G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14:781--798, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....compute G in O(m log n) time. Using this reduction as a preprocessing step, Frederickson s algorithm runs in O(m log n k minfk; mg) time. This is the fastest algorithm known. Frederickson s dynamic best swap structure closely resembles his earlier dynamic minimum spanning tree structure [2]. One can easily imagine in fact, I did imagine when I foolishly assigned this as a homework problem that this data structure can be improved similarly to the dynamic minimum spanning tree algorithm of Holm , de Lichtenberg , and Thorup [4] that we saw in class. However, as far as I know, no ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14:781-798, 1985.

....that each node enters a critical section often) Finally, questions similar to the one asked in this paper can be raised in non distributed contexts as well. e.g. tight mending meaningful also for sequential data structures. There, it forms a generalization of the area of dynamic data structures [Fre83a, T83], in which one assumes that F (rather than 1 or n) changes occurred, and studies the complexity of updating the data structure. Acknowledgments: It is a pleasure to thank Hadas Shachnai and Moti Yung for helpful discussions, and Esther Jennings for her useful comments on an earlier draft. Thanks ....

Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees. In Proc. 15th ACM Symp. on Theory of Computing, April 1983.

....E) lower bound does not apply to amortized communication complexity of dynamic network protocols. Intuitively, we can benefit from the knowledge of the past and thus economize on communication. It is well known that this is the case in sequential computation, e.g. Frederickson shows [Fre83a] how to maintain a dynamic minimum spanning tree with (amortized cost) of O( p E) computations per input change, while constructing a (single) tree from scratch requires O(E) computations. Unfortunately, in the distributed computation model, it is far from obvious how to reduce the incremental ....

Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees. In Proc. 15th ACM Symp. on Theory of Computing. ACM SIGACT, ACM, April 1983.

....of [18] maintains a spanning forest of G. Each tree T in this spanning forest is represented by a top tree T [1] The top tree is a data structure used to represent trees that are subject to insertion and deletion of edges. The top tree structure is a variant of Frederickson s topology trees [8] that directly supports trees of unbounded degree. Top trees are also related to the dynamic trees of Sleator and Tarjan [25] Top trees have been used to help solve several dynamic connectivity problems as well as to maintain centers, medians, and diameters of trees that are subject to insertion ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781-798, 1985.

....number of nodes) Moreover, at any moment, the data structure can answer the following type of query in O(1) time: given two nodes in the graph, are these nodes 2 or 3 edge connected. I Introduction Recently there has been a growing interest in dynamic or on line graph algorithms (see e.g. [3, 8, 9, 10, 18]) A graph algorithm is called dynamic or on line if it maintains some information related to a graph while the graph is being changed (e.g. by inserting or deleting a node or an edge) A dynamic algorithm exploits a suitable data representation for a graph and uses information of the old graph to ....

....from scratch, i.e. by using the new graph as input only, and a better performance may be expected compared to an algorithm that simply ecomputes . Dynamic algorithms are known for e.g. computing transitive closures (cf. 8, 9, 10] or cf. 17] for planar graphs) minimal spanning trees (cf. [3]) incremental planarity testing (cf. 2] and maintaining shortest paths (cf. 18] One sometimes uses the term on line algorithm when only insertions (of nodes or edges) are allowed. This research was partially supported by the ESPRITBasic Research Action No. 3075 (project ALCOM) ....

G.N. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications, SIAM J. Computing 14 (1985), pp.781-798.

....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 ....

....the preprocessing time is O(x m ) Additionally we give an insertionsonly algorithm for maximum cardinality matching with O(n) amortized time per insertion. Assuming that the weight of an edge is arbitrary, but fixed, we show that a modified version of Fredericksoh s topology tree data structure [12] for dynamic minimum spanning forests has an average case update time of O(1 n n x ) plus amortized constant time. The data structure needs linear space and linear expected preprocessing time. The best worst case update time for this problem is O(x ) 10] Dynamic connectivity, 2 edge ....

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Cornput., 14:781 798, 1985.

....problems on planar structures in computational geometry and graph drawing, have been the focal point of dynamic algorithms. Seminal work 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 ....

Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal of Computing, 14(4):781-- 798, 1985.

....to an e#cient solution either, because the intersection graph can have quadratic size, and an insertion deletion of a rectangle can cause a linear number of edge updates in the worst case. Imagine deleting and re inserting a segment like s in Figure 1. Thus, work on dynamic graph connectivity [13, 15, 18, 17, 21, 27, 28] is only the beginning, if we want to tackle the more challenging dynamic connectivity problems from geometry. In this paper, we show that a sublinear time bound is indeed theoretically possible for dynamic connectivity of rectangles, and in fact, axis parallel) boxes in any fixed dimension d. ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14:781--798, 1998.

....and Shiloach s algorithm on the macro tree and in section 3 we will present an algorithm to maintain micro trees of size log n, such that each operation in a micro tree takes constant time. In order to do this we will partition the orignal tree T using the following lemma. The lemma follows from [2, 3] 1 . Lemma 3 Let T be a tree with n nodes, where the degree of a node is at most 3. A linear time algorithm exists which partitions a tree T into O(n= log n) micro trees, where each micro tree includes at most log n nodes and two boundary nodes. 2 To use this lemma we make the following simple ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781-798, 1985.

.... [2] Our presentation of the interface will be somewhat more precise and thorough than that in [2] The more exact understanding of the interface is needed for both our applications, and for our later methodological discussion of top trees versus more classical data structures for dynamic trees [10, 12, 27]. A top tree is de ned based on a pair consisting of a tree T and a set T of at most 2 nodes from T , called external boundary nodes. Given (T ; T ) any connected subtree C of T has a set (T; T ) C of boundary nodes which are the nodes of C that are either in T or incident to an edge in T ....

....new top tree TC . Finally, the new root cluster C is returned. Split(C) where C is the root cluster of a top tree TC and has children A and B. Deletes C, thus turning TC into the two top trees TA and TB . Recall that n denotes the size of the trees involved in a given update operation. From [2, 10] we have: Theorem 1 For a dynamic forest we can maintain top trees of height O(log n) supporting each Link, Cut, or Expose with a sequence of O(log n) Merge and Split. Here the sequence itself is identi ed in O(log n) time. The space usage of the top trees is linear in the size of the dynamic ....

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G.N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SICOMP, 14(4):781-798, 1985.

....(see [8, 29] From the considerations made above, it is simple to derive the time and processor bounds as shown in Table 2. 15 5 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 ....

....maintenance of an MST under single edge updates. In this paper, the employed parallel data structures are designed to handle bounded degree graphs in which no node has degree greater than three. This is not a limitation since, given the input graph G, a well known transformation in graph theory [15, 18] can be used to produce a graph G 0 = V 0 ; E 0 ) with jV 0 j = jE 0 j = O(m) in which each node satisfies the desired degree constraint. Therefore, for the purpose of describing the algorithm, one can assume to deal with bounded degree graphs of n = O(m) nodes, each of degree at ....

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14(4):781--798, 1985.

....that did not become boundary nodes of the merged cluster. Finally the dashed line is the cluster path of the merged cluster. From [2, 3] we know that a top tree over T can be constructed in linear time. The construction from [2, 3] which is based on a reduction to Frederickson s topology trees [6], is very complicated. However, the construction addresses the more general problem of maintaining top trees for a dynamically changing forest. Here, we only need to compute a static top tree once, and this is much simpler. A direct construction is given in the appendix. This direct construction ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781-798, 1985.

....and deletions of edges in the graph. The solution presented in [8] computes a best swap in O(n) time per update, using O(m) space and preprocessing time. For the edge deletion case, that is of interest for this paper, the authors make use of a topology tree and a 2 dimensional topology tree [3], augmented with some extra information. This requires O(m) time and space for initialization, and allows to compute the length of the diameter of any tree obtained as a consequence of a swap in O(n) time. Hence, the approach in [8] is more general than what is needed for solving the ABS problem, ....

....) In the former case, to break the tie, we set d c = d i j , while in the latter case we set d c = d i 1 . Let # T denote a source directed tree obtained by rooting T in d c and orienting the edges towards the leaves. Following [8] we maintain a topology tree and a 2 dimensional topology tree [3, 4], augmented with some extra information, to e#ciently retrieve only O( # m) selected edges among the O(m) replacement edges, whenever an edge e in T is deleted. In fact, among the selected edges, a best swap is contained (for a proof of it, see [8] The general outline of our algorithm is the ....

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G. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Computing, 14(4):781--798, 1985.

.... result, by Katoh et al. [18] is that the k best spanning trees can be found in time O(m log #(m, n) km) This result improved several prior results by Burns and Ha# [6] Camerini et al. [7] and Gabow [15] Apparently, Dov Harel has discovered an O(m log #(m, n) kn log 2 n) time algorithm [13]; this is an improvement for graphs that are not too sparse. Frederickson [13] considered the problem for small k, in particular k = O( # m) His algorithm uses a technique for maintaining a MST in a dynamically changing graph, and runs in time O(m log #(m, n) k 2 # m) Frederickson also gave a ....

....in time O(m log #(m, n) km) This result improved several prior results by Burns and Ha# [6] Camerini et al. [7] and Gabow [15] Apparently, Dov Harel has discovered an O(m log #(m, n) kn log 2 n) time algorithm [13] this is an improvement for graphs that are not too sparse. Frederickson [13] considered the problem for small k, in particular k = O( # m) His algorithm uses a technique for maintaining a MST in a dynamically changing graph, and runs in time O(m log #(m, n) k 2 # m) Frederickson also gave a version of his algorithm for planar graphs that runs in time O(n k 2 log ....

G.N. Frederickson, Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications, SIAM J. Comput. 14(4), 1985, 781--798.

....application to evolutionary trees. The basic idea of transforming a tree into a new tree with logarithmic height is a fundamental approach used in many algorithms. For designing dynamic algorithms on trees several other general tree transformation techniques exist: Frederickson s topology trees [10, 11], Sleator and Tarjan s dynamic trees [24] and Alstrup et al. s top trees [1, 2] One application of such a tree transformation is in Cohen and Tamassia s algorithm for dynamic expression tree evaluation [7] For parallel algorithms on trees related techniques exist, e.g. the centroid ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781798, 1985.

....simpler pivot rules such as cyclically scanning the edges not in T , which tend to find pivots more quickly (although the worst case bound per pivot may still be O(m) making up for the increase in the number of pivots. In this paper we show that a clustering technique developed by Frederickson [6, 7] and applied in a variety of dynamic graph problems [5, 6, 7, 11] can be used to speed up Dantzig s rule for the network simplex algorithm. We describe a method of implementing Dantzig s rule for which the time per pivot is O( # m) improving the previous O(m) bound. To our knowledge this is the ....

....not in T , which tend to find pivots more quickly (although the worst case bound per pivot may still be O(m) making up for the increase in the number of pivots. In this paper we show that a clustering technique developed by Frederickson [6, 7] and applied in a variety of dynamic graph problems [5, 6, 7, 11] can be used to speed up Dantzig s rule for the network simplex algorithm. We describe a method of implementing Dantzig s rule for which the time per pivot is O( # m) improving the previous O(m) bound. To our knowledge this is the first sublinear time bound for pivot selection in the network ....

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput. 14 (1985) 781--798.

.... 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 ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput., 14:781--798, 1985.

....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 ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput., 14:781--798, 1985.

....components of a graph is O(q n log n) 2] and the algorithms given in [2] achieve the O(qff(q; n) bound only in the case of an initially biconnected graph. Our techniques combine a variety of graph properties, data structures and new algorithmic tools. Following the ideas of Frederickson [3], we partition graphs into vertex clusters. However, this by itself is not enough to achieve efficient algorithms for our problems. We find a way to maintain a succinct encoding of each cluster which satisfies nice properties with respect to the 2edge connected components of the graph. This ....

....describe algorithms dealing with connected graphs. All the details of the method as well as the algorithm for unconnected graphs can be found in [5] 2.1 The data structure We will describe a data structure operating on graphs with bounded degree. It is well known (see for example references [3] and [9, page 132] that each graph G = V; E) can be transformed into a graph G 0 = V 0 ; E 0 ) whose vertices have degree no greater than three and G 0 preserves the bridges of G. The transformation is as follows. For each vertex u of G of degree d 4, where v 0 ; v 1 ; v ....

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G. N. Frederickson, "Data structures for on-line updating of minimum spanning trees", SIAM J. Comput. 14 (1985), 781--798.

....y Work partially supported by NSF Grant 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 ....

.... 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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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput., 14:781--798, 1985.

.... be returned as soon as it is available [32, 40, 56] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [27, 33, 35, 36] incremental [21, 22, 49, 59] dynamic [11, 12, 67] and updating [20, 23, 28, 38, 53, 54, 62, 66]. It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for example, the various interpretations of the notion of real time provided in [10, 42, 65] 2.2 Models of ....

....an interesting 18 case arises in connection with the minimum weight spanning tree (MST) problem when corrections to the weights of the edges currently in the MST are received in real time and must be taken into consideration. Sequential and parallel algorithms for this problem are described in [21, 28, 54]. However, while these algorithms update the MST as required, their analyses (much like those of the algorithms in [36] do not allow for the corrections to arrive, or for the results to be produced, at a certain specified rate. Here too an open avenue for research suggests itself quite ....

G. Frederickson, Data structures for on-line updating of minimum spanning trees, Proceedings of the ACM Symposium on Theory of Computing , Boston, Massachusetts, April 1983, 252--257.

....work on dynamic complexity for databases includes the theory of maintaining materialized views upon updates ( J92] GMS93] Io85] and in integrity constraint simplification ( LST87] N82] The design of dynamic algorithms is an active field. See, for example, E 92] E2 92] R94] CT91] [F85], F91] among many others. There is also a large amount of work in the programming language community on incremental computation, see for example [RR93, LT94] 2 This paper is organized as follows. In Section 2, we begin with some background on Descriptive Complexity. In Section 3, for any ....

G.F. Frederickson, "Data structures for on-line updating of minimum spanning trees," SIAM J. Comput. , 14 (1985), 781-798.

....often holds for updates as well, a fact that has been used by Eppstein et al. EGIN92, EGI93] to solve several dynamic graph problems. We shall now summarize one of the key ideas behind sparsification: the use of sparsification trees, which were introduced in [EGI93] see also Frederickson s work [Fre85a, Fre91]) Sparsification trees are built in two steps. In the first, a vertex partition tree is constructed by splitting the vertex set into two equal size parts (to within 1) and then recursively partitioning each half. This results in a complete binary tree of height at most log n where nodes at depth ....

G.N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput. 14:781--798, 1985.

....often holds for updates as well, a fact that has been used by Eppstein et al. EGIN92, EGI93] to solve several dynamic graph problems. We shall now summarize one of the key ideas behind sparsification: the use of sparsification trees, which were introduced in [EGI93] see also Frederickson s work [Fre85a, Fre91]) Sparsification trees are built in two steps. In the first, a vertex partition tree is constructed by splitting the vertex set into two equal size parts (to within 1) and then recursively partitioning each half. This results in a complete binary tree of height at most log n where nodes at depth ....

G.N. Frederickson. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput. 14:781--798, 1985.

....the diameter of the tree the node belongs to. The time complexity is O(logn) for each operation, where n is the number of nodes in the tree(s) involved. We show this, since to the best of our knowledge, no such algorithm has been presented before. All our results are based on topology trees [3, 2] (the terminology of topology trees is recalled in Section 2) Our algorithm for maintaining the diameter is straightforward, based on a simple observation. Our algorithm for finding a best swap is much more involved. One complication is that when we want to merge two clusters, we need to consider ....

....are given. In section 3 we present an algorithm for maintaining the diameters of trees in a dynamic forest. Finally in section 4 we give an algorithm which compute a best swap in O(log 2 n) time. 2 Preliminaries In this section we give a short presentation of the topology trees by Frederickson [3, 2]. Our presentation differ slighty from the original topology trees. We provide a more simple interface in order to simplify the use of the topology trees. Let T be a tree with n nodes. For a connected subtree of T , we call a node which has edges out of the subtree a boundary node. A cluster is a ....

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781--798, 1985.

....each update was known. These bounds apply to problems in which insertions need not respect a fixed embedding of the graph; a number of other papers have worked on dynamic graph problems such as minimum spanning forests, connectivity, and planarity testing for graphs with a fixed planar embedding [12, 14, 15, 18, 19, 22, 21, 24, 32, 33]. Finally, our methods apply to static as well as dynamic graph problems. A general certificate construction method from our companion paper, together with the certificates defined here, gives a unified method of testing 3 and 4 edge , and 2 and SPARSIFICATION II: EDGE AND VERTEX CONNECTIVITY ....

G. N. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications, SIAM J. Comput., 14 (1985), pp. 781--798.

....have been presented in [28] Chapter 5 contains further introduction to these problems and an overview of previous solutions. 3 In many dynamic graph algorithms, a data structure for dynamic trees is used. A well known data structure for dynamic trees is the topology trees of Frederickson [12]. The topology trees are only defined for ternary trees, and it has been an open problem whether topology trees can be generalized to general trees with unbounded degree. Such a variant, called top trees, is provided in this thesis. The top tree data structure is subsequently used to provide O(log ....

....based on a data structure for dynamic trees. Therefore many different data structures have been developed for dynamic trees. Examples are the dynamic trees (here ST trees) of Sleator and Tarjan [35] the ET trees developed in [9] based on Euler tours and finally the topology trees of Frederickson [12, 13, 14, 15]. The ET trees are simple and efficient but limited, in that they cannot maintain information about paths. Both the ST trees and the topology trees partition the tree into node disjoint parts. The ST trees partition the tree according to paths, whereas topology trees use the topology of the tree ....

G.N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781--798, 1985.

.... off this brief introduction to data structures for dynamic trees, in chapter 7 we apply the top trees to the minimum spanning forest problem, thus all of the above the mentioned data structures for dynamic trees, ST trees, topology trees, ET trees and top trees, have been applied to this problem [10, 24, 35]. 2.2 Applications In chapter 4 we give examples of top tree algorithms. In the fully dynamic Maxweight problem, we have a tree with weighted edges. In this tree, we may insert or delete edges and perform path updates (adding a constant to all edges on a specified path) and Maxweight queries ....

....contain the desired edge. After O(log n) decisions, Merge and Split operations, the desired edge is found. To our knowledge, no such tool has been presented before. 1 Now Monica Rauch Henzinger 9 Chapter 3 Top trees In this chapter we will introduce a variant of Fredericksons topology trees [10]. The original topology trees are defined for ternary trees which can then be used to encode trees of unbounded degrees. This is often quite technical, so instead we have developed a variant (first presented in [1] called top trees, which works directly for trees of unbounded degree, and which ....

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781--798, 1985. See also STOC'83.

....at cost O(t(n) log n) per update, then, using dynamic trees [11] we can answer connectivity queries in time O(log n= log t(n) In this paper we study some algorithms in which t(n) is polylogarithmic. The rst non trivial fully dynamic connectivity algorithm was presented in 1985 by Fredrickson [4]. It supported updates in O( p m) time and queries in constant time. In 1992, Epstein et al. 3] improved the update time to O( p n) In 1995, Henzinger and King [6] presented the rst polylogarithmic algorithm for fully dynamic connectivity. The algorithm was randomized, supporting updates ....

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing, 14(4):781-798, 1985. See also STOC'83.

....al. 1998] and compare their performance in practiced on assorted input test families. We also evaluate several heuristics that can improve the performance of these algorithms in practice. 1. 1 History The first non trivial fully dynamic connectivity algorithm was presented in 1985 by Fredrickson [Frederickson 1985]. It supported updates in O( p m) time and queries in constant time. In 1992, Epstein et al. Eppstein et al. 1997] improved the update time to O( p n) In 1995, Henzinger and King [Henzinger and King 1995] presented the first polylogarithmic algorithm for fully dynamic connectivity. The ....

Frederickson, G. N. 1985. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Computing 14, 4, 781--798. See also STOC'83.

....above in time O(n log 2 n) we can apply the algorithm of Berman Ramaiyer for 4 restricted Steiner trees and get an approximation as stated in Theorem 3. The operations of the algorithm itself are mainly update operations on a minimum spanning tree, which can be performed in time O( p n) each [10]. We conclude this subsection with the following theorem: Theorem 5 For any rectilinear Steiner tree problem an 1.271 approximation can be found in time O(n 3=2 ) In the next section, we try to speed up the algorithm to achieve a better running time. We generalize the parameterized method ....

G. Frederickson. Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications. SIAM Journal of Computing, 14, 781-789, 1985.

....realm of graph theory is more elusive. Many basic graph problems like Spanning Trees, Connected Components, Shortest Paths, etc. reduce to Reachability, which seems to be hard in the dynamic case. For undirected graphs, one can hope for polylog time solutions as long as the graph is plane, see [6, 5]. For directed graphs, not even that restriction is enough; the best algorithm for Dynamic Reachability on planar digraphs is due to Subramanian [15] and performs in amortised time O(n 2=3 log n) This is interesting to the theoretician because in the parallel realm, the Reachability Problem is ....

Greg E. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications, SIAM Journal of Computing 14 (1985), no. 4, 781--798.

....time for planar embedded graphs, under the condition that each insertion maintains the planarity of the embedding. The best previous algorithm took time O( # n) per update and O(log n) per query. 1 Insertions or deletions of isolated vertices are usually trivial. 2 Related work. Frederickson [5] gave the first dynamic graph algorithm for maintaining a minimum spanning tree and the connected components. His algorithm takes time O( # m) per update and O(1) per query operation. The first dynamic 2 edge connectivity algorithm by Galil and Italiano [9] took time O(m 2 3 ) per update and ....

....et al. 16, 4] and O( # n) for maintaining biconnected components by Eppstein et al. 4] Outline of the paper. First (Section 2) we study the dynamic biconnectivity problem for general graphs. Our basic approach is to partition the graph G into small connected subgraphs, called clusters (see [5] for a first use of this technique in dynamic graph algorithms) Each biconnectivity query between a vertex u and a vertex v can be decomposed into a query in the cluster of u, a query in the cluster v, and a query between clusters. To test biconnectivity between clusters we use the 2 dimensional ....

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G. N. Frederickson, "Data Structures for On-line Updating of Minimum Spanning Trees", SIAM J. Comput. 14 (1985), 781--798.

....# F , then (a) remove u, v from F and (b) return the minimum cost edge e of G F that reconnects F if e exists or return null if e does not exist. In addition, the data structure permits the following type of query: connected(u,v) Determine if vertices u and v are connected. In 1985 [7], Fredrickson introduced a data structure known as topology trees for the fully dynamic minimum spanning tree problem with a worst case cost of O( # m) per update His data structure permitted connectivity queries to be answered in O(1) time. In 1992, Eppstein et al. 3, 4] improved the update ....

....We present a fully dynamic minimum spanning tree data structure that uses O(n 1 3 log n) amortized time per update and O(1) worst case time per query when update time is averaged over any sequence of #(m in ) updates, for m in the initial size of the graph. Our technique is very different from [7]. The result is achieved in two steps: First, we give a deletions only minimum spanning tree algorithm that uses O(m #1 3 log n n # ) amortized time per update and O(1) worst case time per query when the update time is averaged over any sequence of #(m in ) updates. Here # is any constant ....

G. N. Frederickson, "Data Structures for On-line Updating of Minimum Spanning Trees", SIAM J. Comput., 14 (1985), 781--798.

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14:781--798, 1985.

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G. N. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications,SIAMJ.on Computing, 14 (1985), 781--798.

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Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14(4):781--798, 1985.

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G.N. Frederickson, "Data structures for on-line updating of minimum spanning trees, with applications, " SIAM J. Computing 14 (1985), 781--798.

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Frederickson, G. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput. 14, 4 (1985), 781--798.

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. volume 14, pages 781798, 1985.

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14:781--798, 1985.

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Frederickson G.N. Data structures for on-line updating of minimum spanning trees, with applications SIAM J. Comput, 14(4):781-798, Nov. 1985.

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Frederickson, G. Data structures for on-line updating of minimum spanning trees. SIAM J. Comput. 14, 4 (1985), 781--798.

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781-798, 1985. 18

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G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14:781--798, 1985.

No context found.

G. N. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications, SIAM J. Comput. 14 (1985), 781-798.

No context found.

G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4) (1985), 781--798.

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G. Frederickson, Data structures for on-line updating of minimum spanning trees, Proceedings of the ACM Symposium on Theory of Computing , Boston, Massachusetts, April 1983, 252--257.

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G. N. Frederickson, "Data Structures for On-line Updating of Minimum Spanning Trees", SIAM J. Conput., 14 (1985), 781 798.

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G. N. Frederickson, "Data Structures for On-line Updating of Minimum Spanning Trees", SIAM J. Comput., 14 (1985), 781--798.

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