G. N. Frederickson, Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, in "Proceedings, 32rd Annual Symposium on Foundations of Computer Science, San Juan, 1991," 632--641.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....once, in order from smallest to largest [1] So what we need is a persistent data structure that can maintain the next best swap as we insert or delete edges. Edges can be contracted by deleting them and then reinserting them with weight 1. Such a data structure is described by Frederickson [3]; this data structure has update time O( m) where m is the number of edges in the original graph G. Using this data structure, we can compute the k best spanning trees in O(m log n k m) time. Eppstein [1] showed that when k n, there is a graph G with O(k) vertices, such that the k ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Comput. 26(2):484-538, 1997.

....The boundary nodes of C are denoted as C. A cluster is a connected subgraph of T where jCj 2. A set of clusters CS is a cluster partition of a tree T with root r iff V (T ) C2CSV (C) E(T ) C2CSE(C) and for any C 1 ; C 2 2 CS, E(C 1 ) E(C 2 ) jE(C 1 )j 1, r 2 C if r 2 V (C) From [5, 4, 28] it follows that: 1 x n, there exists a cluster partition CS where jCSj n=x, and for a fixed constant c, jV (C)j cx for all C 2 CS. To each cluster C 2 CS with only one boundary node, we associate the rightmost leaf in C as the second boundary node. For two boundary nodes u; v in a ....

G.N. Frederickson. Ambivalent data structures for dynamic 2--edge--connectivity and k smallest spanning tree. SIAM J. Computing, 26(2):484--538, 1997.

....of a simple polygon where regions have at most three doors. In other words, we make sure that when we split a region with three doors, both subregions contain at least one of the original three doors. Our technique for achieving this is inspired by the topology tree hierarchy of Frederickson [9, 10]. The decomposition can be based either on a fixed triangulation or on the vertical decomposition (trapezoidal map) of the polygon. We believe that this decomposition will prove useful in a number of applications in computational geometry that deal with problems involving paths in simple ....

G.N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Cornput., 26:484-538, 1997.

....n worst case space overhead per server. 4 Logarithmic Dictionary Tree (LDT) 4. 1 Distribution An alternative way to manipulate a random tree T with good operation performance is to build an auxiliary, logarithmic height search tree T on top of it; Frederickson introduced the topology trees [8, 9] in the context of link cut trees and dynamic expression trees: Given a binary tree T = V;E) a topology tree T is a balanced tree constructed on top of the nodes of T by node aggregation or clustering. T consists of multiple levels, with each level i containing a tree structure T i on the ....

....these edges cross edges. Next, clustering is performed on the nodes of T 1 to form V 2 and T 2 of level 2 and so on until we come up with a level consisting of a single cluster node. The cluster nodes V i and the cross edges between cluster nodes of consecutive levels form the topology tree T . [8, 9] proposed the following clustering rules so that the resulting topology tree T is of logarithmic height: a) each cluster of degree 3 contains only one node; b) each cluster of degree less than 3 contains at most two nodes; and (c) no two adjacent clusters i.e. there is an edge with ....

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G.N. Frederickson. Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees. SIAM J. Comput., 26(2):484--538, April 1997.

....from vertices in L(N 1 ) to vertices in L(N 2 ) for each pair N1;N2 of related nodes. The space for this data structure is O(n ) It can either be built as a subset of the data structure of Theorem 11, in time O(n log n) or bottom up (using hierarchical clustering techniques of Frederickson [22] to construct the level structure in T , and then computing each distance matrix from two previously computed distance matrices in time O( in total time O(n ) we omit the details. To answer a query, we form chains of related pairs connecting the home nodes (or small subtrees) of the ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. SIAM J. Comput. 26(2):484{ 538, April 1997.

....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 ....

....tree T of G . In the following whenever we use the term tree edge we refer to an edge of the current minimum spanning tree in case of a deletion or the new minimum spanning tree in case of an insertion. To achieve fast updates for non tree edges we modify the definitions of clusters given in [13]. A basic cluster is a set of vertices that induces a subgraph of T that is connected. An edge is icidet to a cluster if exactly one of its endpoints is in the cluster and it is iteral to a cluster if both endpoints are in the cluster. The tree degree of a cluster is the number of tree edges ....

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G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. In Proc. 32nd Syrup. on Foundations of Computer' Science, pages 632 641, 1991.

....size with the following bounds: 1. worst case update time O(log log n) for both mark and unmark, 2. worst case query time O(log n= log log n) 3. linear space and preprocessing time. To achieve these results we present a new micro macro division of trees. In contrast to standard tree divisions [20, 26], our approach does not limit the number of nodes or the number of boundary nodes in a micro tree. This leads to exponentially better update times. Comparing upper and lower bounds we see that the query time is optimal, even if we would allow polylogarithmic time for each update or consider ....

G.N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. SIAM J. Comput., 26(2):484-538,

....of a simple polygon where regions have at most three doors. In other words, we make sure that when we split a region with three doors, both subregions contain at least one of the original three doors. Our technique for achieving this is inspired by the topology tree hierarchy of Frederickson [9, 10]. The decomposition can be based either on a fixed triangulation or on the vertical decomposition (trapezoidal map) of the polygon. We believe that this decomposition will prove useful in a number of applications in computational geometry that deal with problems involving paths in simple ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Comput., 26:484--538, 1997.

....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 ....

....edge deleted from the tree T belongs to a micro tree, we can in constant time determine if this implies that the boundary nodes in the micro tree are no longer connected using microcon on the boundary nodes. If this is the case, we delete the edge in the macro tree which is incident with the 1 In [3] a micro tree is called a cluster. The external degree of a cluster is de ned as the number of tree edges incident to exactly one node in the cluster. A cluster has external degree at most 3 and if the external degree of a cluster is 3 the cluster consists of only 1 node, implying that the number ....

G.N. Frederickson. Ambivalent data structures for dynamic 2{edge{connectivity and k smallest spanning tree. SIAM J. Computing, 26(2):484-538, 1997. See also FOCS'91.

.... [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 ....

.... we just reverse all Merge and Split to restore the top tree in its original form, and return the edge (v; w) 4 Methodological remarks Our main results in Theorem 4 and 7 could also have been achieved based on either the Sleator and Tarjan s dynamic trees [27] or Frederickson s topology trees [10, 12]. However, we claim that the derivation from these more classical data structures would have been more technical. Sleator and Tarjan s dynamic trees Sleator and Tarjan provide an axiomatic interface for their dynamic trees [27] where the user can choose a root with a so called Evert operation, ....

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G.N. Frederickson. Ambivalent data structures for dynamic 2{edge{connectivity and k smallest spanning trees. SICOMP, 26(2):484-538, 1997. See also FOCS'91.

....Let T be a tree of size n = jV (T )j 1. For a connected subgraph C of T , we call a node in V (C) incident with a node in V (T ) n V (C) a boundary node. The boundary nodes of C are denoted as C. A cluster is a connected subgraph of T where j Cj 2. The de nitions of clusters comes from [3, 2, 5]. A set of clusters CS is a cluster partition of a tree T with root r i V (T ) C2CSV (C) E(T ) C2CS E(C) and for any C 1 ; C 2 2 CS, E(C 1 ) E(C 2 ) jE(C 1 )j 1, r 2 C if r 2 V (C) and if v 2 C has two children a and b, then no clusters include both a and b. From [3, 2, 5] ....

....[3, 2, 5] A set of clusters CS is a cluster partition of a tree T with root r i V (T ) C2CSV (C) E(T ) C2CS E(C) and for any C 1 ; C 2 2 CS, E(C 1 ) E(C 2 ) jE(C 1 )j 1, r 2 C if r 2 V (C) and if v 2 C has two children a and b, then no clusters include both a and b. From [3, 2, 5] it follows that: Lemma 2 Given a tree T , n 1, and a parameter x, where dn=xe 2, it is possible to construct a cluster partition CS in linear time, where jCSj k x, for a constant k, and jV (C)j dn=xe for C 2 CS. In [1] a similarly partitioning is de ned. For the sake of completeness ....

G. Frederickson. Ambivalent data structures for dynamic 2{edge{connectivity and k smallest spanning tree. SIAM J. Computing, 26(2):484-538, 1997. See also FOCS'91.

....) 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 ....

....MDST T # of G # can be derived from a MDST T of G in O(m) time in the following way: we keep in T # all the edges in T , and for each node v # V we add the edges (v i , v i 1 ) i = 1, #(v) 1, of length 0. Then, we associate with T # a topology tree and a 2 dimensional topology tree [3, 4], both augmented with some extra information useful for solving our problem [8] A topology tree # associated with T # is a hierarchical representation of G # based on T # , while a 2 dimensional topology tree # # associated with # is a hierarchical representation of G # based on #. The structures ....

G. Frederickson. Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 632--641, 1991.

.... of removing and contracting edges in algorithms for the related problem of maintaining a minimum spanning tree in a changing graph, subject to an o#ine sequence of edge insertions and deletions, and for a similar dynamic geometric minimum spanning tree problem [10] A recent paper of Frederickson [14] improves the k smallest spanning trees algorithm of [18] and again uses the reduction described here to improve some of our time bounds as well. Throughout this paper we allow graphs to have multiple edges between the same pair of vertices. Therefore we do not denote an edge by its adjacent ....

G.N. Frederickson, Ambivalent Data Structures for Dynamic 2-edge-connectivity and k Smallest Spanning Trees, 32nd IEEE Conf. Foundations of Computer Science, 1991, to appear.

....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. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. SIAM J. Comput., 26(2):484 538, 1997.

....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. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. 32nd IEEE Symp. Foundations of Computer Science (1991) 632--641.

....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 ....

....graph into edge clusters, using the separator theorem of Lipton and Tarjan [40] We keep the time required for updates small by ensuring that each update operation changes at most a constant number of clusters. We remark that clusterization techniques are widely used in fully dynamic algorithms [16, 18, 19]. Our main contribution is that we define and maintain a 2 compressed representation of each cluster. Despite its succinctness, this compressed representation certifies the planarity of a cluster by capturing all the possible ways neighbor clusters interact with each other during changes of the ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. In Proc. 32nd Annual Symp. on Foundations of Computer Science, pages 632-- 641, 1991.

....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. Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees. In Proc. 32nd Annual Symp. on Foundations of Computer Science, pp. 632--641, 1991.

....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. Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees. In Proc. 32nd Annual Symp. on Foundations of Computer Science, pp. 632--641, 1991.

....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. Ambivalent data structures for dynamic 2--edge-- connectivity and k smallest spanning trees. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 632--641, 1991.

....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, Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, in Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 632--641.

....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 ....

....common structures with each other. In the basic version of the algorithm, this collection of heaps forms a bounded degree graph having O(m n log n) vertices. Later we show how to improve the time and space bounds of this part of the algorithm using tree decomposition techniques of Frederickson [25]. 2.1 Preliminaries We assume throughout that our input graph G has n vertices and m edges. We allow self loops and multiple edges so m may be larger than n 2 . The length of an edge e is denoted #(e) By extension we can define the length #(p) for any path in G to be the sum of its edge ....

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G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. Proc. 32nd Symp. Foundations of Computer Science, pp. 632--641. IEEE, 1991.

....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 ....

....degree 3, thus it is necessary to transform the tree to a tree of maximum degree 3 and the solve the problem there. This has given rise to a number of complicated transformations, and previous articles usually start by presenting the entire topology tree data structure from scratch, see e.g. [6, 14, 29]. Our contribution is a variant of the topology trees called top trees. The toptrees are based on Fredericksons idea of partitioning the tree according to topology, but improves over the topology trees in two ways. First, they work directly for trees of unbounded degree. Second, they come with a ....

G.N. Frederickson. Ambivalent data structures for dynamic 2--edge-- connectivity and k smallest spanning trees. In SIAM Journal on computing, volume 26, pages 484--538, 1997. see also FOCS'91.

....queries are supported in O(log n= log log n) time. The update time was further improved to O(log 2 n) in 1996 [27] by Henzinger and Thorup. No randomized technique was known for improving the deterministic O( 3 p n log n) update cost for the minimum spanning tree problem. In 1991 [11], Frederickson succeeded in generalizing his O( p m) bound from 1983 [10] for fully dynamic connectivity to fully dynamic 2 edge connectivity. As for connectivity, the sparsification technique of Eppstein et.al. 7] improved this bound to O( p n) Further, Henzinger and King generalized their ....

