J. Valdes, R.E. Tarjan, and E.L. Lawler, "The Recognition of Series Parallel Digraphs," SIAM J. on Computing 11 (1982), 298--313.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....leaves are associated with the subgraphs of G isomorphic to K 2 . If subgraphs H 1 and H 2 are associated with the descendants of a vertex corresponding to a subgraph H, then H is either a series or a parallel linkage of H 1 and H 2 . A linear algorithm for constructing such a tree is described in [7]. By using these tree, the recursive relations (1) and (2) and Lemma 3, one can successively compute the values Q(H; s; t; k) for all the subgraphs H of G. Since, by Lemma 3, the total number of these quantities is equal to O(n ) the computations can be implemented in O(n ) time. Finally, ....

J. Valdes, R. E. Tarjan, E. L. Lawler, The recognition of series parallel digraphs, SIAM J. Comput. 11 (1982) 298-313. 6

....s, destination vertex t, and with every vertex lying on some s t path. Then the following are equivalent: 1) G is vulnerable (2) G contains a subdivision of the network of Braess s Paradox (Figure 2.2) as a subgraph. By a well known forbidden subgraph characterization of series parallel graphs [57, 182], Fact 7.0.3 implies that the vulnerable graphs are precisely those for which the subgraph induced by the vertices lying on some s t path fails to be two terminal series parallel. This in turn implies that vulnerable graphs can be recognized in linear time [182] Fact 7.0.3 also shows that ....

....of series parallel graphs [57, 182] Fact 7.0.3 implies that the vulnerable graphs are precisely those for which the subgraph induced by the vertices lying on some s t path fails to be two terminal series parallel. This in turn implies that vulnerable graphs can be recognized in linear time [182]. Fact 7.0.3 also shows that vulnerable graphs are ubiquitous (the class of two terminal series parallel directed graphs is a restrictive one) a fact that could be useful in proving that most instances have # value greater than 1; see Open Question 1 above. This characterization of vulnerable ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11(2):298--313, 1982.

....benchmarks. We created two families, one satisfiable and one not, of scheduling instances with deadlines and precedence constraints. The precedence graphs were the same for both families, hierarchical structures based on N graphs, forbidden subgraphs of vertex series parallel dags see, e.g. [39] for further discussion. Deadlines were designed so that the satisfiable instances had only a small number of feasible solutions based on scheduling along specific critical paths first. The unsatisfiable instances di#ered only in the deadlines of two tasks, making them barely infeasible . Sizes ....

J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11:298 -- 313, 1982.

.... shown to be NP complete [15] Only for very restricted problems efficient algorithms are known: the execution lengths of all tasks are equal, the communication delays for information exchange are neglected, the number of processors is two [4] or special classes of precedence graphs are considered [9, 13, 16]. In parallel architectures large delays occur before the result of the execution of a task on one processor can be used by a task on another processor. If these communication delays are not neglected, the scheduling problems known to be solvable in polynomial time are even more restricted. The ....

....: G k ) is shown in figure 11. G 1 G 2 : G k Figure 10: SER(G 1 ; G k ) t s G k Figure 11: PAR(G 1 ; G k ) This definition is similar to the definitions of two terminal series parallel graphs by Lawler [10] and edge series parallel graphs by Valdes, et al. [16]. The class of series parallel graphs, as we will consider them, is almost a subclass of the class of two terminal series parallel graphs: the graph consisting of a single node is the only series parallel graph, that is not a two terminal series parallel graph. Two terminal series parallel graphs ....

J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11(2):298--313, May 1982. 26

....before the top one and that right parallel composition groups before the left one. Given such constraints the right binary representation tree is uniquely determined for each SP graph. An algorithm to recognize in linear time if a graph is an SP graph and to build its SPQ tree can be found in [5]. 5.1 The algorithm for SP graphs Our algorithm to test if a matching cut exists in a given SP graph relays on a simple post order traversal of the binary SPQ tree for such graph. Before describing the details of the algorithm we must state some definitions. We say that a matching cut is ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM J. Comput., 11(2):298--313, 1982. 13

....benchmarks. We created two families, one satisfiable and one not, of scheduling instances with deadlines and precedence constraints. The precedence graphs were the same for both families, hierarchical structures based on Ngraphs, forbidden subgraphs of vertex series parallel dags see, e.g. [39] for further discussion. Deadlines were designed so that the satisfiable instances had only a small number of feasible solutions based on scheduling along specific critical paths first. The unsatisfiable instances di#ered only in 15 Table 5: Three solver comparisons on PC class instances of the ....

