Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20(3):242--276, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... to the uniform case comes from the fact that our packings are rooted at particular vertex v (which will be chosen to be the vertex with the largest requirement value rv ) Our algorithm maintains the cactus tree under edge insertions (a problem previously studied by Dinitz and Westbrook [6]) 2. OVERVIEW OF THE ALGORITHMS In this section, we describe the broad frameworks for Gabow s algorithm for directionless spanning tree packing, for our directionless tree packing and cactus construction algorithms, and for the WGMV Survivable Network construction algorithm. 2.1 Gabow s ....

....minimal cut by drawing in more black supervertices. In fact, this growth, which is computed by a closure procedure, may require working into black supervertices recursively. The changes to the cactus tree resulting from edge addition are standard apart from the e#ect of black supervertices (see [6] for the e#ect on a standard cactus tree) The overall time taken for iteration i turns out to be O( i n im) log n) giving a total time of O( maxv rv maxv rv m) log n maxv rv m log n) which improves to n log n maxv rv n) using the construction of Nagamochi and Ibaraki ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20, pp. 242--276, 1998.

....to a small component of the graph that is separated from the rest of the graph by an edge cut of size less than k. Similarly to the k = 1 case, it can be shown that if a graph is ffl far from being k connected then it has many such components. This can be shown by defining an auxiliary graph [DW98] whose nodes are components of the graph and that is based on the cactus structure of [DKL76] In addition, there are efficient procedures for recognizing such a component given a vertex that resides in it. In what follows we sketch these procedures, for the different values of k. For simplicity ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20(3):242--276, 1998.

....Sec. 3.2.6) remove this assumption. In Appendix A we describe in more detail the structure of (k Gamma 1) connected graphs in terms of their k classes. Here we only state the facts necessary for our algorithms. Let G be a (k Gamma 1) connected graph. Then we can define an auxiliary graph TG [DW95] based on the cactus structure of [DKL76] which is a tree, such that for every k class in G there is a corresponding (unique) node in TG . The tree TG might include additional auxiliary nodes, but they are not leaves and we shall not be interested in them here. If G is k connected, then TG ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Technical Report #871, Technion, Department of Computer Science, 1995.

....for k connectivity are (k Gamma 1) connected. We later (in Sec. 3.2.5) remove this assumption. We next state some facts, necessary for our algorithms, concerning the structure of (k Gamma 1) connected graphs. Let G be a (k Gamma 1) connected graph. Then we can define an auxiliary graph TG [9] (based on the cactus structure of [8] which is a tree, such that for every k class in G there is a corresponding (unique) node in TG . The tree TG might include additional auxiliary nodes, but they are not leaves and we shall not be interested in them here. If G is k connected, then TG ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4edge -connectivity in a graph on-line. Technical Report #871, Technion, Department of Computer Science, 1995.

....connectivities. 1 Introduction Edge connectivity is a fundamental structural property of a graph. In the last decade, not only the properties of minimum cuts but also the number [14, 12, 9] and structure [1] of near minimum cuts were examined. Galil and Italiano [8] and Dinitz and Westbrook [3, 7] developed models for all 1 and 2 cuts and all 2 and 3 cuts, respectively. Based on these two models, Dinitz and Nutov introduced the so called 2 level cactus model a data structure that represents the minimum and minimum 1 edge cuts of an undirected multi graph with connectivity 3 in a ....

....minimum 1 edge cuts of an undirected multi graph with connectivity 3 in a compact way [5] There is no other so compact model, and no other compact graph model for these cuts known, for the best of our knowledge. The above models imply, in particular, fast incremental maintenance algorithms [8, 7, 5]. The 2 level cactus model (or 2 level cactus , for simplicity) generalizes the cactus model of all minimum cuts [4] In case of odd connectivity 3, the 2 level cactus is really a cactus, that is a connected graph in which every edge is contained in at most one simple cycle. Some cuts, ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20:242-276, 1998.

....these connectivities. 1 Introduction Edge connectivity is a fundamental structural property of a graph. In the last decade, not only the properties of minimum cuts but also the number [13, 11] and structure [1] of near minimum cuts were examined. Galil and Italiano [8] and Dinitz and Westbrook [3, 7] developed models for all 1 and 2 cuts and all 2 and 3 cuts, respectively. Based on these two models, Dinitz and Nutov introduced the so called 2 level cactus model a data structure that represents the minimum and minimum 1 edge cuts of an undirected multi graph with connectivity 3 in a ....

....minimum 1 edge cuts of an undirected multi graph with connectivity 3 in a compact way [5] There is no other so compact model, and no other compact graph model for these cuts known, for the best of our knowledge. The above models imply, in particular, fast incremental maintenance algorithms [8, 7, 5]. The 2 level cactus model (or 2 level cactus , for simplicity) generalizes the cactus model of all minimum cuts [4] In case of odd connectivity 3, the 2 level cactus is really a cactus, that is a connected graph in which every edge is contained in at most one simple cycle. Some cuts, ....

Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20:242-276, 1998.

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Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20(3):242--276, 1998.

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Y. Dinitz and J. Westbrook. Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica, 20(3):242-276, 1998.

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