H. Nagamochi and T. Ibaraki, "Linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica, pp. 583--596, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bound in the above theorem can be improved by precomputing a sparse certificate for (k 1) connectivity, G )#E E,see[NI92, CKT 93,FIN93] G has jE (k 1) n 1) O(kn) and if G is (k 1) connected, then G is (k 1) connected. Moreover, G can be computed in linear time, NI 92] In detail, we construct a legal ordering v 1 v 2 : OE v n of V , and retain an edge v i v j , i j,inE iff igj k 1. Also, we need an extension of [CKT 93, Corollary 2.17] and [FIN 93, Corollary 2.3] Proposition 3.4 (1) S V with jSjk is a shredder (or separator) of G iff S ....

.... from (1) replacing the input graph G by a sparse certificate, and (2) using our fast dynamic algorithm for maintaining b(G ) At the start of the algorithm, we replace the k connected input graph G = V#E)by(V# E) where E is a sparse certificate for the (k 1) connectivityofG, see [NI 92, CKT 93, FIN 93] The cardinalityof E is (k 1)n = O(kn) and E can be computed in linear time by finding a so called legal ordering of the nodes. The key point is that for every node set Q V , Q is a tight set (or a k separator, or a k shredder) of (V#E)iffQ is a tight set (or a ....

H. Nagamochi and T. Ibaraki, "A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica 7( 583--596.

....is the global connectivity. For undirected graphs, the same result holds (typically, one just converts an undirected graph to directed by orienting the edges in both directions) with the additional improvement that m can be held down to O(Cminn) using a construction due to Nagamochi and Ibaraki [20]. Gabow s result is based on an e#cient construction of spanning tree packings, as opposed to previous approaches which were based on network flows and Menger s theorem; the best of these had running time O(min C , mn ) on directed graphs and O(Cminn ) on undirected graphs. Thus, the ....

H. Nagamochi, T. Ibaraki. Linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7, 1992, pp. 583--596.

....to intplement. In order to simplify the algorithm, we use a single sink realizer for a triconnected planar graph G, i.e. a realizer in which a common vertex s of degree three is chosen as sb, sa and st. If G has no vertex of degree three, we first apply the algorithm of Nagamochi and Ibaraki [34] to obtain a sparse triconnected spanning subgraph G of G, which is guaranteed to have a vertex of degree three (see Lemma 2.6 of [34] Otherwise, G is identical to G. Then, a realizer of G is computed, as shown in the proof of Lemma 3, with v: s. A realizer of G is also a realizer of G. ....

....in which a common vertex s of degree three is chosen as sb, sa and st. If G has no vertex of degree three, we first apply the algorithm of Nagamochi and Ibaraki [34] to obtain a sparse triconnected spanning subgraph G of G, which is guaranteed to have a vertex of degree three (see Lemma 2. 6 of [34]) Otherwise, G is identical to G. Then, a realizer of G is computed, as shown in the proof of Lemma 3, with v: s. A realizer of G is also a realizer of G. Finally, a single sink realizer of G is obtained in the following way: let (sa, wa) be the edge following ( b) in the clockwise order ....

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H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583 596, 1992.

....cut in time O(z 2 min(x , To be precise, let 6 be the minimum degree in G. Note that 6. Our algorithm computes the exact size of if 6 2 and it returns 6 2 if 6 2. This is a speed up of a factor at least over the fastest exact algorithm for computing , which takes time O( 2I 2 d 3I1 5) [10, 1, 17]. Using the new 2 approximation algorithm as subroutine gives an incremental algorithm with total time O(wo r;z2 wz) Since z = O(wo w) the amortized time per insertion is if the initial graph is empty. Section 2 presents the basic structure that is common to all incremental algorithms in ....

....3 Basic definitions A maximal spanning forest decomposition (msfd) of order k is a decomposition of a graph G into k edge disjoint spanning forests Fi, 1 i k, such that Fi is a maximal spanning forest of G(FUF2U. UFi ) Let Gi be the multigraph (V, FUF2U, Fi) Nagamochi and Ibaraki [17] give a linear time algorithm that given a graph G with n nodes and m edges computes a msfd of order m in time O(m q n) We will refer to this algorithm as the decomposition algorithm (DA) This algorithm also determines an linear order on the nodes, called the mazimum cardinality search order ....

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H. Nagamochi and T. Ibaraki, "Linear time algorithms for finding a sparse k-connected span- ning subgraph of a k-connected graph", Algorithmica 7, 1992, 583 596.

....is able to answer a query whether the current graph is edge connected in constant time. The data structure needs O(m kn) space and preprocessing time. We create a dynamic vertex connectivity algorithm, using the algorithm by Nagamochi and Ibaraki for finding sparse k vertex certificates [23] and the O(k3tz 1 5 d k2tz 2) minimum vertex cut algorithm by Galil [15] A query takes constant time. The average update time is O(min(1, kn m) k3nLa k2n2) which is O(min(n 2, n3 m) for constant k. The preprocessing time and the space requirement is linear. Note that our algorithms are ....

....k Vertex Connectivity Eppstein et al. 11] give a dynamic algorithm for edge connectivity with worst case update time O( 2n log(n ) which we slightly modify in order to speed up the good case. It uses an algorithm by Gabow [14] for the static problem and the following lemma. Lemma 9. 1 [23, 31] Let G be a graph and T] U] a spanning forest of G. Let i be a spanning forest of G Ui 1 and let Ui be Ui 1 U i. Then G is l edge connected if and only if U is l edge connected. For notational convenience let U0 be the empty graph. For each i we store G U,i in the above minimum spanning tree ....

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H. Nagamochi and T. Ibaraki. Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583 596, 1992.

....with maximum vertex degree 2D T Gamma 2. 4.2.1 Theorem Every biconnected graph G = V;E) has a D T spanning tree T such that (v) 2e 1 for all vertices v of G. Such a D T spanning tree can be found in time O(n m) Proof: We first sparsify G, i.e. we reduce the size of the graph. In [NI92] an O(n m) time algorithm is given that outputs a spanning subgraph G of G that contains the same k connected components as G and has fewer than kn edges. Now, for each edge e = x;y) 2 E, place a new vertex v e in the middle of e, i.e. replace each edge by a path of length two. Let G = ....

....T Gamma 2) 2, then we can easily verify that this inequality holds only if d D T . Observe that the last lemma is valid for k edge connected graphs. and that every k connected graph is also k edge connected. Thus by combining Lemma 7.3. 1, the sparsification algorithm of Nagamochi and Ibaraki [NI92] see also the proof of Theorem 4.2.1) and the algorithm of Furer and Raghavachari [FR94] see Section 7.2) we obtain: 7.3.2 Theorem Every k connected graph G of maximum degree k(D T Gamma 2) 2 has a D T spanning tree. One can find such a tree in time k a(kn;n) logk) 87 Finally we ....

H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583--596, 1992.

.... = feg, r(y) w(e) q(e) since r(y) c(A; y) otherwise, there exists a vertex z 6= x such that e 0 = z; y) 2 E(y) Then, r(y) w(e) w(e 0 ) q(e) showing that w(e) q(e) ii) The proof of q(e) e) is based on the concept of sparse k connected spanning subgraph of a k connected graph [14]. Interested reader is referred to [14, 13] 2 If q(e) G) then it is always true that (e) G) since q(e) is the lower bound of (e) according to Lemma 3, which in turn indicates that e is contractible according to Definition 2. Consequently, for a given graph G(V; E; s; w) where q(e) for all ....

.... c(A; y) otherwise, there exists a vertex z 6= x such that e 0 = z; y) 2 E(y) Then, r(y) w(e) w(e 0 ) q(e) showing that w(e) q(e) ii) The proof of q(e) e) is based on the concept of sparse k connected spanning subgraph of a k connected graph [14] Interested reader is referred to [14, 13]. 2 If q(e) G) then it is always true that (e) G) since q(e) is the lower bound of (e) according to Lemma 3, which in turn indicates that e is contractible according to Definition 2. Consequently, for a given graph G(V; E; s; w) where q(e) for all edges in E is computed by CAPFOREST, a set ....

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H. Nagamochi and T. Ibaraki. A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, pages 583--596, 1992. 16

....optimal edges as forming a certificate of the shortest path structure of the graph, that must be revealed. The philosophy of the Hidden Paths Algorithm is thus similar to recent algorithms for connectivity, which work by first finding a sparse subgraph (or certificate) with the same connectivity [2, 17]. We have shown a lower bound of Omega Gamma mn) on the running time of comparison based algorithms for allpairs shortest paths. It is of particular interest that the construction and verification algorithms have the same worst case complexity. Compare this to the situation for the minimum ....

H. Nagamochi and T. Ibaraki. "Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph". Algorithmica, to appear, 1991.

....devised a very nice algorithm for edge connectivity. His algorithm, unlike previous algorithms for connectivity, does not appeal to Menger s theorem. It runs in O(km log(n 2 =m) time [9] The algorithms for vertex connectivity for 3 k p n currently require O(k 2 n 2 ) time [14] 2] [20]. The subject of this paper is the parallel complexity of 3 vertex connectivity. The importance of 3 vertex connectivity stems from the fact that if a planar graph is 3vertex connected (triconnected) then it has a unique embedding on a sphere. Hence an efficient algorithm that divides a graph ....

H. Nagamochi and T. Ibaraki, "Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica, to appear.

....that the problems of finding a minimum k vertex connected or k edge connected spanning subgraph are NP hard for any fixed k 2. For the more relaxed problem of finding sparse but not necessarily minimal k edge connected and k vertex connected spanning subgraphs, linear time algorithms are known ([18]) Yannakakis ( 24] see also [15] showed that the related problem of deleting a minimum set of edges so that the resulting graph has a given property is NP hard for several graph properties (e.g. planar, outerplanar, transitive digraph) There is a natural sequential algorithm for finding a ....

H. Nagamochi AND T. Ibaraki, Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica, vol. 7, 1992, pp. 583-596.

....vertex connectivity, planar graphs AMS subject classifications. 68P05, 68Q20, 68R10 PII. S0097539794269072 1. Introduction. Sparse certificates, small graphs that retain some property of a larger graph, appear often in graph theory, especially in problems of edge and vertex connectivity [2, 13, 31, 35]. The main motivation for studying sparse certificates lies in the fact that they are e#ective tools for speeding up many graph algorithms. To check whether a graph G has a given property P, one can first compute a sparse certificate C for property P and then run an algorithm for P on the ....

....C for property P and then run an algorithm for P on the certificate rather than on G itself. This is favorable whenever computing certificates is faster than checking property P. This method has led to improved algorithms for testing k edge and k vertex connectivity sequentially [16, 31, 35] and in parallel [2] for finding three independent spanning trees [1] and for reliability in distributed networks [26] With the sparsification technique [8] sparse certificates additionally became an important tool for speeding up dynamic graph algorithms, in which edges may be inserted into ....

H. Nagamochi and T. Ibaraki, A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica, 7 (1992), pp. 583--596.

....them. 2.3 A Recursive Algorithm for 2 Tree Problem The recursive algorithm presented in this section has a preprocessing step in which the edges whose removal do not change the connectivity of the network are removed. The preprocessing is based on a linear time algorithm by Nagomochi and Ibaraki [15]. The algorithm constructs a sparse k connected spanning subgraph of a k connected graph such that the subgraph G 0 = V; E 0 ) has jE 0 j = O(kjV j) edges. As a result of the preprocessing, the number of edges in the network is reduced to O(N) which yields to O(N 3 ) time complexity of ....

H. Nagamochi and T. Ibaraki. A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, pages 583--596, 1992.

....) be the connectivity requirement vector and the vector of roots, respectively. We use k = c 1 to denote the largest node requirement. Our first algorithm simply finds a sparse certificate for local node connectivity in G. In detail, it employs the polynomial algorithm of Nagamochi and Ibaraki [12] to find k edge disjoint forests F 1 ; F k of G such that in the graph H = V; F 1 [ F k ) we have H (u; v) minfk; G (u; v)g for every two nodes u; v. This graph H has at most k(n Gamma 1) edges, while the optimal subgraph has at least nk=2 edges, since it has minimum degree at ....

H.Nagamochi and T.Ibaraki, "A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica 7 (1992), 583--596.

....and the B ; hB values are integers in the interval [0; n] Thus, Step 6 can be implemented in linear time. 2 The time bound in the above theorem can be improved by precomputing a sparse certificate for (k 1) connectivity and local (k 1) node connectivities, G 0 = V; E 0 ) E 0 E, see [NI 92, CKT 93, FIN 93] The number of edges in G 0 , jE 0 j, is (k 1) n Gamma 1) O(kn) If G is (k 1) connected, then G 0 is (k 1) connected, and moreover, G 0 (v; w) min(k 1; G (v; w) for all node pairs v; w, where H (v; w) denotes the maximum number of openly disjoint v w paths in ....

.... Gamma 1) O(kn) If G is (k 1) connected, then G 0 is (k 1) connected, and moreover, G 0 (v; w) min(k 1; G (v; w) for all node pairs v; w, where H (v; w) denotes the maximum number of openly disjoint v w paths in the graph H . A linear time algorithm for computing G 0 is known, see [NI 92] In detail, we construct a legal ordering v 1 ; v 2 ; v n of V , and retain an edge v i v j , i j, in E 0 iff jfv : v v j 2 E; igj k 1. An ordering v 1 ; v 2 ; v n of the nodes of G is called legal if d(V i Gamma1 ; v i ) V i Gamma1 ; v j ) 81 i j n, where V ....

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H. Nagamochi and T. Ibaraki, "A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica 7 (1992), 583--596.

....use flow. They grew out of early work of Mader [20] that showed that every k edge connected graph has a subgraph that is k edge connected and has only O(kn) edges. Subsequent work of Nishizeki and Poljak [25] showed how this subgraph can be constructed as a union of forests. Nagamochi and Ibaraki [22, 23] gave fast algorithms for constructing such subgraphs. A short proof of a generalization of the results of Nagamochi and Ibaraki to mixed cuts containing both edges and nodes was presented by Frank, 1 Actually, you don t really need to know about the Gomory Hu tree to show that you can find the ....

H. Nagamochi and T. Ibaraki, A Linear Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph, Algorithmica 7 (1992), pp. 583-596.

....by # . Our most efficient Monte Carlo algorithm has a running time 1 of O(n 2 log(U c) where n is the number of vertices and U is the highest edge capacity in the graph. In comparison the deterministic algorithm of Gabow [18] runs in O(n 2 m) time while a very recent algorithm [33] improves this to O(nm) The best algorithm for graphs A.A. Benczr: Computer and Automation Institute, Hungarian Academy of Sciences and Department of Operations Research, Etvs University, Budapest, e mail: benczur cs.elte.hu Supported from EC grant ALTEC KIT, OTKA grants T 16503, T 16524, ....

.... the extreme system: we present the first RNC algorithm by a new analysis of Karger s [24] algorithm, as well as a very efficient sequential algorithm based on a representation of near minimum cuts [3] By recent breakthroughs in the design of algorithms for finding edge connectivity and min cuts [23,33,15,26], it turns Parallel and fast sequential algorithms for undirected edge connectivity augmentation 597 out that several min cut structures are much easier to build than to find even a single source sink min cut. Our algorithms take advantage of these results and, in particular, avoid maxflow ....

[Article contains additional citation context not shown here]

Nagamochi H., Ibaraki, T. (1992): A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583--596

....In order to simplify the algorithm, we use a single sink realizer for a triconnected planar graph G, i.e. a realizer in which a common vertex s of degree three is chosen as s b , s g and s r . If G has no vertex of degree three, we first apply the algorithm of Nagamochi and Ibaraki [34] to obtain a sparse triconnected spanning subgraph G 0 of G, which is guaranteed to have a vertex of degree three (see Lemma 2.6 of [34] Otherwise, G 0 is identical to G. Then, a realizer of G 0 is computed, as shown in the proof of Lemma 3, with v 1 = s. A realizer of G 0 is also a ....

....a common vertex s of degree three is chosen as s b , s g and s r . If G has no vertex of degree three, we first apply the algorithm of Nagamochi and Ibaraki [34] to obtain a sparse triconnected spanning subgraph G 0 of G, which is guaranteed to have a vertex of degree three (see Lemma 2. 6 of [34]) Otherwise, G 0 is identical to G. Then, a realizer of G 0 is computed, as shown in the proof of Lemma 3, with v 1 = s. A realizer of G 0 is also a realizer of G. Finally, a single sink realizer of G 0 is obtained in the following way: let (s g ; w g ) be the edge following (s g ; s b ) ....

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H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583--596, 1992.

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H. Nagamochi and T. Ibaraki, "Linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica, pp. 583--596, 1992.

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N. Nagamochi and T. Ibaraki. Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583--596, 1992.

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H. Nagamochi and T. Ibaraki, A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica, 7 (1992), pp. 583--596.

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H. Nagamochi and T. Ibaraki, "Linear-time Algorithm for Finding a Sparse k- connected Spanning Subgraph of a k-connected Graph," Algorithmica, 7 (5-6), 583-- 596, 1992. 149

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H. Nagamochi and T. Ibaraki, "Linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph," Algorithmica, pp. 583--596, 1992.

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Nagamochi, H. and Ibaraki, T. Linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7 (1992), 583--596

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H. Nagamochi and T. Ibaraki, "Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph", Algorithmica 7, 1992, 583--596.

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H. Nagamochi and T. Ibaraki, "Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph", Algorithmica 7, 1992, 583--596.

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