....This is the first polylogarithmic bound for the problem, even when we include randomized algorithms. Finally, our connectivity techniques are generalized to 2 edge and biconnectivity, leading to O(log 4 n) algorithms for both of these problems. The generalization uses some of the ideas from [11, 21, 23] of organizing information around a spanning forest. However, finding a generalization that worked was rather delicate, particularly for biconnectivity, where we needed to make a careful recycling of information, leading to the first polylogarithmic algorithm for this problem. The reader is ....

[Article contains additional citation context not shown here]

G. N. Frederickson. Ambivalent data structures for dynamic 2-EdgeConnectivity and k smallest spanning trees. SIAM Journal on Computing, 26(2):484--538, April 1997. See also FOCS'91.

.... Gamma F , CoverSet F (e) feg, and for a cutedge e 2 F , CoverSet F (e) We sometimes use cover F (e) to denote jCoverSet F (e)j. Throughout this paper, unless otherwise mentioned, covering will be with respect to F only and the subscript F will be dropped when there is no ambiguity. Fact 1 [5] Two vertices u and v are 2 edge connected iff cover F (e) 1 for every edge e 2 F uv . We store F in a dynamic tree data structure ( 19] with edge costs representing the cover values. The basic idea behind our algorithm is that the cover values of the tree edges (which are sufficient to answer ....

G.N. Frederickson. Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees. In Proceedings of 32nd Symp. on Foundations of Computer Science, pages 632-- 641, 1991.

....the following bounds: 1 1. worst case update time O(log log n) for both mark and unmark, 2. worst case query time O(log n log log n) 3. linear space and preprocessing time. To achieve these results we present a new micro macro division of trees. In contrast to standard tree divisions [15, 20], our approach does not limit the number of nodes or the number of boundary nodes in a micro tree. This leads to exponentially better update times than Amir et al. 5] who achieved O(log n log log n) per operation. 1.1. Variants and extensions Existential queries. In the existential marked ....

G. Frederickson. Ambivalent data structures for dynamic 2edge -connectivity and k smallest spanning trees. SIAM J. Comput., 26(2):484--538, 1997.

....called topological tree of G, that each level is a decomposition of the level above it. Using this technique, Frederickson maintains a MST of general graph with O( p m) time per update and O(log n) time per query. The variant for planar graphs achieves O(log 2 n) time per query. Frederickson [3, 4], using the technique for handling MST dynamically, showed an algorithm that achieve, for general graphs, O( p m) time per update and O(log n) time per query. For this, he developed the ambivalent data structure. ambivalent data structure maintains, for each tree edge e, some candidate edges ....

....used T 1 [ T 2 , where T 1 is a breadth first forest of the given graph, G, and T 2 is a breadth first forest of G n E(T 1 ) Rauch [13] improved her results from [12] and achieved O( p m log n) amortized time per update and O(1) time per query. It is done by using ambivalent data structure [3, 4] and sparsification to enable recomputing the high level spanning tree in just O( z m=z) log n) amortized time, which yields the desired update time. The algorithm s variant for planar graph works in O(log 2 n) time per operation. Rauch [13] also showed a lower bound for k edge and k vertex ....

G.N. Frederickson. Ambivalent Data Structure for Dynamic 2-edge Connectivity and k-Smallest Spanning Trees. Technical Report CSD-TR-91-048, Purdue University, 1991.

....called topological tree of G, that each level is a decomposition of the level above it. Using this technique, Frederickson maintains a MST of general graph with O( p m) time per update and O(log n) time per query. The variant for planar graphs achieves O(log 2 n) time per query. Frederickson [3, 4], using the technique for handling MST dynamically, showed an algorithm that achieve, for general graphs, O( p m) time per update and O(log n) time per query. For this, he developed the ambivalent data structure. ambivalent data structure maintains, for each tree edge e, some candidate edges ....

....used T 1 [ T 2 , where T 1 is a breadth first forest of the given graph, G, and T 2 is a breadth first forest of G n E(T 1 ) Rauch [13] improved her results from [12] and achieved O( p m log n) amortized time per update and O(1) time per query. It is done by using ambivalent data structure [3, 4] and sparsification to enable recomputing the high level spanning tree in just O( z m=z) log n) amortized time, which yields the desired update time. The algorithm s variant for planar graph works in O(log 2 n) time per operation. Rauch [13] also showed a lower bound for k edge and k vertex ....

G.N. Frederickson. Ambivalent Data Structure for Dynamic 2-edge Connectivity and k-Smallest Spanning Trees. FOCS 32:632--641, 1991.

