Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs, Theoret. Comput. Sci., 47 (1986), 277--298.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is based on Kao and Klein s algo rithm [11] but is not obtained using PRAM simulation. It uses the SSSP algorithm developed in Section 2. We also consider the problem of computing an open directed ear decomposition. An efficient PRAM algorithm is known for the undirected version of this problem [15]; but to our knowledge the directed version of the problem has not been considered before. We present an O(sort(N) I 0 algorithm that com putes an open directed ear decomposition for a biconnected strong planar digraph. In the full paper, we also show that a digraph has an open directed ear ....

....computed decomposition to obtain an open directed ear decomposition, provided that the graph is also biconnected. 3. 1 Directed Ear Decomposition An ear decomposition of an undirected graph G can be obtained as a collection of appropriate subpaths of funda mental cycles of a spanning tree of G [15]. The idea in the PRAM algorithm of Kao and Klein [11] for obtaining a directed ear decomposition of a strong digraph G is similar; but the construction makes use of two spanning trees Tc and Td rooted at the same vertex r. The edges in Tc are directed towards r (T is called convergent) the edges ....

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Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (eds) and st-numbering in graphs. Theoretical Computer Science, 47:277-298, 1986.

....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 and each vertex is assigned a level and parent. Each non tree edge corresponds to a distinct ear, since an arbitrary spanning tree and the non tree edges form a cycle ....

....edge s endpoints and a unique serial number for that edge. Finally, the tree edges are assigned ear labels by choosing the smallest label of any non tree edge whose cycle contains it. This algorithm runs in O( m n) log n) time. 4. 2 The PRAM Algorithm The PRAM algorithm for ear decomposition [32, 33] is based on the second sequential algorithm. The first step computes a spanning tree in O(log n) time, using O(n m) processors. The tree can then be rooted and levels and parents assigned to nodes by using the Euler tour technique. Labelling the nontree edges uses an LCA algorithm, which runs ....

Y. Moan, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 47(3):277--296, 1986.

....[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 and each verO] is assigned a level andparF t. Eachnon trp edge cor r esponds to a distinct ear , since an ar bitr ar y spanningtr ee and the non tr ee edges for a ....

....of the edge s endpoints and a unique ser ial number for that edge. Finally, the trO edges a r assignedear labels by choosing the smallest label of anynon trp edge whose cycle contains it. This algor thm r ns in O( m n)logn)time. 4. 2 The PRAM Algorithm The PRAM algor thm for ear decom osition [32,33] is based on the second sequential algor ithm. Thefir st ste com utes a s anningtr ee in O(log n)time, using O(n m) rcessorp Thetr] can then ber ooted and levels and arp ts assigned to nodes by using the Euler tour technique. Labelling the nontr ee edges uses an LCA algor ithm, whichr uns ....

Y. Moan, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 4 (3):2 --296, 1986. 134, 135 David A. Bader et al.

.... A A A A A A A A 1=s 3 2 4 5 6=t Q Q Q Q Q Q P P P P P P Figure 1: An st numbered Graph We observe that a parallel st numbering method can be obtained by combining the CGM graph algorithms of [3] and the PRAM algorithm by Maon, Schieber and Vishkin [13]. We now give a brief outline of the method. Let G = V; E) be a graph with an open ear decomposition G = P 0 ; P 1 ; P l ] such that P 0 = s; t) and the endpoints L(P i ) and R(P i ) of each P i are in P j and P k where i j k 0. The vertex in an ear P i , i 1, which is adjacent ....

....P i , i 1, which is adjacent to L(P i ) respectively R(P i ) is denoted by LS(P i ) respectively RS(P i ) The vertex L(P i ) is called the anchor of P i and if v 2 P i , v 6= L(P i ) and v 6= R(P i ) then v is called an internal vertex. Algorithm 1 Computing the st Numbering of a Graph [13] Input: An open ear decomposition of graph G = V; E) G = P 0 ; P 1 ; P l ] where P 0 = s; t) Output: A valid st numbering of graph G. 1) Compute the ear tree ET (V T ; E T ) where V T = fP i jP i is an ear in Gg and E T = f(P i ; P j )j L(P i ) is an internal vertex of P j ; i ....

[Article contains additional citation context not shown here]

Y. Maon, B. Schieber, and U. Vishkin. Parallel Ear Decomposition Search (EDS) and st-Numbering in Graphs. Theoretical Computer Science, 47:277-298, 1986.

