V. Ramachandran, Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity, in [32], pp. 276--340.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... that a graph has an open ear decomposition if and only if it is biconnected [46] Lovasz showed that the problem of computing an open ear decomposition in parallel is in NC [30] Ear decomposition has also been used in designing efficient sequential and parallel algorithms for triconnectivity [37] and 4 connectivity [23] In addition to graph connectivity, ear decomposition has been used in graph embeddings (see [9] The sequential algorithm: Ramachandran [37] gave a linear time algorithm for ear decomposition based on depth first search. Another sequential algorithm that lends itself to ....

.... is in NC [30] Ear decomposition has also been used in designing efficient sequential and parallel algorithms for triconnectivity [37] and 4 connectivity [23] In addition to graph connectivity, ear decomposition has been used in graph embeddings (see [9] The sequential algorithm: Ramachandran [37] gave a linear time algorithm for ear decomposition based on depth first search. Another sequential algorithm that lends itself to parallelization (see [22, 33, 37, 42] finds the labels for each edge as follows. First, a spanning tree is found for the graph; the tree is then arbitrarily rooted ....

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V. Ramachandran. Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity. In J. H. Reif, editor, Synthesis of Parallel Algorithms, pages 275--340. Morgan Kaufman, San Mateo, CA, 1993.

.... and showed that a g r h has an o enear decom osition if and only if it is biconnected [46] Lovasz showed that therep1O of com uting an o enear decom osition inar;1O3 is in NC [30] Ear decom osition has also been used in designing e#cient sequential andarO lel algor thms for tr iconnectivity [37] and 4 connectivity [23] In addition togr h connectivity,ear decom osition has been used in gr h embeddings (see [9] The sequential algorithm: an [37] gave a linearX ime algor thm for ear decom osition based on de th firp sear h. Another sequential algor thm that lends itself to ar ....

.... osition inar;1O3 is in NC [30] Ear decom osition has also been used in designing e#cient sequential andarO lel algor thms for tr iconnectivity [37] and 4 connectivity [23] In addition togr h connectivity,ear decom osition has been used in gr h embeddings (see [9] The sequential algorithm: an [37] gave a linearX ime algor thm for ear decom osition based on de th firp sear h. Another sequential algor thm that lends itself to ar allelization (see [22,33,37,42] finds the labels for each edge as follows. Fir st, a spanningtr ee is found for the gr aph; the tr ee is then ar bitrOx [ r oted ....

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V. Ramachandran. Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity. In J. H. Reif editor, Synthesis of Parallel Algorithms, pages 275--340. Morgan Kauf man, San Mateo, CA, 1993. 134

....on a CGM with p processors and O( n p ) local memory per processor, n p p ffl (ffl 0) using O(log p) communication rounds and O( n p ) local computation per round. 12 4. 4 Open Ear Decomposition We first recall the definition of an ear decomposition and open ear decomposition (see e.g. [30]) Consider an undirected graph G = V#E) with n vertices and m edges. For the remainder, we assume that G is connected. An ear decomposition of G is an ordered partition of E into r simple paths P 1 #: #P r such that P 1 is a cycle, and, for each2 i r, P i is a simple path with endpoints ....

....created by nontree edges incident to a descendant of v in T . Let low(v) be the minimum preorder number of a node w which lies on any such fundamental cycle. If no such w exists then let low(v) n. The classical O(log n) time PRAM algorithms for ear decomposition and open ear decomposition [26, 28, 30] consist of a constant number of the following operations: 1) find a spanning tree T for G# (2) find the lowest common ancestor lca(e)ofevery nontree edge e = u# v)# (3) number the vertices of T in preorder from 0 to n ; 1# (4) compute low(v) for eachvertex v of V # (5) find the connected ....