....while the input is being modified. Obviously, to make things interesting, it is required that the dynamic algorithm update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [4, 5, 1, 2]) and graph properties (e.g. planarity [7] connectivity [6, 5] and can be classified according to the type of operations that are allowed on graph. In this paper we will consider weight change operations. Thus, our working scenario will be the following. A weighted graph is presented to the ....

....it is required that the dynamic algorithm update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [4, 5, 1, 2] and graph properties (e.g. planarity [7] connectivity [6, 5]) and can be classified according to the type of operations that are allowed on graph. In this paper we will consider weight change operations. Thus, our working scenario will be the following. A weighted graph is presented to the algorithm. The algorithm performs some preprocessing during which ....

G. N. Frederickson, Ambivalent Data Structure for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, in Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 632--641.

....c ) 4 Improved Sparsification In the original sparsification technique, the edge partition was arbitrary, and the graphs induced at each node of the sparsification tree could have many vertices. We use a di#erent partition technique, similar to the 2dimensional topology tree of Frederickson [7, 8] to partition the edges in a way that induces subgraphs with few vertices. We start with a partition of the vertices of the graph, as follows: we split the vertices evenly in two halves, and recursively partition each half. Thus we end up with a complete binary tree in which nodes at distance i ....

....# The best swap in a graph is the pair of edges such that if they are respectively added to and removed from the minimum spanning forest, the result is again a spanning forest with minimum possible weight. Best swaps have been used to enumerate the spanning forests of a graph in order by weight [8]. For this application one needs a fully persistent data structure (one in which any update does not modify previous versions of the data structure, but instead creates a new separately existing version, and in which each update or query can be performed in any of the versions that have been so ....

[Article contains additional citation context not shown here]

G. N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. In Proc. 32nd Symp. Foundations of Computer Science, pages 632--641, 1991.

.... k k min(k,n) 1 2 log(k n) In three or four dimensions our time bound is O(n 4 3 # k min(k,n) 1 2 log(k n) and in higher dimensions the bound is O(n 2 2 (#d 2# 1) # kn 1 2 log n) 1 Introduction The k smallest spanning tree problem for graphs has been studied extensively [5, 6, 7, 8, 9, 10, 11, 12], but it was only recently that the author introduced the corresponding geometric problem [7] In this problem, one is given a point set as input, and one must construct k di#erent spanning trees that have the minimum total edge lengths among all possible spanning trees of the set. The trees need ....

....Then we solved the graph problem on the graph formed by adding O(k 2 ) possible edges to the n 1 MST edges. The time for this procedure is O(k 2 kn log(n k) 1 This is always an improvement on the O(kn 1 # ) Voronoi diagram based algorithm. Since our paper appeared, Frederickson [9] and Eppstein et al. 8] improved the algorithm for general graphs, to the bounds stated above. However, their improvements only help the non geometric algorithm for our 1 We adopt the convention that log x = log 2 (2 x) so log x is always #(1) even if x 1. 1 problem (valid for any metric ....

G.N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. 32nd IEEE Symp. Foundations of Computer Science (1991) 632--641.

....last result. Let # be the number of line segments formed by the intersection of two primitives; then we solve the 3d CSG problem in time O(n 2 # log n # log n log log n) 2 Tree partitions Our algorithms use a technique of partitioning trees into smaller subtrees, introduced by Frederickson [7]. We will apply this technique to trees representing CSG formulae. Definition 1 (Frederickson [7] A restricted partition of order z with respect to a binary tree T is a partition of the vertices of V such that: 1. Each set in the partition contains at most z vertices. 2. Each set in the ....

....then we solve the 3d CSG problem in time O(n 2 # log n # log n log log n) 2 Tree partitions Our algorithms use a technique of partitioning trees into smaller subtrees, introduced by Frederickson [7] We will apply this technique to trees representing CSG formulae. Definition 1 (Frederickson [7]) A restricted partition of order z with respect to a binary tree T is a partition of the vertices of V such that: 1. Each set in the partition contains at most z vertices. 2. Each set in the partition induces a connected subtree of T . 3. For each set S in the partition, if S contains more ....

[Article contains additional citation context not shown here]

G.N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. 32nd IEEE Symp. Foundations of Computer Science (1991) 632--641.

....log P S factor by allowing a constant fraction of the simulated processors to fail to make progress at each step [6] 3.4. Restricted Partitions Our algorithms use a technique of partitioning trees and forests into smaller subtrees, or clusters of vertices, that was introduced by Frederickson [15,16] and used by him and others as part of various dynamic graph algorithms. We will combine this clustering technique with some geometric data structures (primarily, planar convex hulls) in a manner similar to techniques used in our previous paper on speedups in the network simplex method [10] We ....

....as G undergoes edge insertions or deletions: each update in G causes a constant number of updates to G # . Thus, for the remainder of the description of our kinetic algorithm, we assume our input graph has all vertex degrees at most three. The following definition is due to Frederickson [16]. Definition 1. A restricted partition of order z with respect to a tree T in which all vertex degrees are at most three is a partition of the vertices of V such that: 1. Each set in the partition contains at most z vertices. 2. Each set in the partition induces a connected subtree of T . 3. ....

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G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Computing, 26 (1997), 484--538.

....describes a simple O(kn 2 ) time dynamic programming solution to this problem. For the k shortest simple paths problem, the best known time bound is O(k(m n log n) 21] A similar problem is that of finding the k minimum weight spanning trees in a graph. Recent algorithms for this problem [11, 13] 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 [8] 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 [11, 13]. Further, previous papers on the k shortest path problem give time bounds omitting the O(k 2 ) term in the lower bound 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 ....

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G. N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. In Proc. 32nd IEEE Symp. Foundations of Computer Science, pages 632--641, 1991.

....from vertices in L(N 1 ) to vertices in L(N 2 ) for each pair N1;N2 of related nodes. The space for this data structure is O(n ) It can either be built as a subset of the data structure of Theorem 11, in time O(n log n) or bottom up (using hierarchical clustering techniques of Frederickson [22] to construct the level structure in T , and then computing each distance matrix from two previously computed distance matrices in time O( 3 ) in total time O(n 2 ) we omit the details. To answer a query, we form chains of related pairs connecting the home nodes (or small subtrees) of the ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees. SIAM J. Comput. 26(2):484{ 538, April 1997.

....and Shiloach s algorithm on the macro tree and in section 4 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 original 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. An linear time algorithm exist 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. To use this lemma we make the following simple ....

....be an edge incident with boundary nodes from two different micro trees, and is therefore deleted in the macro tree. Maintaining the universe we can, according to lemma 2, answer connectivity queries in T using a bounded number of connectivity queries in the macro tree and micro trees. 2 1 In [3] a micro tree is called a cluster. The external degree of a cluster is defined as the number of tree edges incident to exactly one node in the cluster. A cluster has external degree at most 3, and if the external degree of a cluster is 3 the cluster consist of a single node, implying that the ....

[Article contains additional citation context not shown here]

Greg N. Frederickson. Ambivalent data structures for dynamic 2-edge- connectivity and k smallest spanning trees. SIAM Journal on Computing, 26(2):484--538, April 1997.

....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 ....

....common structures with each other. In the basic version of the algorithm, this collection of heaps forms a bounded degree graph having O(m n log n) vertices. Later we show how to improve the time and space bounds of this part of the algorithm using tree decomposition techniques of Frederickson [25]. 2.1. Preliminaries. We assume throughout that our input graph G has n vertices and m edges. We allow self loops and multiple edges, so m may be larger than # n 2 # . The length of an edge e is denoted #(e) By extension we can define the length #(p) for any path in G to be the sum of its ....

[Article contains additional citation context not shown here]

<F3.774e+05> G. N.<F3.828e+05> Frederickson,<F4.018e+05> Ambivalent data structures for dynamic 2-edge-connectivity and<F3.572e+05> k<F4.018e+05> smallest spanning<F3.828e+05> trees, in Proc. 32nd Symp. Foundations of Computer Science, IEEE, 1991, pp. 632--641.

.... what is the value of output k Requests of both types are mixed in the request stream, and we would like a fast guaranteed worst case response time. The machine servicing the requests may be either a sequential or parallel machine. While incremental versions of many graph problems (for example [9, 11, 20]) and geometry problems (for example [1, 6, 21] have been studied, very little has been done in incremental versions of algebraic problems. Two notable exceptions are the incremental maintenance of prefix sums studied by Fredman [10] and the incremental maintenance and evaluation of size n ....