....[22] show that for any biconnected graph G = V; E) and for any s; t 2 V , there exists an st ordering of G. Cheriyan and Reif [6] extended this result to directed graphs. Even and Tarjan [17] develop a linear time algorithm to compute an st ordering of an undirected biconnected graph (also see [16, 24, 29]) Under the guise of bipolar orientations, st orderings have also been studied in [7, 10, 26] In related work, Papakostas and Tollis [25] describe an algorithm for producing so called bst orderings of graphs with maximum degree four; these are st orderings with a lower bound on the number of ....

Y. MAON, B. SCHIEBER, AND U. VISHKIN, Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoret. Comput. Sci., 47(3):277--298, 1986.

....exploration. In order to connect the newly explored ear to the explored subgraph the robot methodically traverses the ear in the opposite direction. 2 More specifically, first the robot picks 1 Any connected planar graph can be decomposed into a set of cycles which are called ears (Maon et al. [MSV88]) 2 The following step could be eliminated by adding a second marker. The second marker is dropped after following the first up the marker and backtracks to the previous vertex visited where it drops the marker, then the robot continues backtracking until it either reaches the marked vertex or ....

Yael Maon, Baruch Schieber, and Uzi Vishkin. Parallel ear decomposition search (eds) and st-numbering in graphs, 1988.

....ear decomposition starting from a single edge. Theorem 15 (Folklore) A directed graph is strongly connected if and only if it has an ear decomposition starting from a single vertex. Lovasz [44] gave a parallel algorithm for finding the ear decompositions described in these theorems. Maon et al. [46] gave more e#cient algorithms for both undirected cases, using Euler tours. Miller and Ramachandran [50] independently discovered similarly e#cient algorithms for ear decomposition and open ear decomposition, instead using centroid decomposition. We describe here the algorithm of Maon et al. for ....

....ties above. Least common ancestors can be computed using an algorithm of Schieber and Vishkin [61] which uses prefix computation on Euler tours of the tree. The remaining steps can be performed by further Euler tour computations together with centroid decomposition. Thus we have Theorem 16 [46]. Given a spanning tree of a bridgeless graph, and assuming that the ears need not be sorted, the remaining steps of the ear decomposition algorithm above can be computed in time O(log n) time with O( m n) log n) CREW processors. Maon et al. 46] also give an algorithm for finding an open ear ....

[Article contains additional citation context not shown here]

Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs. VLSI Alg. and Arch., Springer-Verlag LNCS 227, 1986, 34--45.

....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 ....

....and shared memory. In our parallel algorithms, we assume, without loss of generality, that the number of processors is a power of two and that number of processors evenly divides the number n of vertices in the input graph. 3. 1 PRAM Algorithm The PRAM algorithm for ear decomposition [24] uses an approach to that of the sequential algorithm but applies concurrent operations whenever possible. We next analyze the complexity bounds of the algorithm using the Concurrent Read Exclusive Write (CREW) PRAM Model [19] Step (1) requires a spanning tree algorithm, which can be found in ....

Y. Moan, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 47(3):277--296, 1986.

....[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 ....

.... 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 connected ....

Y. Maon, B. Schieber, and U. Vishkin, Parallel ear decomposition search (EDS) and st- numbering in graphs, Theoret. Comput. Sci. (1986), 277--298.

....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 [KR91a, KR90, MSV86, MR92, Ram93, RR89] Combined with an efficient parallel routine for finding connected components [AS87] and the Euler tour technique [TV85] this gives efficient parallel algorithms for several important problems on undirected graphs which include various connectivity problems [KR91a, FRT93, ....

.... with an efficient parallel routine for finding connected components [AS87] and the Euler tour technique [TV85] this gives efficient parallel algorithms for several important problems on undirected graphs which include various connectivity problems [KR91a, FRT93, MR92, Ram93] st numbering [MSV86] planarity testing and embedding [RR89] finding a strong orientation and finding a minimum cost spanning forest. Figure 10.2 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and the routine for finding connected ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoret. Comput. Sci., pages 277--298, 1986.

....[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 ....

.... 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 connected ....

Y. Maon, B. Schieber, and U. Vishkin, Parallel ear decomposition search (EDS) and st- numbering in graphs, Theoret. Comput. Sci. (1986), 277--298.

....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] or using the planar embedding algorithm of Tutte [17] ....

Y. Maon, B. Schieber, U. Vishkin, Parallel Ear Decomposition Search (EDS) and st-Numberings in Graphs, Theoretical Computer Science 47 (1986), pp. 277-296.

....i are oriented to form a path from u to v. If ear(u) ear(v) then all edges on P i are oriented to form a path from u to v if u comes before v in the orientation on P ear(v) and are oriented to form a path from v to u otherwise. G st is acyclic, and every vertex lies on a path from s to t [18]. We show that G st can be computed from G and D in logarithmic space. Orienting the edges in ear P i is easy if ear(u) #= ear(v) The routine shown in Figure 3 shows how to orient the edges if ear(u) ear(v) i # . It is clear that this can be implemented in L. 3.5 Constructing the local ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 47:277--296, 1986.

....even close to linear in the number of nodes and edges in the input graph. Our work is part of an effort by researchers to design fast and efficient parallel decompositions more suitable for use in parallel computation. Other such decompositions that have proven fruitful include ear decompositions [14, 15], and Euler tours [20, 2, 13, 6] 2 Finding a Maximal Set of Edge Disjoint Cycles We begin by reviewing an algorithm to find a maximal set of edge disjoint cycles in O(log n) time on (m n) log n processors. A slight variation on this algorithm will be an important part of our algorithm to ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. In VLSI algorithms and architectures, Lecture notes in computer science 227, pages 34--45. Springer-Verlag, 1986.

....of 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 ....

....(1.2) 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 ....

[Article contains additional citation context not shown here]

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (eds) and st- numbering in graphs. Theoret. Comput. Sci., 47:277--298, 1986.

....the connectivity and related problems as well) but also have done so using the weakest of the PRAM models. We note that among the problems having running times depending on the connectivity algorithm are ear decomposition [MR86] biconnectivity [TV85] strong orientation [Vis85] st numbering [MSV86] and Euler tours [AV84] Computing the MST of a weighted graph has attracted much attention in both the sequential and parallel settings. The best known sequential algorithm runs in time O(n 2 ) for dense graphs [Pri57] and in time O(m log 2 log 2 log d n) for sparse graphs [GGS89] where d = ....

....derive a connectivity algorithm from the MST algorithm we described, by assigning arbitrary distinct weights on the edges of the graph. In particular, we can assign weight(e) id(e) where id(e) is the id of the processor assigned on edge e. For the remaining problems, we note that the results in [MR86, TV85, Vis85, MSV86, AV84] use a connectivity algorithm as the most expensive subroutine. 5 Conclusions We have presented a new, simple and implementable parallel algorithm for computing the minimum spanning tree (MST) of an undirected weighted graph G = V; E) of n = jV j vertices and m = jEj edges on an EREW PRAM, the ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and s-t numbering in graphs. Theoretical Computer Science, 47:277-- 298, 1986.

....The end points of each ear are anchored on previous paths. Once an ear decomposition of a graph is found, it is not difficult to determine if two edges lie on a common cycle. This information can be used in algorithms for determining biconnectivity, triconnectivity, 4 connectivity, and planarity [52, 55]. An ear decomposition can be found in parallel using linear work and logarithmic depth, independent of the structure of the graph. Hence, this technique can be used to replace the standard sequential technique for solving these problems, depth first search. 2.4 Other techniques Many other ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (eds) and stnumbering in graphs. Theoretical Computer Science, 47:277--298, 1986.

.... time with linear or sub linear number of processors have been developed for several fundamental problems on undirected 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 ....

.... been developed for several fundamental problems on undirected 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 ....

Y. Maon, B. Schieber, and U. Vishkin, Parallel ear decomposition search (EDS) and st- numbering in graphs, Theoret. Comput. Sci. (1986), 277--298.

....any graph into a set of outerplanar subgraphs (called hammocks) We call this technique the hammock on ears decomposition. As the name indicates, our technique is based on the sequential hammock decomposition method of Frederickson [16, 17] and on the well known ear decomposition technique [31], and nontrivially extends our previous work for planar graphs [36] to any graph. We demonstrate its applicability by using it to improve the parallel (and in one case the sequential) bounds for a variety of problems in a significant class of graphs, namely that of sparse (di)graphs. This class ....

