G. N. Frederickson, "Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees," SIAM J. Comput., Vol. 26, No. 2, pp. 484-538, April 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in its forwarding database the next node in the tree. Because the nodes are moving, there are discrete times at which the present spanning tree is no longer optimal, and a new shortest path spanning tree should be used. This tree is typically updated using a distributed algorithm cf. 2] 3] [4], and it is important that the nodes be able to determine when to change their forwarding databases. To do so, messages must be exchanged among neighboring nodes. We propose a distributed algorithm that adapts techniques from the theory of kinetic spanning trees [5] 6] to maintain the correct ....

G. N. Frederickson, "Ambivalent data structures for dynamic 2-edgeconnectivity and k smallest spanning trees," SIAM J. Comput., Vol. 26, No. 2, pp. 484-538, April 1997.

....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 static ....

G.F. Frederickson, "Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees," IEEE Found. of Comp. Sci. Symp. (1991), 632641.

....fully dynamic 2 edge connectivity problem is to maintain a data structure under an arbitrary sequence of the following update operations: insert(u,v) Add the edge u, v to G. delete(u,v) Remove the edge u, v from G. query(u,v) Return true iff u and v are 2 edge connected in G. In 1991 [5], Fredrickson introduced a data structure known as topology trees for the fully dynamic 2edge connectivity problem with a worst case cost of O( # m) per update, where m is the number of edges in the graph at the time of the update. His data structure permitted 2 edge connectivity queries to be ....

....edges and the tree path between u and v is denoted by #(u, v) A nontree edge u, v covers a tree edge e iff e lies on the tree path between u and v. A bridge is an edge of F that is not covered by nontree edge of G. Two nodes u and v are 2 edge connected iff all edges on #(u, v) are covered [5]. Throughout the algorithm, the nontree edges of G are partitioned into levels E 1 , E l , l = #2 log n#. We let F i denote a forest of 2 edge connected components of G i = V , # j #i E j # F) where F i # F i 1 . The level of a tree edge e is defined to be the smallest i ....

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees" Proc. 32nd Annual IEEE Symposium on Foundation of Comput. Sci., 1991, 632--641.

....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 query operation. It was consequently improved to O( # m) per update and O(log n) per query operation [6]. The sparsification technique of Eppstein et al. 3, 2] improves the running time of an update operation to O( # n) Very recently, a deterministic dynamic connectivity algorithm was given with O(n 1 3 log n) update time and O(1) query time [13] and a randomized dynamic connectivity algorithm ....

....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 topology tree data structure [5] in a novel way and extend the ambivalent data structure [6]. These data structures were used before to test connectivity and 2 edge connectivity. To test biconnectivity within a cluster we need to know how the vertices outside the cluster are connected with each other. Thus, we build two graphs, called internal and shared graph. Each graph contains all ....

[Article contains additional citation context not shown here]

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees", Proc. 32nd Annual IEEE Symposium on Foundation of Comput. Sci., 1991, 632--641.

....techniques appears instead to be lacking in the area of dynamic graph algorithms. The goal of this research is exactly to provide such generalized techniques in the realm of dynamic tree and graph problems. Our approach is motivated by the observation that a number of dynamic graph algorithms [10,11,13,14,15,16,19,23,31], developed mostly for connectivity problems, appear to be based on the following fundamental idea: Decompose a graph into subgraphs with limited overlap, and represent such a decomposition by means of a tree so that dynamic operations on the graph are reflected into corresponding dynamic tree ....

G.N. Frederickson, "Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees," Proc. 32th IEEE Symp. on Foundations of Computer Science (1991).

....counterparts. Graph connectivity is one of the most basic problems with numerous applications and various algorithms in different settings. The area of dynamic graph algorithms has been a blossoming field of research in the last years, and it has produced a large body of algorithmic techniques [4, 8, 11, 12, 15, 20]. Most of the results [8, 11, 12, 14] on efficient fully dynamic structures for general graphs were based on clustering techniques. This has led to solutions of an inherent time bound of O(n ffl ) for some ffl 1, since the key problem encountered by these techniques is that the algorithm must ....

....of the most basic problems with numerous applications and various algorithms in different settings. The area of dynamic graph algorithms has been a blossoming field of research in the last years, and it has produced a large body of algorithmic techniques [4, 8, 11, 12, 15, 20] Most of the results [8, 11, 12, 14] on efficient fully dynamic structures for general graphs were based on clustering techniques. This has led to solutions of an inherent time bound of O(n ffl ) for some ffl 1, since the key problem encountered by these techniques is that the algorithm must somehow balance the work investing ....

[Article contains additional citation context not shown here]

G. Frederickson, "Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees," Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science (FOCS'91), pp. 632-641, 1991. i

....of its children. Then we can find the k smallest weights of vertices in T , in time O(k) Note that the time bound of Lemma 4 does not depend on T , but only on k. For instance, Frederickson uses this lemma to find the k smallest spanning trees in a graph in time O(m log #(m, n) k 3 2 ) [14], even though in this application T has exponential size. In our application, the breadth first search trees are not binary, but they have bounded degree, which is sufficient for the lemma (one can translate a degree # tree into a binary tree by expanding each vertex into a subtree of # 1 ....

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

....research in recent years. Currently, several results exist for incremental and fully dynamic graph problems, like for maintaining spanning trees, the 2 edge or the 2 vertex connected components of a graph, or the planarity of a graph under the insertions and or deletions of edges and vertices [3, 4, 5, 7, 8, 9, 10, 11, 12, 14]. In [4, 5, 12] algorithms for maintaining minimum spanning trees and the connectivity, 2 edge connectivity and the 2 vertex connectivity relations in fullydynamic graphs were presented that run in O( # m) or O( # m log n) time per operation (amortized time for 2 vertex connectivity) In this ....

.... several results exist for incremental and fully dynamic graph problems, like for maintaining spanning trees, the 2 edge or the 2 vertex connected components of a graph, or the planarity of a graph under the insertions and or deletions of edges and vertices [3, 4, 5, 7, 8, 9, 10, 11, 12, 14] In [4, 5, 12], algorithms for maintaining minimum spanning trees and the connectivity, 2 edge connectivity and the 2 vertex connectivity relations in fullydynamic graphs were presented that run in O( # m) or O( # m log n) time per operation (amortized time for 2 vertex connectivity) In this paper, n is the ....

[Article contains additional citation context not shown here]

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees" Proc. 32nd Annual IEEE Symp. on Foundation of Comput. Sci., 1991, 632--641.

....algorithm for fully dynamic 2 edge connectivity. For general graphs, their algorithm takes O(m 2=3 ) time per operation (update or query) where m is the current number of edges in the graph. For planar graphs, their time complexity improves to O( p n log log n) Soon afterwards, Frederickson [5] improved the time bound in [7] to O( p m) per operation. Frederickson also presented a faster algorithm for planar embedded graphs, with query time O(log n) and update time O(log 3 n) Rauch [15] gave a fully dynamic algorithm for maintaining 2 vertex connectivity. An update operation takes ....

....Tree edges are solid and non tree edges are dashed. Covered tree edges are drawn heavy. Thin solid edges are bridges. 3 Topology trees We build a hierarchical representation of G based on the spanning tree T . The representation is a tree, called the topology tree, that has depth O(log n) [5, 6]. Each level of the topology tree partitions the vertices of G into connected subsets called clusters. Two clusters are said to be adjacent if they are joined by an edge in the spanning tree T . The external degree of a cluster is the number of tree edges with exactly one endpoint inside the ....

[Article contains additional citation context not shown here]

G. N. Frederickson. "Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees." Proc. of 32nd FOCS, 1991.

....the tree. However, by exactly maintaining the number of descendants in a balanced tree, we can quickly (in O(log n) sequential time) compute the current pre order number of any node in the tree. Previous work on maintaining dynamic trees has been done by Sleator and Tarjan [16] and by Frederickson [5, 6]. In particular, Frederickson notes that his algorithm for dynamic tree maintenance clusters tree nodes in a manner very similar to that of tree contraction, and performs sequential updates in O(log n) time. The problem of maintaining dynamic expression trees was studied by Cohen and Tamassia [3] ....

