D. Eppstein, Z. Galil, and G. F. Italiano. Improved sparsification. Technical Report 9320, Dept. of Information and Computer Science, University of California, Irvine, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The Edge Update Problem Sequential algorithms for updating the MST of a graph under edge updates, have received considerable attention in the past. Frederickson [15] designed an O( p m) time algorithm for the single edge update problem, which is recently improved to O( p n) by Eppstein et al. [11]. They used a new technique called sparsification. Although the edge update problem has been well studied in parallel computation, work optimal parallel algorithms have not been proposed. Till now, the best work bound [10, 13] is even far from the O( p n) sequential time, and thus the parallel ....

....G 0 and G 00 . It is simple prove that the computation of the certificate is an associative and a transitive operation. 17 We are ready now to recall the sparsification data structure. The sparsification data structure has been recently proposed by Eppstein et al. 12] and later improved in [11]. From now on, we will refer to the data structure provided in [12] considering only the case of the maintenance of the minimum spanning tree of a graph. Therefore, if not explicitly written, the notion of certificate of G and MST of G will be used equivalently, since the certificate for the MST ....

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D. Eppstein, Z. Galil, and G. F. Italiano. Improved sparsification. Technical Report 9320, Dept. of Information and Computer Science, University of California, Irvine, 1993.

.... [62] and Ravi, Sundaram, Marathe, Rosenkrantz and Ravi [43] Ravi etal proposed a polynomial time approximation algorithm for the planar k MST (d = 2) with approximation ratio O(k 1=4 ) which has been successively improved to O(log(k) by Garg and Hochbaum [28] O(log(k) log log(n) by Eppstein [23], O(1) by Blum, Chalasani and Vempala [13] 2 p 2 by Mitchell [40] 3 by Garg [27] and 1 by Arora [3] 4] While it is di cult to establish general and useful properties of optimal graphs for a xed set of points X n , interesting properties of many classes of optimal graphs can be ....

D. Eppstein, \Faster geometric k-point MST approximation," Technical Report 95-13, Dept. of Information and Computer Science, University of California, Irvine, March 1995.

....tree problem. In 1985 [5] Frederickson introduced a data structure known as topology trees for the fully dynamic minimum spanning tree problem with a worst case cost of O( p m) per update, permitting connectivity queries in time O(log n= log( p m= log n) O(1) In 1992, Epstein et al. [3, 2] improved the update time to O( p n) using the sparsification technique. Finally in 1997 Henzinger and King [8] gave an algorithm with O( 3 p n log n) update time and O(log n= log log n) time per connectivity query. Randomization has been used to improve the bounds for the connectivity ....

.... get a fully dynamic algorithm with update cost O 0 log 3 n 3 log 2 m X i=1 i X j=1 log 2 (minfn; 2 j g) 1 A = O(log 4 n) Note for comparison, that in [8] Henzinger and King had t(k; l) O( 3 p l log k) giving them an update cost of O( 3 p m log n) Then sparsification [2, 3] reduces the cost to O( 3 p n log n) From the combination of Theorem 4 and Lemma 5, we conclude Theorem 6 There is a fully dynamic MSF algorithm that for a graph with n nodes and starting with no edges maintains a minimum spanning forest in O(log 4 n) amortized time per edge insertion or ....

David Eppstein, Zvi Galil, and Giuseppe F. Italiano. Improved sparsification. Technical Report 93-20, Univ. of California, Irvine, Dept. Information and Computer Science, 1993.

....connectivity, and that the biconnectivity result is considered the main contribution of this paper. Relating to previous work In 1991 [5] Fredrickson succeeded in generalizing his O( p m) bound from 1983 [4] for fully dynamic connectivity to fully dynamic 2 edge connectivity. In 1992 1993 [3, 2], this was improved by Eppstein, Galil, Italiano, and Nissenzweig to O( p n) In 1995 1997 [7, 8] these bounds were improved to O(log 5 n) expected amortized time per operation, generalizing the randomized O(log 3 n) bound for connectivity from [7] In 1996 [10] Henzinger and Thorup ....

David Eppstein, Zvi Galil, and Giuseppe F. Italiano. Improved sparsification. Technical Report 93-20, Univ. of California, Irvine, Dept. Information and Computer Science, 1993.

....improvements in Dijkstra s algorithm] Thus all previous algorithms took time O(n log n) or more per path. We improve this to constant time per path. A similar problem to the one studied here is that of finding the k minimum weight spanning trees in a graph. Recent algorithms 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 ....

....algorithm: we can list the edges of any path we output in time proportional to the number of edges, and simple properties (such as the length) are available in constant time. Similar implicit representations have previously been used for related problems such as the k minimum weight spanning trees [21, 22, 25]. Further, previous papers on the k shortest path problem give time bounds omitting the O(k 2 n) term above, and so these papers must tacitly or not be using an implicit representation. Our representation is similar in spirit to those used for the k minimum weight spanning trees problem: for ....

D. Eppstein, Z. Galil, and G. F. Italiano. Improved sparsification. Tech. Rep. 93-20, Univ. of California, Irvine, Dept. Information and Computer Science, 1993. http://www.ics.uci.edu:80/ TR/UCI:ICS-TR-93-20.

....and Zelikovsky and Lozevanu [ZL93] and proved NP complete by those authors and also by Ravi et al. RSM 94] Ravi et al. devised an approximation algorithm with ratio O(k 1 4 ) for the Euclidean k MST. The ratio was improved to O(logk) by Garg and Hochbaum [GH94] O(logk loglogn) by Eppstein [Epp95] O(1) by Blum et al. BCV95] and 2 # 2 by Mitchell [Mit96] Mitchell builds on the approach of Blum et al. who built on the approach of Hochbaum and Garg. He considers only trees of a special type, and shows that the shortest tree of this type can be found using dynamic programming.A tree ....

D. Eppstein. Faster geometric k-point MST approximation. Technical Report 95-13, Dept. of Information and Computer Science, UC-Irvine, 1995.

....We described e#cient algorithms for o#ine computation of MSTs in a changing graph or point set. We use reduction and contraction, which we introduced in our paper on finding several small spanning trees [2] and have recently applied to a persistent query version of the dynamic MST problem [3]. Reduction and contraction have thus proven useful for a variety of MST problems; perhaps they can be used in other graph algorithms. The most pressing open problem suggested by this work is maintenance of geometric MSTs. No algorithm was known that achieved sublinear time bounds for both point ....

D. Eppstein. Persistence, o#ine algorithms, and space compaction. Tech. Rep. 91-54, Dept. of Information and Computer Science, Univ. of California, Irvine, CA 92717.

....an unweighted problem of the same type. 1.2 Related Work The problem of listing all minimum spanning trees is a special case of finding the k minimumweight spanning trees for some input parameter k, which has been well studied. The best bounds known for this problem are O(M k min(n, k) 1 2 ) [9, 10], where M denotes the time to find a single minimum spanning tree. However this is slower than the bound we give for the problem of listing all minimum spanning trees, and does not extend to the problems of counting or randomly generating minimum spanning trees. Gavril [15] considered the same ....

.... for the k best spanning trees can also be adapted to list all spanning trees with total weight better than some given bound; the best methods for this problem of bounded spanning tree generation take time O(M min k 3 2 ,kn 1 2 ) where M is the time to compute a single minimum spanning tree [9, 10]. This bound therefore can be applied to minimum spanning tree generation. Note that the time per tree is o(n) so there is not time to output the trees explicitly; instead an implicit representation 6 is generated in which the edge set of each tree is described by its symmetric di#erence with ....

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D. Eppstein, Z. Galil, and G. F. Italiano. Improved sparsification. Tech. Rep. 93-20, Dept. of Information and Computer Science, Univ. of California, Irvine, 1993.

....one is following may cross an edge of the graph, or a face of the graph; in either case the path must go around these obstacles. The two properties above imply that neither type of detour can force the dilation of the pair of vertices to be high. For a survey of further results on dilation, see [7]. Our interest here is in another geometric graph, the # skeletons [11, 13] which have been of recent interest for their use in finding edges guaranteed to take part in the minimum weight triangulation [1, 9, 14] As a special case, # = 1 gives the Gabriel graph, a subgraph of the Delaunay ....

D. Eppstein. Spanning trees and spanners. Tech. Report 96-16, Dept. Information and Computer Science, University of California, Irvine, 1996.

....improvements in Dijkstra s algorithm) Thus all previous algorithms took time O(n log n) or more per path. We improve this to constant time per path. A similar problem to the one studied here is that of finding the k minimum weight spanning trees in a graph. Recent algorithms for this problem [22, 21, 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 ....

....algorithm: we can list the edges of any path we output in time proportional to the number of edges, and simple properties (such as the length) are available in constant time. Similar implicit representations have previously been used for related problems such as the k minimum weight spanning trees [22, 21, 25]. Further, previous papers on the k shortest path problem give time bounds omitting the O(k 2 n) term above, so these papers must tacitly or not be using an implicit representation. Our representation is similar in spirit to those used for the k minimum weight spanning trees problem: for that ....

<F3.774e+05> D. Eppstein, Z. Galil, and G. F.<F3.828e+05> Italiano,<F4.018e+05> Improved<F3.828e+05> sparsification, Tech. Report 93-20, Univ. of California, Dept. Information and Computer Science, Irvine, 1993.

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