....of the dividing vertices, which by Lemma 3.5 is accomplished in O(log n) time using O(n m) CREW PRAM processors. Therefore, the time and processor bounds are dominated by the bounds of the connected components algorithm [5] which also dominates the bounds for finding an open ear decomposition [31]) and which are those stated in the theorem. We end this section by showing how to remove the restriction in algorithm Find Outgrowths as promised in the proof of Lemma 3.2. We remind the reader that the problem was that the algorithm correctly identifies an outerplanar outgrowth, if the first ....

[Article contains additional citation context not shown here]

Y. Maon, B. Schieber and U. Vishkin, Parallel ear decomposition search (EDS) and st-numbering in graphs, Theoretical Computer Science 47 (1986) 277-298.

....with the algorithms of Awerbuch, Israeli and Shiloach [AIS87] and Atallah and Vishkin [AV84] we obtain optimal randomized EREW PRAM algorithm for finding Euler tours in both directed and undirected Eulerean graphs. Our algorithm can be plugged into the algorithm of Maon, Schieber and Vishkin [MSV86] for finding ear decompositions and open ear decompositions yielding optimal randomized CREW PRAM algorithms for these problems. Concurrent reads are used by this algorithm for finding the LCA s of all the non tree edges using the algorithm of Schieber and Vishkin [SV88] As a consequence we also ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (eds) and st-numbering in graphs. Theoretical Computer Science, 47:277--298, 1986.

.... The techniques presented in this paper have been used to design new parallel algorithms for the minimum spanning tree problem [17] Other algorithms having running times that depend on the connectivity algorithm include the Euler tour on graphs [3, 4] biconnectivity [28] the ear decomposition [20, 22] and its applications on 2 edge connectivity, triconnectivity, strong orientation, s t numbering etc. See the surveys by Karp and Ramachandran [19] and by Vishkin [31] for more details on this. We should also mention that, with a minor modification our algorithm works on the weaker CREW PPM ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and s-t numbering in graphs. Theoretical Computer Science, 47:277--298, 1986.

....the input graph. Thus, in order to design fast and efficient parallel algorithms for graph theoretic problems, we must consider new, or at least different, types of graph decompositions and algorithmic techniques. Examples of such decompositions that have proven fruitful include ear decompositions [35, 36], and Euler tours [44, 8, 34, 17] examples of useful general techniques include divide and conquer [32, 28] and dynamic programming [1] The novel use of these decomposition and algorithmic techniques has led to efficient parallel algorithms for a variety of problems, and in some cases to ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. In VLSI algorithms and architectures, Lecture notes in computer science 227, pages 34--45. Springer-Verlag, 1986.

....technique known for designing sequential algorithms for graph problems. Unfortunately, it is not known how to implement DFS efficiently in parallel. A technique called ear decomposition search (EDS) was suggested as a replacement for DFS in the context of efficient and fast parallel algorithms [MSV86] and [MR86] after an earlier suggestion in [Lov85] for computing EDS in parallel in a fast but inefficient manner. The EDS method implies alternative algorithms for biconnectivity and strong orientation. More powerful applications were for finding an st numbering of a graph, again in [MSV86] ....

....[MSV86] and [MR86] after an earlier suggestion in [Lov85] for computing EDS in parallel in a fast but inefficient manner. The EDS method implies alternative algorithms for biconnectivity and strong orientation. More powerful applications were for finding an st numbering of a graph, again in [MSV86] as well as for triconnectivity algorithms [MR87] and [RV88] An st numbering is used in the planarity testing algorithm of [KR88b] The most recent algorithms for triconnectivity [FRT89] and planarity testing [RR89b] are very nice examples of reaching target problems by building an even ....

Y. Maon, B. Schieber, and U. Vishkin. Parallel ear-decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 47:277--298, 1986.

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Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs, Theoret. Comput. Sci., 47 (1986), 277--298.

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Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs, Theoret. Comput. Sci., 47 (1986), 277--298.

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Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs, Theoret. Comput. Sci., 47 (1986), 277--298.

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Y. Maon, B. Schieber, and U. Vishkin, Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoret. Comput. Sci., 47(3):277-298, 1986.

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Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (eds) and st-numbering in graphs. Theoretical Computer Science, 47:277--298, 1986.

No context found.

Y. Maon, B. Schieber, and U. Vishkin, Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs, Theoret. Comput. Sci., 47 (1986), 277--298.

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Y Maon, B Schieber, and U Vishkin. Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoretical Computer Science, 47:277--296, 1986.

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