G. N. Frederickson, Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, in "Proceedings, 32rd Annual Symposium on Foundations of Computer Science, San Juan, 1991," 632--641.

....URL: http: www.info.uniroma2.it italiano . Part of this work was done while visiting the Max Planck Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. 1 Introduction In the last years research in dynamic graph algorithms has been a blossoming field (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24]) The main dynamic model that has been considered in the literature is the following. We are given a graph G = V; E) and we wish to maintain some property P in G during edge deletions and edge insertions. We refer to this as the dynamic edge model . If the graph represents a communication ....

G.N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM Journal on Computing, 26 (1997), 484--538.

....implies that at any level a cluster of tree degree 3 consists always of a single vertex. Once again, several multi level partitions are possible. But each restricted multi level partition has the nice property of having only logarithmic depth, as implied by the following lemma of Frederickson [15]. Lemma 1 (Frederickson [15] For any 0, the number of clusters at level 1 is at most 5=6 times the number of clusters at level . The topology tree is a hierarchical representation of T . Each level of the topology tree partitions the vertices of T into connected subsets called clusters. ....

....a cluster of tree degree 3 consists always of a single vertex. Once again, several multi level partitions are possible. But each restricted multi level partition has the nice property of having only logarithmic depth, as implied by the following lemma of Frederickson [15] Lemma 1 (Frederickson [15]) For any 0, the number of clusters at level 1 is at most 5=6 times the number of clusters at level . The topology tree is a hierarchical representation of T . Each level of the topology tree partitions the vertices of T into connected subsets called clusters. More precisely, given a ....

[Article contains additional citation context not shown here]

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. In Proc. 32nd Symp. Foundations of Computer Science, pages 632--641, 1991.

....while the input is being modified. Obviously, to make things interesting, it is required that the dynamic algorithm update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [4, 5, 1, 2]) and graph properties (e.g. planarity [7] connectivity [6, 5] and can be classified according to the type of operations that are allowed on graph. In this paper we will consider weight change operations. Thus, our working scenario will be the following. A weighted graph is presented to the ....

....it is required that the dynamic algorithm update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [4, 5, 1, 2] and graph properties (e.g. planarity [7] connectivity [6, 5]) and can be classified according to the type of operations that are allowed on graph. In this paper we will consider weight change operations. Thus, our working scenario will be the following. A weighted graph is presented to the algorithm. The algorithm performs some preprocessing during which ....

G. N. Frederickson, Ambivalent Data Structure for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, in Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 632--641.

....some constraints may be required. One possible approach for this problem is to generate all the spanning trees in nondecreasing order of weight until one satisfying the additional constraints is obtained. There exist some algorithms which generate spanning trees in nondecreasing order of weight [6, 9, 11, 14, 15, 20]. In [14] Kapoor and Ramesh proposed an algorithm for ranking all the spanning trees which requires O(nm log n) time and O(nm ) space. Recently, Eppstein, Galil, Italiano and Nisenzweig [8] developed the sparsification technique and proposed an algorithm for generating k smallest spanning ....

G.N. Frederickson, Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, In 32th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico (1991), pp.632--641.

....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 ....

....deleted from the tree T belongs to a micro tree, we can in constant time determine if this implies that the boundary nodes in the micro tree are no longer connected using microcon on the boundary nodes. If this is the case, we delete the edge in the macro tree which is incident with the 1 In [3] a micro tree is called a cluster. The external degree of a cluster is defined as the number of tree edges incident to exactly one node in the cluster. A cluster has external degree at most 3 and if the external degree of a cluster is 3 the cluster consists of only 1 node, implying that the number ....

G.N. Frederickson. Ambivalent data structures for dynamic 2--edge--connectivity and k smallest spanning tree. SIAM J. Computing, 26(2):484--538, 1997. See also FOCS'91.

....and biconnectivity problem in O(log 4 n) amortized time per operation. It should be noted that biconnectivity has a history of being much harder than 2 edge 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 ....

Greg N. Frederickson. Ambivalent data structures for dynamic 2-EdgeConnectivity and k smallest spanning trees. SIAM Journal on Computing, 26(2):484--538, April 1997. See also FOCS'91.