V. Ramachandran. "Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity", in [32], pp. 276--340.

....paradigms, for message passing and shared memory architectures. We experimentally test our algorithms on a variety of inputs, described in Section 4, and offer some conclusions on our results. 2 Sequential Algorithm In this section, we give a well known sequential algorithm for ear decomposition [19, 24, 27, 25]. There are five major steps in this algorithm. First, a spanning tree is found for the graph, a root vertex is arbitrarily chosen, and each vertex is then assigned a level and parent using the spanning tree. Next, the non tree edges are examined and uniquely labeled. Finally the tree edges are ....

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J.H. Reif, editor, Synthesis of Parallel Algorithms. Morgan Kaufman, San Mateo, CA, 1991.

....in computing A 0;3 and A 3;4 together with additional information we found to derive A 0;4 . We also show that the same approach can be used to construct A 0;3 from A 0;2 and A 2;3 . The algorithmic notation used is pseudo Pascal and is similar to the notation of Tarjan [Tar83] and Ramachandran [Ram93]. We enclose comments between f and g . The organization of this paper is as follows. In Section 2, we survey related work. In Section 3, we give definitions used in this paper. We then describe properties of blocks in Section 4, properties of separating sets in Section 5, and properties of ....

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. H. Reif, editor, Synthesis of Parallel Algorithms, pages 275--340. Morgan-Kaufmann, 1993.

....protocols for concurrent write, discussed above for the CRCW models, 5 also apply here. Not too many results are known for the ERCW PRAM models [MR93] Algorithmic Notation The algorithmic notation used is pseudo Pascal and is similar to the notation of Tarjan [Tar83] and Ramachandran [Ram93] We enclose comments between f and g . Parameters are called by value unless they are declared with the keyword modifies in which case they are called by value and result. We use the following pfor statement for executing a loop in parallel. pfor iterator do statement list rofp The effect ....

....be the set of vertices in X; let X be the induced subgraph of G on (V Gamma VX ) fa 1 ; a 2 g. Let B be a bridge of G a such that B and B both contain at least two edges and either B or B is biconnected. The Tutte split operation on fa 1 , a 2 g defined in Tutte [Tut66] see also Ramachandran [Ram93] on a biconnected graph G forms two graphs G 1 = B [ f(a 1 ; a 2 )g and G 2 = B [ f(a 1 ; a 2 )g. The edge (a 1 , a 2 ) added in G 1 and G 2 is called a virtual edge. Tutte Component Let G 0 be the graph obtained from a biconnected graph G by performing the Tutte split operation successively ....

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V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. H. Reif, editor, Synthesis of Parallel Algorithms, pages 275--340. MorganKaufmann, 1993.

....algorithm to perform this step is also fairly complex, and space limitations prevent us from presenting all of the details for this step. Therefore, we present this section so that it can be used as a companion to the presentation of the open ear decomposition algorithm of Ramachandran as given in [21]. Finally, we go through the other steps of the algorithm of [22] 3.1 Elementary graph computations in SL Our method of exposition in this subsection is to give a statement of the subproblem to be solved, and then in parentheses give an indication of how this subproblem can be restated in a way ....

....w, x where (1) u, w are attachment points of the bridge containing f , 2) v, x are attachment points of the bridge containing g, and (3) u, v, w, x occur in cyclic order on C e . To check cyclic order, use the previous test. 3. 2 Finding an open ear decomposition We follow the exposition from [21]. The algorithm in Figure 1 finds an open ear decomposition of a graph: it labels each edge e by the number of the first ear containing e. input: biconnected graph G; vertices v, r; edge e 1: Find a spanning tree T , and number the vertices in preorder from 0 to n 1 with respect to root r. ....

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. Reif, editor, Synthesis of Parallel Algorithms. Morgan Kaumann, 1993.

....G, H 1 and H 2 , we can construct the specified 2 edge block graph, the specified 2 block graph and the specified bi level block graph in O(log 2 n) time using n m processors. Proof. It takes O(log 2 n) time using n m processors to construct the 2 edge block forest and the 2 block forest [18, 21]. To construct the specified 2 edge block (respectively, 2block) forest we need to root each tree in the forest. We also need to check, for each vertex v in the rooted tree, whether there is a 2 edge block (respectively, 2 block) containing a vertex of H 2 (respectively, H 1 ) in the subtree ....

