log n) update time using O(n) space. In [19] Kapoor and Smid gave data structures of size S(n) that maintain the closest pair in U(n) amortized time per update, where for k 3, size S(n) O(n) and time U(n) O(log k Gamma1 n log log n) ....

....insertions in O(log k Gamma1 n) amortized time. Schwarz, Smid and Snoeyink [26] presented a data structure of size O(n) that maintains the closest pair in O(log n) amortized time per insertion. Several algorithms are obtained for the dynamic closest pair problem [19, 20, 22, 24, 29 31] In [20, 22, 29] the problem is solved with O( p n log n) update time using O(n) space. In [19] Kapoor and Smid gave data structures of size S(n) that maintain the closest pair in U(n) amortized time per update, where for k 3, size S(n) O(n) and time U(n) O(log k Gamma1 n log log n) for k = 2, size ....

M. Smid. Maintaining the minimal distance of a point set in less than linear time. Algorithms Rev., 2, 1991, pp. 33-44.

....uses linear space and maintains the closest pair in O(log n) amortaized time per insertion. It also solves the problem of computing on line the closest pair that existed over the history of a fully dynamic point set in O(log n) amortized time per insertion or deletion, using linear space. Smid [153,156] gives two fully dynamic techniques for maintaining the minimal L r distance of a point set in d dimensional space. The data structure of [153] uses O(n) space and supports updates in O(n 2=3 log n) time, by giving a method to compute the O(n 2=3 ) smallest distances defined by a set of n ....

....closest pair that existed over the history of a fully dynamic point set in O(log n) amortized time per insertion or deletion, using linear space. Smid [153,156] gives two fully dynamic techniques for maintaining the minimal L r distance of a point set in d dimensional space. The data structure of [153] uses O(n) space and supports updates in O(n 2=3 log n) time, by giving a method to compute the O(n 2=3 ) smallest distances defined by a set of n points in O(n log n) time. By [49,141] which show how to compute the n smallest distances in O(n log n) time, the update time is improved to O(n ....

M. Smid, "Maintaining the Minimal Distance of a Point Set in Less Than Linear Time," Algorithms Review 2 (1) (1991), 33--44.

....with O(log D n) amortized update time which uses O(n log D 1 n) space. When both insertions 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 ....

M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Rev., 2 (1991), pp. 33--44.

....with O(log D n) amortized update time that uses O(n log D Gamma1 n) space. When both insertions and deletions are allowed, Smid [Smi92] described a data structure that 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 [Smi91], with improvements stemming from results of Salowe [Sal92] and Dickerson and Drysdale [DD91] uses O(n) space and requires O( p n log n) time for updates. Very recently, after a preliminary version of this paper was presented, Kapoor and Smid [KS94] devised a deterministic data structure of ....

M. Smid. Maintaining the minimal distance of a point set in less than linear time. Algorithms Rev., 2:33--44, 1991.

....depend on the dimension D and, in the last two lines, on D and k. The update times are either worst case (w) or amortized (a) mode dimension update time space reference insertions D 2 log n (w) n [17, 18] deletions D 2 (log n) D (a) n(log n) D Gamma1 [23] dynamic D 2 p n log n (w) n [16, 20] dynamic D 2 (log n) D log log n (a) n(log n) D [22] dynamic D 3 (log n) D Gamma1 log log n (a) n this paper dynamic 2 log n log log n (a) n log n= log log n) k this paper dynamic 2 (log n) 2 = loglog n) k (a) n this paper It seems that in higher dimensions it is impossible to ....

M. Smid, Maintaining the minimal distance of a point set in less than linear time, Algorithms Review, 2 (1991), pp. 33--44.

....problem in O(n log n) time. 2.5.3 The k closest pairs problem In this version of the problem, we are given a set S of n points in IR D and an integer k, 1 k i n 2 j , and we have to compute the k smallest distances in the set S. The first algorithms for this problem are due to Smid [118]. He gives an incremental algorithm using a space efficient variant of range trees [119] that solves the problem in O(n 4=3 log n n p k log k) time. For the planar case, this can be improved to O(n log n n p k log k) by using a straightforward generalization of the sweep algorithm given in ....

....be maintained in O(n) time per insertion and deletion (see Overmars [96, 97] and Aggarwal et al. 4] we can also maintain the closest pair in a planar point set in O(n) time per update. The first fully dynamic data structure that maintains the closest pair in sublinear time was given by Smid [118]. It is based on the following idea. Instead of maintaining only the minimum distance, we start with the sorted list L of n smallest distances. If a point p is deleted then all distances in L in which p occurs have to be deleted. It can easily be shown that there are at most O( p n ) such ....

M. Smid. Maintaining the minimal distance of a point set in less than linear time. Algorithms Review 2 (1991), pp. 33--44.

....existed over the history of a fully dynamic point set in O(log n) amortized time per insertion or deletion. Note that recently there has been much interest in the dynamic closest pair problem. For the case where only deletions are allowed, see Supowit [14] For the fully dynamic case, see Smid [11, 13], Salowe [8] and Dickerson and Drysdale [3] The algorithm in this paper is based on the algorithm of Smid [12] To update the closest pair when a point is inserted, that algorithm makes some queries into a data structure for the k dimensional rectangular point location problem. In this data ....

M. Smid. Maintaining the minimal distance of a point set in less than linear time. Algorithms Review 2 (1991), pp. 33-44.

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