V. Ramachandran and J. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517--561, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... biconnectivity algorithm of [30] There are no direct results on computing triconnected components I O efficiently, although one may apply the PRAM simulation of [9] to the triconnectivity algorithm of [13] A number of PRAM algorithms for planarity testing and planar embedding have been proposed [18, 19, 26, 28]. In [19] the first such algorithm using a linear number of processors was presented; the algorithm runs in O(log 2 2 N) time. The algorithm of [28] runs in O(log 2 N) time using O(C(N) processors, where C(N) is the number of processors required to compute the connected components of a graph ....

....to the triconnectivity algorithm of [13] A number of PRAM algorithms for planarity testing and planar embedding have been proposed [18, 19, 26, 28] In [19] the first such algorithm using a linear number of processors was presented; the algorithm runs in O(log 2 2 N) time. The algorithm of [28] runs in O(log 2 N) time using O(C(N) processors, where C(N) is the number of processors required to compute the connected components of a graph in O(log 2 N) time. Using the PRAM simulation technique of [9] one can obtain I O efficient, but suboptimal, embedding algorithms from the algorithms ....

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V. Ramachandran and J. H. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49(3):517--561, 1994.

....graph G f consisting of terminal and nonterminal edges with an edge uv. After replacement of f by G f , the edge f is removed from G f . Each two connected graph can be produced in that way in O(n) time [3] see also for example [14] or in O(log n) time on a CRCW PRAM with O(n) processors [15]. To get al..l connected planar graphs, one has to allow the following additional production rules. For a vertex v, one can do the following. 1. add a terminal or nonterminal edge e with one endpoint v and a new vertex as the other endpoint, 2. add a two connected graph G v where one vertex in G ....

....analogously. Theorem 3 To check whether a clustering of a connected graph with connected clusters has a clustered embedding can be done in linear time sequentially and in logarithmic time with a linear processor number on a CRCW PRAM. Proof: We always get any planar embedding in these bounds [14, 15]. We also get the derivation tree TD in these bounds [13] We can transform any planar embedding into a clustered embedding as described above in these bounds. Q.E.D. 6 Conclusions It might also be of interest to get a quite nice planar clustered embedding, e.g. by using lmc orderings [10] 2] ....

V. Ramachandran, J. Reif, Planarity Testing in Parallel, Journal of Computer and Systems Sciences 49 (1994), pp. 517-561.

....of the existence of a parallel planarity testing and embedding algorithm, or of a parallel separator algorithm without requiring an embedding as part of the input. We are not aware of any parallel algorithm for the latter case. Work efficient, NC algorithms for planarity testing have been given in [10, 14], but neither of these algorithms seems to be easily implementable. Designing a simple, easily implementable, parallel algorithm for planarity testing and embedding is an interesting open problem. Acknowledgement. We are grateful to Hillel Gazit for providing us with [6] ....

V. Ramachandran and J. Reif, Planarity testing in parallel, Journal of Computer and System Sciences, 49(3):517-561, 1994.

....in the context of (nonuniform) circuit complexity, since L poly is equal to SL poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degree three graphs had been shown to be in SL [23, 20] 1 Introduction The problem of determining if a graph is planar has been studied from several perspectives of algorithmic research. From most perspectives, optimal algorithms are already known. ....

....optimal algorithms are already known. Linear time sequential algorithms were presented by Hopcroft and Tarjan [10] and (via another approach) by combining the results of [16, 4, 8] In the context of parallel computation, a logarithmic time CRCWPRAM algorithm was presented by Ramachandran and Reif [22] that performs almost linear work. From the perspective of computational complexity theory, however, the situation has been far from clear. The best upper bound on the complexity of planarity that has been published so far is the bound of AC 1 that follows from the logarithmic time CRCW PRAM ....

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V. Ramachandran and J. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517--561, 1994.

....of the graph, though not necesarily one suitable for the above algorithm. A related question that immedately arises is: what is the complexity of planarity testing itself Can this be done in GapL The best known result so far is that planarity testing can be done on a CRCW PRAM in O(log n) time [RR94], and hence is in AC 1 . Of course, the big question still remains open: what exactly is the complexity of both the decision and counting versions of perfect matchings Acknowledgments The authors would like to thank Gunter Rote for very helpful comments on an earlier version of the paper. ....

V. Ramachandran and J. Reif, Planarity Testing in Parallel, Journal of Computer and System Sciences, vol. 49, pp 517-561, 1994.

....for finding the LCA s of all the non tree edges using the algorithm of Schieber and Vishkin [SV88] As a consequence we also get an optimal randomized CREW PRAM algorithm for st numbering. Further applications of our algorithm may include triconnectivity [FRT93] Ram93] and planarity testing [RR94] 8 Concluding remarks and open problems The main remaining open problem is the question whether there exists an optimal speed up O(log n) time deterministic EREWPRAM algorithm for finding spanning forests. Note that such a deterministic algorithm is not even known for the much stronger CRCW ....

V. Ramachandran and J. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517--561, 1994.

....of the existence of a parallel planarity testing and embedding algorithm, or of a parallel separator algorithm not requiring an embedding as part of the input. We are not aware of any parallel algorithm for the latter case. Work efficient, NC algorithms for the former case have been given in [12, 17], but neither of these algorithms seems to be easily implementable. Designing a simple, easily implementable, parallel algorithm for planarity testing and embedding is an interesting open problem. Acknowledgement. We are grateful to Hillel Gazit for providing us with [7] ....

Vijaya Ramachandran and John H. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49(3):517--561, 1994.

....optimal logarithmic time parallel algorithms for graph biconnectivity and triconnectivity. Open ear decomposition has been used to obtain efficient parallel algorithms for several other important graph problems such as graph four connectivity [KR91] st numbering [MSV86] and graph planarity [RR89]. Algorithmic Notation The algorithmic notation in this report is from Tarjan [Ta83] We enclose comments between a pair of curly brackets with asterisks ( f and g ) We incorporate parallelism by use of the following statement that augments the for statement. pfor iterator statement list ....

....P . This property is useful in determining planarity of G(P ) for the case when every star has to be embedded completely on one side of P . While this is not needed for the triconnectivity algorithm, it is an important step in the parallel algorithm for graph planarity given in Ramachandran Reif [RR89]. In Section 4.2.3 we describe a simple efficient algorithm to form the star embedding of G c (P ) and to extract from it the candidate lists of G(P ) 4.2.1 Determining Interlacings of Chords on a Path Let G(P ) be a star graph in which each star has exactly two attachments on P . For ....

V. Ramachandran, J. H. Reif, "Planarity testing in parallel," TR-90-15, Dept. of Computer Sciences, The University of Texas, Austin, TX,

....[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 ....

.... 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 connected components. Our parallel implementations ....

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V. Ramachandran and J. Reif, Planarity testing in parallel, Jour. Comput. and Sys. Sci. 49 (1994), no. 3, 517--561, Special Issue for FOCS '89.

.... graphs including connected components and spanning forest x [2, 5, 7, 13, 16, 17, 24, 26, 42] minimum spanning forest (MSF) 2, 5, 6] ear decomposition and 2 edge connectivity [32, 37, 43] open ear decomposition and biconnectivity [32, 37, 43, 52] triconnectivity [12, 36] and planarity [44]. All of these algorithms (with the exception of some algorithms for MSF) have the additional feature that they serialize into linear time sequential algorithms. However, these algorithms are quite different from earlier linear time algorithms based on depthfirst search [51] in that they are very ....

V. Ramachandran and J. Reif, Planarity testing in parallel, Jour. Comput. and Sys. Sci. 49 (1994), no. 3, 517--561, Special Issue for FOCS '89.

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V. Ramachandran and J. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517--561, 1994.

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Vijaya Ramachandran and John Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517--561, 1994.

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V. Ramachandran and J. Reif. Planarity testing in parallel. Journal of Computer and System Sciences, 49:517-561, 1994.

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