....dynamic minimum spanning tree (MST) problem consists of mantaining a minimum spanning tree of a connected graph subject to edge insertions and edge deletions. In this library, we included three different data structures for this problem: Sparsification [5] Frederickson s clustering algorithms [8, 9] and a simple dynamic algorithm which we called adhoc. We have already performed an extensive empirical study on the performance of these data structures, and we refer the interested reader to reference [2] for the details. All our algorithms support adding and deleting edges, MST membership ....

....the running time of Min Spanning Tree in case of a tree edge deletion. One would expect that adhoc has very low implementation constants. This was exactly confirmed by our experiments. Let G = V; E) be a graph, with minimumspanning tree T . The first ingredient of the algorithms of Frederickson [8, 9] is clustering. Frederickson gives three different algorithms for maintaining the minimum spanning tree of a graph. Algorithm FredI is based on clustering only and obtains time bounds of O(m 2=3 ) per update. If the partition is applied recursively and a topology tree is associated with each ....

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G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Comput., 26:484--538, 1997. A Software Library of Dynamic Graph Algorithms 136

....insertion and deletion of nodes are not taken into consideration. It makes sense only to consider algorithms that update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [3, 4, 1, 2]) and graph properties (e.g. planarity [6] connectivity [5, 4] The mechanics of a dynamic algorithm can be described as follows. Aweighted graph is presented to the algorithm. The algorithm performs some preprocessing during which it builds some data structure; for our algorithms this ....

....It makes sense only to consider algorithms that update the solution faster than it would take to recompute it from scratch. Dynamic algorithms are known for several optimization problems on graphs (e.g. minimum spanning tree [3, 4, 1, 2] and graph properties (e.g. planarity [6] connectivity [5, 4]) The mechanics of a dynamic algorithm can be described as follows. Aweighted graph is presented to the algorithm. The algorithm performs some preprocessing during which it builds some data structure; for our algorithms this preprocessing takes linear time and the size of the data structure is ....

G. N. Frederickson, Ambivalent Data Structure for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, in Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 632--641.

....clustering operation, and the last one specifies the maximality property of clustering at each level. Based upon these clustering rules, it is fairly straightforward to show that the number of levels in a topology tree is O(log N ) where N is the number of nodes in the base tree T . Frederickson [9] proves the following (stronger) lemma which relates the number of clusters at one level with the previous level: Lemma 1. 9] For any level l 0 in a topology tree, the number of clusters at level l is at most 5=6 of the number of clusters at level l Gamma 1. 2.1 Implementation of Primitives ....

....rules, it is fairly straightforward to show that the number of levels in a topology tree is O(log N ) where N is the number of nodes in the base tree T . Frederickson [9] proves the following (stronger) lemma which relates the number of clusters at one level with the previous level: Lemma 1. [9] For any level l 0 in a topology tree, the number of clusters at level l is at most 5=6 of the number of clusters at level l Gamma 1. 2.1 Implementation of Primitives on Topology Tree In this section, we discuss how to implement the dynamic operations on the topology tree. Our methods are very ....

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 632--641, 1991.

....will be able to incrementally maintain the graph property at a cost (per update operation) that is significantly smaller than the cost of computing the graph property from scratch. While much of the prior research in the design of dynamic graph algorithms involved problem specific approaches [11, 12, 13, 22, 27], recently Eppstein, Galil, Italiano, and Nissenzweig [9] presented a general paradigm called sparsification that is useful in both designing new dynamic graph algorithms, as well as speeding up the existing ones. The sparsification technique is based on the notion of a sparse strong certificate ....

G.N. Frederickson. Ambivalent Data Structures for Dynamic 2-edge Connectivity and k Smallest Spanning Trees. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (1991), pp. 632--641.

....the diameter of the tree the node belongs to. The time complexity is O(log n) 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 ....

[Article contains additional citation context not shown here]

G. N. Frederickson. Ambivalent data structures for dynamic 2--edge--connectivity and k smallest spanning trees. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 632--641, 1991.

No context found.

G. N. Frederickson, Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, in "Proceedings, 32rd Annual Symposium on Foundations of Computer Science, San Juan, 1991," 632--641.

No context found.

G. N. Frederickson. Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees. SIAM J. Comput. 26(2):484--538, 1997. Note that the users always overpay under N . By scaling with (log n) -1/2 , we could N even make O((log n) 1/2 )-BB.

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