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. H. Reif, editor, Synthesis of Parallel Algorithms, pages 275--340. Morgan-Kaufmann, 1993.

....a cutpoint. If v is not a cutpoint, let d 2 (v) 1. For more on properties of 2 blk(G) see [10] An example of a graph and its 2 blk(G) is shown in Figure . Tutte component The Tutte components of a biconnected graph are the triconnected components defined in Tutte [20] see also Ramachandran [15]) Each Tutte component can be a single vertex, a triconnected component, a simple cycle (a polygon) or a pair of vertices with at least three edges between them (a bond) 3 block graph Given a 2 block H of G, we define the 3 block graph, 3 blk(H) as follows. The 3 block graph contains three ....

....is oe vertex corresponding to a pair of vertices z 1 and z 2 in Q; iv) interchanging u and v in any one of the previous three conditions. We call the pairs of vertices that correspond to oe vertices Tutte pairs. It is known that the number of Tutte pairs in an n node biconnected graph is O(n) [13, 15]. From [9, 15, 20] we know that 3 blk(H) is a tree if H is a 2 block. We call this tree a 3 block tree. We call the set of trees corresponding to 2 blocks in G the 3 block graph of G or 3 blk(G) Each Tutte component that corresponds to a fi vertex in the 3 block graph is a 3 block of G. Given a ....

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Vijaya Ramachandran, "Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity, " invited chapter for Synthesis of Parallel Algorithms, J. H. Reif, editor, Morgan-Kaufmann.

....where z is a new vertex not in V ; E(G 0 ) is the subset of those edges in E that have at least one endpoint outside V 0 ; OE G 0 (e) OE G (e) if no endpoint of e belongs to V 0 and OE G 0 (e) z; v) if OE G (e) u; v) where u 2 V 0 and v 2 V Gamma V 0 . An ear decomposition ([21]) D = P 0 ; P 1 ; P r Gamma1 ] of a graph G is a partition of E(G) into an ordered collection of edge disjoint paths P 0 ; P r Gamma1 such that P 0 is a cycle and the two endpoints of P i , for i 1, are contained in some P j , j i, and none of the internal vertices of P i are ....

....claim. 2 Theorem 10 Algorithm 4 computes a minimal 2 edge connected spanning subgraph of any 2 edgeconnected graph on n vertices and m edges in time and space O(n m) Proof. Consider one iteration of the while loop. We compute an ear decomposition in linear time and space using the algorithm of [21]. We compute an optimal tree TH and S critical edges in linear time and space as described in the proof of theorem 6. We compute a minimal augmentation in linear time and space using the algorithm of [13] To compute the S critical edges, we determine the connected components of (V (H) S) ....

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V. Ramachandran, Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity, in Synthesis of Parallel Algorithms, J. Reif, ed., Morgan-Kaufmann, 1993.

....[7] Hence we are unable to obtain efficient parallel algorithms by parallelizing sequential algorithms based on depth first search or breadth first search. Instead, an alternative search technique called ear decomposition has proved to be a very useful tool for designing parallel graph algorithms [7, 6, 10, 17, 20, 19]. Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19] st numbering [10] planarity testing and ....

.... for designing parallel graph algorithms [7, 6, 10, 17, 20, 19] Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19], st numbering [10] planarity testing and embedding [20] finding a strong orientation and finding a minimum cost spanning forest z . Figure 1 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and the routine for finding ....

[Article contains additional citation context not shown here]

V. Ramachandran, Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity, Synthesis of Parallel Algorithms (J. H. Reif, ed.), MorganKaufmann, 1993, pp. 275--340.