G. N. Frederickson. "Ambivalent Data Structures for Dynamic 2-Edge-connectivity and k Smallest Spanning Trees", FOCS, pp. 632--641, 1991.

....fully dynamic 2 edge connectivity problem is to maintain a data structure under an arbitrary sequence of the following update operations: insert(u,v) Add the edge u, v to G. delete(u,v) Remove the edge u, v from G. query(u,v) Return true iff u and v are 2 edge connected in G. In 1991 [5], Fredrickson introduced a data structure known as topology trees for the fully dynamic 2edge connectivity problem with a worst case cost of O( # m) per update, where m is the number of edges in the graph at the time of the update. His data structure permitted 2 edge connectivity queries to be ....

....edges and the tree path between u and v is denoted by #(u, v) A nontree edge u, v covers a tree edge e iff e lies on the tree path between u and v. A bridge is an edge of F that is not covered by nontree edge of G. Two nodes u and v are 2 edge connected iff all edges on #(u, v) are covered [5]. Throughout the algorithm, the nontree edges of G are partitioned into levels E 1 , E l , l = #2 log n#. Let F i denote a forest of subtrees of F which are 2 edge connected in G i = V , # j #i E j ) # F) Then F i # F i 1 and an edge is a bridge in G iff it is in F F l . ....

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees" Proc. 32nd Annual IEEE Symposium on Foundation of Comput. Sci., 1991, 632--641.

....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] E 92b] 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 [RR96, LT94] This paper is organized as follows. In Section 2, we begin with some background on Descriptive Complexity. In Section 3, for any static ....

G.F. Frederickson, "Ambivalent data structures for dynamic 2-edge connectivity and k smallest spanning trees," IEEE Found. of Comp. Sci. Symp. (1991), 632641.

....and deletions of edges. It is harder than the versions that allow only insertions or only deletions. Dynamic graph algorithms have applications in communication networks, computer aided design [2] database systems [20] logic programming [3] and incremental data flow analysis [4] Frederickson [9] gave an algorithm for maintaining 2 edge connected components in a graph which takes O( p m) time per update, where m is the number of edges in the graph. This bound was consequently reduced to O( p n) by Eppstein et al. 7] and to O(log 6 n) by Henzinger and King [12] The running time ....

....G 0 can be generated in time O(m n) A minimum spanning tree T of G 0 can be determined in time O(m n) by computing a spanning forest of all edges of cost 1 and then connecting the spanning forest into a spanning tree with cost 2 edges. Next we find a partition of G 0 along the lines of [9]. A cluster with respect to T is a set of vertices such that the subgraph of T induced on the cluster is connected. The external degree of a cluster is the number of tree edges with precisely one endpoint in the cluster. A restricted partition of order k with respect to T is a partition of the set ....

[Article contains additional citation context not shown here]

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees" Proc. 32nd Annual IEEE Sympos. on Foundation of Comput. Sci., 1991, 632--641.

....E mail: italiano unive.it. URL: http: www.dsi.unive.it italiano. tions (i.e. either insertions or deletions, but not both) is allowed. The area of dynamic graph algorithms has been a blossoming field of research in the last years, and it has produced a large body of algorithmic techniques [2, 5, 6, 7, 8, 9, 10, 11, 14]. Perhaps one of the most studied problems in this area is the fully dynamic maintenance of the minimum spanning tree of a graph [2, 5, 7, 8] This problem is important on its own, and it finds applications to other problems as well, including many dynamic vertex and edge connectivity problems. ....

....problem is important on its own, and it finds applications to other problems as well, including many dynamic vertex and edge connectivity problems. Many elegant solutions have been proposed for the dynamic minimum spanning tree problem, such as the partitions and topology trees of Frederickson [8, 9], and sparsification [5] In [1] we tried to make a first step toward bridging the gap between the design and theoretical analysis of dynamic graph algorithms on the one side, and their implementation, experimental tuning and practical performance evaluation on the other side. In particular, in ....

