D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig, Sparsification --- A technique for speeding up dynamic graph algorithms, in "Proceedings, 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, 1992," 60--69.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... with some classical strength reduction rules [3] dynamic mappings maintained by finite differencing rules for aggregate primitives in SETL [36] and INC [52] and auxiliary data structures for problems with certain properties like stable decomposition [40] and other decomposition properties [15]. However, until now, the systematic discovery of auxiliary information for general problems has been a subject left completely open for study. Methods for dealing with incremental programs, intermediate results, and auxiliary information together have also been needed. This Paper. This paper ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. In Proceedings of the 33rd Annual IEEE Symposium on FOCS, Pittsburgh, Pennsylvania, October 1992.

....worst case analysis. Furthermore we consider an intermediate model between worst case analysis and average case analysis: the semi random adversary introduced in [3] 1 Introduction Significant progress has been recently made in the design of algorithms and data structures for dynamic graphs [1, 5, 6, 8, 11, 12, 13, 16, 17, 18, 19, 20, 21, 24]. These data structures support insertions and deletions of edges and or nodes in a graph, in addition to several types of queries. The goal is to compute the new solution in the modified graph without having to recompute it from scratch. Usually, the sequence of insertions deletions of edges is ....

D.Eppstein, Z.Galil, G.F.Italiano, A.Nissenzweig, Sparsification - a technique for speeding up dynamic graph algorithms, Proc. 33rd Annual Symp. on Foundations of Computer Science, 1992.

.... an amortized cost of ff(n) per operation by Tarjan [16] with a matching lower bound by Fredman and Saks [7] Up till recently, deletions (with or without insertions) were considered much harder, with an operation cost of O( n) for fully dynamic connectivity by Eppstein, Galil, and Italiano [4,5]. All the above upper bounds were based on deterministic algorithms. However, using a randomized method Henzinger and King got the cost down to an amortized expected n) per operation [9] This bound was further improved to O(log n) by Henzinger and Thorup. A lower bound of Omega Gamma 20 n= ....

D. Eppstein, Z. Galil, G. F. Italiano, A. Nissenzweig, Sparsification - A Technique for Speeding up Dynamic Graph Algorithms. Proc. 33rd Symp. on Foundations of Computer Science, 1992, 60--69.

....algorithm to maintain this system under edge insertions, in O(q m n log n) time. Using the reduction from edge connectivity problems to vertex connectivity problems given in [GI91] one can in principle derive an algorithm to maintain the 4 edge connectivity classes of a graph (see Th. 10 in [EGIN]) The text of [KTBC] however, deals exclusively with graphs that are initially 3 vertex connected, and is already quite complex. For the general case of a graph that is not initially 3 vertex connected, which is significantly more complicated to solve, the result is only stated (Th. 7) So, up ....

Eppstein, D., Galil, Z., and Italiano, G. F., Nissenzweig, A.: Sparsification - a technique for speeding up dynamic graph algorithms. Proc. 33rd Annual Symp. on Foundations of Computer Science, 1992, 60--69

....spanning tree, and bipartiteness testing. 1.2 Previous Work Dynamic graph algorithms are compared using the (amortized or worst case) time per operation. The best deterministic algorithms for the above graph properties take time O( p n) per update operation and O(1) or O(log n) per query [3, 4]. Recently [6] Henzinger and King gave algorithms with polylogarithmic amortized time per operation using (Las Vegas type) randomization. Their algorithms achieve the following running times: n) to maintain a spanning tree in a graph (the connectivity problem; n) to maintain the bridges in a ....

D. Eppstein, Z. Galil, G. F. Italiano, A. Nissenzweig. Sparsification - A Technique for Speeding up Dynamic Graph Algorithms. Proc. 33rd Symp. on Foundations of Computer Science, 1992, 60--69.

....algorithms that solve various nearest common ancestor, network flow, and constrained minimum spanning tree problems. The Fibonacci heaps of Fredman and Tarjan [FT87] yield fast algorithms that solve certain shortest path, assignment, and minimum spanning tree problems. Recently, Eppstein et al. EGIN92] developed sparsification as an algorithmic tool to improve the running time of many dynamic graph algorithms. In each case, the authors isolate a problem common to several different algorithms and produce one data structure or technique that addresses that problem. Similarly, data structure ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification--- A technique for speeding up dynamic graph algorithms. In Proc. 33rd IEEE Symp. on Foundations of Computer Science, pages 60--9, 1992.

....cardinality. In the case of maximum matching a query outputs a current maximum matching. Alternatively, a query could also be: Is the edge e in the current graph in the current maximum matching Recently, a lot of work has been done on dynamic algorithms for various connectivity proper ties [10, 11, 12, 13, 16, 24, 25, 26]. The current best deterministic bound for maintaining connected or 2 edge connected components of a graph is O(x ) 10] The best randomized algorithm achieves O(1 3 resp. O(1 4 per update [17] It is an open problem if the connected or 2 edge con nected components of a graph can be ....

....updated within the same bounds for space and time. In the worst case the best determinis tic bound is O(x ) 10] and the best randomized algorithms take polylogarithmic time per update [17] In the case of , edge and vertex connectivity we slightly improve the known bounds: Eppstein et al. [11] describe an algorithm for dynamic k edge connectivity with worst case update time O(k2nlog(n k) using a minimum edge cut algorithm by Gabow [14] We show that (with a slight modification) its average case update time is O(min(1, kn m)k2n log(n k) plus O(k) amortized time. This gives time ....

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D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification a technique for speeding up dynamic graph algorithms. In Pvoc. 33r'd Syrup. on Foundations of Computer' Science, pages 60 69, 1992.

....ask whether a pair of vertices are connected by two edge disjoint paths. This 2 edge connectivity problem has a fairly long history. It was a long standing open problem whether deterministic polylogarithmic update time was possible. Previously, the best worst case result was O( n) update time [12], and the best randomized result was O(log n) expected update time [21] Recently, Holm, de Lichtenberg, and Thorup [22] developed a data structure with O(log n) worst case update and query time. 9 Therefore, we can find perfect matchings in 3 regular bridgeless graphs in O(n log time, ....

David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification--a technique for speeding up dynamic graph algorithms. Journal of the ACM, 44(5):669--696, September 1997.

....ask whether a pair of vertices has two edge disjoint paths between them. This 2 edge connectivity problem has a fairly long history. It was a long standing open problem whether deterministic polylogarithmic update time was possible. Previously, the best worst case result was O( n) update time [11], and the best randomized result was O(log n) expected update time [17] Recently, Holm, de Lichtenberg, and Thorup [18] developed a data structure with O(log n) worst case update and query time. Therefore, we can find perfect matchings in 3regular bridgeless graphs in O(n log n) time, ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- a technique for speeding up dynamic graph algorithms. Journal of the ACM, 44(5):669--696, 1997.

.... includes the dynamic convex hull algorithm by Overmars and van Leeuwen [57] Frekerickson s algorithm for maintaining a minimumspanning tree in a plane graph [23] and the dynamic tree data structure of Sleator and Tarjan [62] Recent successes in the field include the technique of sparsification [18] and the efficient algorithm for dynamic reachability in undirected graphs of Henzinger and King [32] Albers, Cattaneo, and Italiano [4] report empirical results for dynamic graph algorithms. A forthcoming handbook chapter [17] provides an introduction with focus on undirected graphs, because ....

David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification---a technique for speeding up dynamic graph algorithms. In Proc. 33rd FOCS, pages 60--69, 1992.

....order to match the sequential bound of O( p n) Such an approach was first proposed by Ferragina and Luccio [14] who showed how to combine simple parallel techniques with good dynamic data structures maintained on the current MST. Indeed, the sparsification data structure due to Eppstein et al. [12], designed to speed up fully dynamic sequential algorithms, has been used in [14] to provide efficient dynamic parallel algorithms, too. Subsequently, this approach has been followed (see [10, 13] providing more and more efficient techniques to manage in parallel the sparsification data ....

....to provide efficient solutions. 5.1 Background on Sparsification Data Structures We introduce the notion of certificate [3] and other relevant definitions related to the sparsification tree that will be useful in the subsequent sections. For details on the sparsification technique, refer to [12]. 16 C G G C C U C C Figure 1. A generic internal node of the sparsification tree. C 0 and C 00 are the sparse certificates (i.e. MSTs) of G 0 and G 00 , respectively. Each internal node contains a sparse subgraph formed by merging the two MSTs of its two children (i.e. C 0 ....

[Article contains additional citation context not shown here]

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification - a technique for speeding up Dynamic Graph Algorithms. In IEEE Symposium on Foundations of Computer Science, pages 60--69, 1992.

....involving d edges. In comparison, the best known algorithms for fully dynamic connectivity in general graphs require O(log n(log log n) 3 ) expected amortized time per edge operation [17] or O(log 2 n) amortized time per edge operation [11] or O( p n) worst case time per edge operation [5]. Furthermore, we show that the lower bound of Fredman and Henzinger [10] of 39 n= log log n log b) amortized time per edge operation (in the cell probe model with word size b) for fully dynamic connectivity in general graphs, applies also to the problem of maintaining connectivity in proper ....

D. ESTEIN, Z. GALIL, G. F. ITALIANO, AND A. NISSENZWEIG, Sparsification --- a technique for speeding up dynamic graph algorithms, Journal of the ACM, 44 (1997), pp. 669--696.

....graph and halts. We show how to implement this procedure in O(log n) time per operation. In comparison, the best known algorithm for maintaining connectivity in general graphs requires O(log 2 n) amortized time per operation [9] or O( p n) worst case (deterministic) time per operation [4]. We also show that the lower bound of Omega Gamma 19 n= log log n log b) amortized time per operation (in the cell probe model with word size b) 5] for maintaining connectivity in general graphs, applies also to the problem of maintaining connectivity in proper interval graphs. The paper is ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- A technique for speeding up dynamic graph algorithms. In Proc. 33rd Symp. Foundations of Computer Science, pages 60--69. IEEE, 1992.

....worst case fully dynamic connectivity was (for quite a number of years) the very basic algorithm due to Frederikson ( 10] which takes O( p jEj) update time and which initiated a clustering technique. Very recently this was improved by a novel sparsification technique to O( p n log(jEj=n) by ([7]) We remark that all the previous results ( 10] 14] 11] 15] and [7] on efficient fully dynamic structures for general graphs were based on clustering techniques. This has led to solutions of an inherent time bound of O(n ffl ) for some ffl 1, since the key problem encountered by ....

....very basic algorithm due to Frederikson ( 10] which takes O( p jEj) update time and which initiated a clustering technique. Very recently this was improved by a novel sparsification technique to O( p n log(jEj=n) by ( 7] We remark that all the previous results ( 10] 14] 11] 15] and [7]) on efficient fully dynamic structures for general graphs were based on clustering techniques. This has led to solutions of an inherent time bound of O(n ffl ) for some ffl 1, since the key problem encountered by these techniques is that the algorithm must somehow balance: i) the work ....

D. Eppstein, Z. Galil, G. Italiano and A. Nissenzweig, "Sparsification- A technique for speeding up Dynamic Graph Algorithms", FOCS 1992.

....which has the property, proving that G as it as well. The advantage is that since the certificate is sparse, the property can be verified more quickly. The skeleton is a kind of sparse approximate certificate. Sparse certificates have already been used successfully in several areas. Eppstein et al. [4] give sparsification techniques which improve the running times of dynamic algorithms for numerous graph problems such as connectivity, bipartitioning, and and minimum spanning trees. In [14] we developed a randomized sparsification technique for minimum spanning trees which led to the ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. "Sparsification---a technique for speeding up dynamic graph algorithms". In Proceedings of the 33 rd Annual Symposium on Foundations of Computer Science, pages 60--69, Oct. 1992.

.... e Sistemistica, Universit a di Roma La Sapienza , Via Salaria 113 00198 Roma, Italy, ffrigioni,ioffreda,nanni,pasqualog dis.uniroma1.it 1 1 Introduction A lot of efforts have been done in the last years in order to devise efficient algorithms for dynamic graph problems (e.g. see [6, 9, 13, 14, 15, 16, 18, 20, 23, 24, 25, 26, 30, 31, 32]) motivated by theoretical as well as practical applications. In the literature, the most used dynamic model is the following: we are given a graph G and we want to answer queries on a property P of G, while the graph is changing due to insertions and deletions of edges. For instance, if the ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- a technique for speeding up dynamic graph algorithms. In IEEE Symposium on Foundations of Computer Science, pages 60--69, 1992. 19

.... planar minimum spanning tree problem can be reduced to a graph problem in a graph formed by a number of bichromatic closest pair problems, which could then be solved with the same techniques used for diameter [2] By combining this idea with clustering techniques for graph minimum spanning trees [20, 21, 23], we were able to use this idea to solve the minimum spanning tree problem in worst case time O(n 1 2 log 2 n) per update [2] But in the average case, the minimum spanning tree can be maintained much more easily in expected time O(log n) per update by combining a dynamic Delaunay ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- A technique for speeding up dynamic graph algorithms. In Proc. 33rd IEEE Symp. Foundations of Computer Science, pages 60--69, 1992.

....Minimum spanning tree. The planar minimum spanning tree problem can be reduced to a graph problem in a graph formed by a number of bichromatic closest pair problems, which could then be solved with the same techniques used for diameter. Using clustering techniques for graph minimum spanning trees [14, 15, 18], we were able to solve the minimum spanning tree problem in time O(n 1 2 log 2 n) per update [2] In the average case, the minimum spanning tree can be maintained much more easily in time O(log n) per update by combining a dynamic Delaunay triangulation algorithm [23] with a dynamic planar ....

D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- A technique for speeding up dynamic graph algorithms. In Proc. 33rd IEEE Symp. Foundations of Computer Science, pages 60--69, 1992.

....CCR 9014605. z On leave from Universit a di Roma, Italy. 1 Introduction In the last decade there has been a growing interest in dynamic problems on graphs. In particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53], transitive closure [29, 30, 31, 37, 47, 55] planarity [7, 8, 46] shortest paths [2, 5, 13, 39, 44] and minimum spanning trees [10, 11, 16] In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and ....

.... particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53] transitive closure [29, 30, 31, 37, 47, 55] planarity [7, 8, 46] shortest paths [2, 5, 13, 39, 44] and minimum spanning trees [10, 11, 16]. In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, so queries can be ....

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D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- A technique for speeding up dynamic graph algorithms. In Proc. 33rd Annual Symp. on Foundations of Computer Science, 1992.

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D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig, Sparsification --- A technique for speeding up dynamic graph algorithms, in "Proceedings, 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, 1992," 60--69.

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David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - A technique for speeding up dynamic 11 graph algorithms. Journal of ACM, 44(1):669--696, 1997.

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D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification -- A technique for speeding up dynamic graph algorithms. In IEEE Symposium on Foundations of Computer Science, pages 60--69, 1992.

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David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig, Sparsification -- a technique for speeding up dynamic graph algorithms, J. ACM 44 (1997), no. 5, 669--696.

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D. Eppstein, Z. Galil, G. F. Italiano, and A. Nissenzweig. Sparsification---a technique for speeding up dynamic graph algorithms. J. Assoc. Comput. Mach., 44(5) (1997), 669--696.

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D. Eppstein, Z. Galil, G. F. Italiano, A. Nis- senzweig, "Sparsification - A Technique for Speeding up Dynamic Graph Algorithms" Proc. 33rd Syrup. on Foundations of Computer Science, 1992, 60 69.

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