....[7] Hence we are unable to obtain efficient parallel algorithms by parallelizing sequential algorithms based on depth first search or breadth first search. Instead, an alternative search technique called ear decomposition has proved to be a very useful tool for designing parallel graph algorithms [7, 6, 10, 17, 20, 19]. Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19] st numbering [10] planarity testing and ....

.... for designing parallel graph algorithms [7, 6, 10, 17, 20, 19] Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19], st numbering [10] planarity testing and embedding [20] finding a strong orientation and finding a minimum cost spanning forest y . Figure 1 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and the routine for finding ....

[Article contains additional citation context not shown here]

V. Ramachandran, Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity, Synthesis of Parallel Algorithms (J. H. Reif, ed.), MorganKaufmann, 1993, pp. 275--340.

....is within a polylog factor of the best possible. The parallel computation model we use here is the EREW PRAM model. For PRAM models and techniques for designing efficient algorithms on a PRAM, see Karp Ramachandran [8] The algorithmic notation in this paper is from Tarjan [15] and Ramachandran [14]. 2 Preliminaries Definition 2.1 A contour is a simple closed curve in the plane. A contour divides the plane into two parts: the part that is unbounded is called the outer part of the contour, the part that is bounded is called the inner part of the contour. We say contour c 1 encloses contour ....

Ramachandran, Vijaya "Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity" Invited chapter for Synthesis of Parallel Algorithms, J. H. Reif, editor, MorganKaufmann.

....G by examining the edges of G one at a time and removing an edge if the resulting graph is a 2 edge connected spanning subgraph Work supported in part by NSF Grant CCR 89 10707. of G. The total time is dominated by m calls to the algorithm for testing 2 edge connectivity ( 21] 14] 13] [18]) giving a time bound of O(m(n m) The time can be brought down to O(m n 2 ) by first finding a sparse 2 edge connected spanning subgraph of G (see section 3) There is a similar sequential algorithm with the same time bound for finding a minimal biconnected spanning subgraph of G. None of ....

....Determine a minimal subset B of edges in H such that the graph TH B is 2 edgeconnected. Let H = TH B. The purpose of step (0) is to speed up subsequent iterations of the while loop by computing a sparse subgraph of the input graph. In this step we compute an ear decomposition of G ( 14] 13] [18]) and eliminate all trivial ears. Let H be the resulting graph. H is clearly a 2 edge connected spanning subgraph of G. Let m 0 denote the number of edges of H and let q be the number of (nontrivial) ears in the above ear decomposition. A proof by induction over q establishes m 0 = n q Gamma ....

[Article contains additional citation context not shown here]

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. Reif, editor, Synthesis of Parallel Algorithms. Morgan-Kaufmann, New York, NY, 1992. To appear.

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V. Ramachandran, Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity, in [32], pp. 276--340.

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V. Ramachandran. "Parallel open ear decomposition withapplications to graph biconnectivityandtriconnectivity", in [28], pp. 276 - 340.

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V. Ramachandran, Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity, in [32], pp. 276--340.

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V. Ramachandran. "Parallel open ear decomposition withapplications to graph biconnectivityandtriconnectivity", in [28], pp. 276 - 340.

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V. Ramachandran, Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity, in [32], pp. 276--340.

No context found.

V. Ramachandran. "Parallel open ear decomposition withapplications to graph biconnectivityandtriconnectivity", in [28], pp. 276 - 340.

No context found.

V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J.H. Reif, editor, Synthesis of Parallel Algorithms, chapter 1, pages 275--340. Morgan Kaufmann, 1993.

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V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. Technical Report UTEXAS.CS//CS-TR-92-02, University of Texas at Austin, Department of Computer Sciences, 1992.

No context found.

V. Ramachandran, Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity, in [32], pp. 276--340.

No context found.

V. Ramachandran. "Parallel open ear decomposition withapplications to graph biconnectivityandtriconnectivity", in [28], pp. 276 - 340.

No context found.

V. Ramachandran. Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity. Technical Report, University of Texas, Austin, Number CS-TR-92-02, January 1, 1992.

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