[Article contains additional citation context not shown here]

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

....trees, either biased search trees [3] or splay trees [21] With the former representation the time per tree operation can be made O(log n) in the worst case; with the latter representation the time per tree operation is O(log n) amortized over a worst case sequence of operations. Frederickson [7, 8] proposed a rather different representation, called the topology tree, which is related to the rake and compress operations used in parallel tree processing [17] This representation, too, has an O(log n) worst case time bound per tree operation. Both the Sleator Tarjan representation and the ....

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

....requires linear space. If a query(u; v) returns true, the data structure returns at no extra cost the exact points in the adjacency lists of u and v where the new edge can be inserted without violating the current planar embedding. Our data structure is a novel application of the topology trees [5, 6, 9], augmented with some additional information to maintain efficiently dynamic information about the embedding of an arbitrary plane graph. For the ease of description, we only present here our data structure for connected graphs. If the graph is not connected, we maintain a separate data structure ....

....maximum vertex degree 3 obtained from the above transformation. Throughout the sequence of updates, we maintain a spanning tree T of b G containing all of the dashed edges of b G. We use different data structures for tree edges and non tree edges. To store tree edges, we use topology trees [5, 6, 9], which represent a tree in a hierarchical way by partitioning the set of vertices into subsets, called clusters. Similar to [9] we group non tree neighboring edges that are incident to the same clusters into edge bundles. Additionally, we keep bits, called coverage bits, at each face that tell ....

[Article contains additional citation context not shown here]

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2--Edge--Connectivity and k Smallest Spanning Trees" Proc. 32nd Annual Symp. on Foundations of Computer Science, 1991, 632--641.

....Award. y Department of Computer Science, University of Victoria, Victoria, BC. Email: val csr.uvic.ca. This research was supported by an NSERC Grant. 1 Throughout the paper the logarithms are base 2. Previous Work. In recent years a lot of work has been done in fully dynamic algorithms (see [1, 3, 4, 6, 7, 8, 10, 11, 12, 15, 16, 18] for connectivity related work in undirected graphs) There is also a large body of work for restricted classes of graphs and for insertions only algorithms. Currently the best time bounds for fully dynamic algorithms in undirected n node graphs are: O( p n) per update for a minimum spanning ....

G. N. Frederickson, "Ambivalent Data Structures for Dynamic 2-Edge-connectivity and k Smallest Spanning Trees", Proc. 32nd Symp. on Foundations of Computer Science, 1991, 632--641.

....answering a query. For the dynamic connectivity problem, for example, a query takes two nodes u and v as its arguments and returns True , if there is a path connecting u and v in the current graph. The field of dynamic graph algorithms has been a blossoming field of research in the last years [4, 9, 11, 13, 14, 15, 17, 19, 31, 33], motivated by theoretical and practical questions (see for instance [29] However, despite this blend of theoretical and practical interests, we are aware of no implementations and experimental studies in this field. In this paper, we aim at bridging this gap by studying the practical properties ....

.... and implemented a version of simple sparsification that operates on dynamic algorithms (the version described in this paper only operates on static algorithms) 5] We used this version of simple sparsification on top of existing dynamic algorithms, such as Frederickson s dynamic data structures [14, 15] for minimum spanning trees. The algorithms by Frederickson are sophisticated algorithms, for which no implementation was previously available, and that required a great deal of implementation effort. We refer the interested reader to reference [5] for details on how the asymptotically efficient ....

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

....Least Common Ancestor return the least common ancestor of two tree nodes. ffl Find Minimum Find the minimum weight node on a path. Dynamic trees were introduced as internal data structures in sequential maximum flow algorithms [24,55,100] Since then, a large number of dynamic algorithms [27,28, 48,49,50,51,69,83,122] have used dynamic trees as part of their data structures. Initial data structures [24,55] based on balanced binary trees (e.g. AVL trees [1] or Red Black trees [57] take O(log 2 n) time per operation. Sleator and Tarjan improve this to O(log n) time per operation by basing their data ....