J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11:298 -- 313, 1982.

....before the top one and that right parallel composition groups before the left one. Given such constraints the right binary representation tree is uniquely determined for each SP graph. An algorithm to recognize in linear time if a graph is an SP graph and to build its SPQ tree can be found in [5]. 5.1 The algorithm for SP graphs Our algorithm to test if a matching cut exists in a given SP graph relays on a simple post order traversal of the binary SPQ tree for such graph. Before describing the details of the algorithm we must state some definitions. Wesay that a matching cut is ....

J. Valdes,R.E.Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM J. Comput.,11(2) 1982. 13

....benchmarks. We created two families, one satisfiable and one not, of scheduling instances with deadlines and precedence constraints. The precedence graphs were the same for both families, hierarchical structures based on Ngraphs, forbidden subgraphs of vertex series parallel dags see, e.g. [31] for further discussion. Deadlines were designed so that the satisfiable instances had only a small number of feasible solutions based on scheduling along specific critical paths first. The unsatisfiable instances di#ered only in the deadlines of two tasks, making them barely infeasible . Sizes ....

J. Valdes, R. Tarjan, and E. Lawler, The recognition of series parallel digraphs,SIAMJournal on Computing, 11 (1982), pp. 298 -- 313.

....1 generalizations of the problem. The cotree decomposition of cographs and the series parallel decomposition of series parallel partial orders are special cases on graphs and digraphs, respectively, for which linear time solutions have been given, see Corneil et al. 1985) Valdes et al. 1982). O(n m log n) bounds for arbitrary undirected graphs were then given in Cournier and Habib (1993) and an O(n m (m;n) bound was given in Spinrad (1992) The rst linear time algorithm was given in McConnell and Spinrad (1994) An altogether di erent linear time was given shortly thereafter in ....

Valdes, J., Tarjan, R. E., , and Lawler, E. L. (1982). The recognition of series-parallel digraphs. Siam J. Comput., 11:299-313. 26

.... takes time O(2k.k(k 1) 2) Since each iteration step removes a node, there are at most n iteration steps, yielding a total of n # k=1 k 2 (k 1) steps, which is O(n 4 ) Valdes Tarjan Lawler An early improvement on the naive approach is the Valdes Tarjan Lawler (VTL) algorithm [76] which has execution time O(m n) This algorithm can be applied to any directed acyclic graph, whereas most other algorithms require that the dag be represented either as its equivalent transitive reduction or its transitive closure. The VTL algorithm proceeds by first performing a ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11(2), pp. 298--313, May 1982.

....of any SP order is unique modulo the associativity of 4; and the commutativity of . We write T = for the corresponding quotient set, which is in bijection with the class of SP orders. We end this section by giving a famous characterisation of SP orders in term of forbidden conguration [3] (see [5] for an alternative proof) Denition 2.3 A digraph R = VR ; ER ) is said to be N free whenever its restriction to any four element set E = fa; b; c; dg ae VR is dioeerent from (E; f(a; b) c; d) a; d)g) Proposition 2.4 The class of series parallel order is exactly the class of ....

E.L. Lawler, R.E. Tarjan, and J. Valdes. The recognition of Series-Parallel digraphs. SIAM Journal of Computing, 11(2):298313, May 1982.

....work for evaluating computation tasks to examine if a task digraph is an AOSP digraph. Previously, the recognition for Edge Series Parallel (ESP) digraphs (sometimes called two terminal series parallel digraphs) which arise in the analysis of electrical networks [12] 14] was proposed in [15]. The ESP digraph is a special case of the AOSP digrpah, and contains only two types of subgraphs: sequential and Fork to Join. Namely, ESP digraphs do not take the logic structures among modules into consideration. Obviously, ESP digraphs can not satisfy modern varieties of distributed ....

....AOSP digraphs. For our overall algorithm, the input is a task digraph with a set of Boolean formulas attached to vertices. The ow chart of the overall recognition algorithm is depicted in Figure 6. This algorithm is generalized from the recognition algorithm for ESP digraphs proposed in [15]. The latter comprises the series reduction and the parallel reduction. By applying series and parallel reductions until no more is applicable, an ESP digraph will be reduced to a digraph with only one single arc, but other digraphs will not. We redesign the 9 Given a digraph G= V(v) E,F(v) ....

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J. Valdes, R. E. