Eppstein, D. Average case analysis of dynamic geometric optimization. Comput. Geom. Theory Appl. 6 (1996), 45-68.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....when we expect only a small number of nearest neighbors to change in an update. This result is not surprising, but it makes us appreciate our earlier worstcase algorithms better. Still, the special case algorithm would be interesting in applications where the update sequence is random (e.g. see [15]) The subsequent corollary mentions one particular consequence involving the all nearest neighbors graph, which connects each point p 2 P to its nearest neighbor in P n p. Theorem 4.3 We can maintain H (Q) for a set Q of at most n points in IR and a set H of at most n hyperplanes in IR in ....

D. Eppstein, Average case analysis of dynamic geometric optimization, Comput. Geom. Theory Appl., 6:45-68, 1996. 17

....costs O(log n) time. Janardan, Rote, Schwarz, and Snoeyink [15, 20] solved the weaker approximate problem ( nding a value within a factor 1 of the width for a xed 0) in O(log n) time per update, using known data structures for dynamic convex hulls [18] see also [5, 6] Eppstein [11] studied the exact problem in the case of a random sequence of insertions and deletions and obtained an O(log n) expected update time bound. Recently, Eppstein [12] gave another solution to the exact problem that given any xed 0, requires O(n ) amortized time per update for the ....

D. Eppstein. Average case analysis of dynamic geometric optimization. Comput. Geom. Theory Appl., 6:45-68, 1996.

....if the sequence of update operations is long enough and the graphs are not dense, but since the model is weaker, the results are incomparable. The rr model is a variation of a model for random update sequences used before in computational geometry (see, e.g. 6, 22, 27] Eppstein [8] considers the dynamic (geometric) maximum spanning tree problem and related problems for points in the plane. Exploiting their geometry, he gives data structures with polylogarithmic expected update times for these problems. New Results We show that a conceptually simple dynamic algorithm for ....

....is 1 2. In contrast, we do not make any assumptions on the distribution of types of update operations. Thus, our analysis also applies if an adversary provides the (worst case) types of update operations. We adopt a generic model for random update sequences from computational geometry (see, e.g. [6, 8, 22, 27]) The dynamically changing object is a set E which is a random subset of a fixed set E, the universe. An update is arbitrarily either a deletion of an element of E which has to be chosen uniformly at random from the elements which are currently in the set E, or an insertion of an element chosen ....

D. Eppstein. Average case analysis of dynamic geometric optimization. In Pvoc. 5th Syrup. on Discr'ete Algorithms, pages 77 86, 1994.

....when we expect only a small number of nearest neighbors to change in an update. This result is not surprising, but it makes us appreciate our earlier worstcase algorithms better. Still, the special case algorithm would be interesting in applications where the update sequence is random (e.g. see [15]) The subsequent corollary mentions one particular consequence involving the all nearest neighbors graph, which connects each point p 2 P to its nearest neighbor in P n p. Theorem 4.3 We can maintain H (Q) for a set Q of n points in IR d 1 and a set H of n hyperplanes in IR d in e O(kn 1 ....

D. Eppstein. Average case analysis of dynamic geometric optimization. Comput. Geom. Theory Appl., 6:45-68, 1996.

....costs O(log 3 n) time. Janardan, Rote, Schwarz, and Snoeyink [13, 17] solved the weaker approximate problem ( nding a value within a factor 1 of the width for a xed 0) in O(log 2 n) time per update, using known data structures for dynamic convex hulls [15] see also [5, 6] Eppstein [10] studied the exact problem in the case of a random sequence of insertions and deletions and obtained an O(log n) expected update time bound. Recently, Eppstein [11] gave another solution to the exact problem that given any xed 0, requires O(n ) amortized time per update for the ....

D. Eppstein. Average case analysis of dynamic geometric optimization. Comput. Geom. Theory Appl., 6:45-68, 1996.

....)j 2 k 0 Gamma1 jL(T 0 )j c 0 2 jL(T 0 )j. 2 We now turn to the analysis of the expected running time of simple sparsification. We adopt Mulmuley s expected case model of dynamic geometric computation [26] a model which was already used for fully dynamic graph problems by Eppstein [10] and by Alberts and Henzinger [4] This model makes only very weak assumptions about the input distribution. Namely, the order in which edges are inserted and deleted is assumed to be random, but the set of edges to be inserted or deleted, and the times at which insertions and deletions occur are ....

D. Eppstein, "Average Case Analysis of Dynamic Geometric Optimization", Proc. 5th Symp. on Discrete Algorithms (1994), 77--86.

....our bound in the case of connectivity if the sequence of update operations is long enough and the graphs are not dense, but since the model is weaker, the results are incomparable. The rr model is a variation of a model for random update sequences used before in computational geometry (see, e.g. [6, 8, 25, 30]) Eppstein [8] considers the dynamic (geometric) maximum spanning tree problem and related problems for points in the plane. Exploiting their geometry, he gives data structures with polylogarithmic expected update times for these problems. Average Case Analysis of Dynamic Graph Algorithms 3 1.2 ....

....connectivity if the sequence of update operations is long enough and the graphs are not dense, but since the model is weaker, the results are incomparable. The rr model is a variation of a model for random update sequences used before in computational geometry (see, e.g. 6, 8, 25, 30] Eppstein [8] considers the dynamic (geometric) maximum spanning tree problem and related problems for points in the plane. Exploiting their geometry, he gives data structures with polylogarithmic expected update times for these problems. Average Case Analysis of Dynamic Graph Algorithms 3 1.2 New Results ....

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D. Eppstein. Average case analysis of dynamic geometric optimization. In Proc. 5th Symp. on Discrete Algorithms, pages 77 -- 86, 1994.

....only occur at certain prespecified times the algorithms are not fully dynamic. A number of other papers have considered dynamic computational geometry problems under an average case model that assumes that among a given set of points each point is equally likely to be inserted or deleted next [10, 19, 20, 21, 24]. However we are interested here in worst case bounds. We are particularly interested in the dynamic geometric MST problem. If only insertions are allowed, it is not hard to maintain the MST in time O(log 2 n) per update. The same bound has recently been achieved for offline updates consisting ....

D. Eppstein. Average case analysis of dynamic geometric optimization. In Proc. 5th ACM-SIAM Symp. Discrete Algorithms, pages 77--86, 1994. To appear in Computational Geometry Theory & Applications.

....Lemma 11, finding the longest edge between each adjacent pair of trees in the cyclic ordering, and removing the shortest such edge. The farthest neighbor forest and the longest edge between adjacent trees can be computed easily via point location in the farthest point Voronoi diagram. Eppstein [55] considered the same problem from the average case dynamic viewpoint discussed above. His algorithm performs a similar sequence of steps dynamically: maintaining a dynamic farthest neighbor forest, keeping track of the intervals induced on the convex hull and of the cyclic ordering of the ....

D. Eppstein. Average case analysis of dynamic geometric optimization. Comp. Geom. Theory & Appl., to appear. Available online at http://www.ics.uci.edu/Document/UCI:ICS-TR-93-18.

....in which only partial information about the future sequence of updates is known. For completeness, we mention that the fully dynamic circumradius problem can be solved in worst case time O(n # ) per update [2] and in expected time O(1) per update for a certain class of input distributions [9]; however we will not use these results here. 5 Figure 2. Far apart disks can be separated by a line. 4 Well separated disks We now start our discussion of our two center algorithm with the case in which the disks D 1 and D 2 are far apart. Strictly speaking this case is unnecessary as it is ....

D. Eppstein. Average case analysis of dynamic geometric optimization. 5th ACM-SIAM Symp. Discrete Algorithms, 1994, 77--86. Comp. Geom. Theory & Applications, to appear.

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Eppstein, D. Average case analysis of dynamic geometric optimization. Comput. Geom. Theory Appl. 6 (1996), 45-68.

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