....To perform q dynamic operations takes O(qff(q; n) time. A fully dynamic data structure for 2 edge connectivity is presented in [50,53] In general, dynamic operations take each O(m 2=3 ) time. For planar graphs this time bound is reduced to O( p n log log n) Recently, Frederickson [49] presented a fully dynamic data structure such that queries take O(log n) time, and edge insertions and deletions are performed in O( p m) time. If we restrict the class of graphs to planar graphs, then edge insertions and deletions can be performed in O(log 3 n) time. The only dynamic result ....

[Article contains additional citation context not shown here]

G.N. Frederickson, "Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees," Proc. 32th IEEE Symp. on Foundations of Computer Science (1991).

....play any role in the data structures of Chapter 2. They are only required to find small cycle separators using Miller s algorithm. We therefore have the following lemma. Lemma 2 Given an n node planar graph G, a face restricted k cluster partition can be obtained in O(n log n) time. Frederickson [28,30] used the idea of partitioning a graph into clusters to develop dynamic data structures for maintaining minimum spanning trees, connected components, and two edge connected components. Galil and Italiano [36] used a k cluster partition to develop a fully dynamic data structure for two and ....

....both additions and deletions of edges while it is said to be semi dynamic if it supports only one of them. Unfortunately designing fully dynamic algorithms seems to be considerably harder than designing their sequential counterparts, and very few graph problems have fully dynamic solutions. See [28,30,36,37] for fully dynamic data structures to various graph problems. See [81] for a complexity theoretic approach to dynamic computation. As we discussed in Chapter 1 the shortest path problem is not very well understood in the dynamic realm. Though there are many algorithms for the dynamic problem [7, ....

G.N. Frederickson, "Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees," Proc. of the 32nd Symposium on Foundations of Computer Science (1991), 632--641.

....edge; 2b) test the cycleequivalence between a tree edge in the cluster and a tree edge outside of the cluster; and (2c) test the cycle equivalence between two tree edges in the cluster. Two non tree edge cannot be cycle equivalent. For testing (2a) we combine the ambivalent data structure of [7] with the recipe technique of [10] For testing (2b) we introduce the following new technique, called fast non tree updates: We give each edge outside the cluster cost 1 and each other edge cost 0 and maintain a minimumspanning tree of this graph using a data structure that implements insertions ....

.... edges incident to C whose endpoint in C does not lie in sub(e 0 ) Since the degree of proj(e 0 ) is 3, the projection of an edge in other(e 0 ) is not proj(e 0 ) Data structure For each pair of clusters C and C 0 and each tree edge e incident to C we maintain ambivalent information [7] in the form of 3 non tree edges ambiv i (C; C 0 ; e) for i = 1; 2; 3 (if they exist) Assuming that e lies on (C; C 0 ) let ambiv 1 (C; C 0 ; e) ambiv 2 (C; C 0 ; e) be the edge e 0 between C and C 0 that (1) covers the maximum number of edges on TP (C) and that (2) is the first ....

[Article contains additional citation context not shown here]

G. N. Frederickson, "Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees" Proc. 32nd Annual IEEE Sympos. on Foundation of Comput. Sci., 1991, 632--641.

....from a strong orientation algorithm into an alternative biconnectivity algorithm. Its use as a general technique for parallel graph algorithms came at a later stage. The fact that EDS yields an alternative biconnectivity algorithm is noted in [MR86] as well. 4) Application of Graph Decompositions [GI91, Fr91] to dynamic 2 edge and 3 edge connectivity. Improving approximation factors: Considerable attention has been given to improving constant approximation factors. For example, Johnson [Jo82] reports a series of 8 papers that give such improvements for bin packing, starting from an approximation ....

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

No context found.

G. N. Frederickson, "Ambivalent Data Struc- tures for Dynamic 2-edge-connectivity and k smallest spanning trees", Proc. 32rid Syrup. on Foundations of Computer Science, 1991, 632 641.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC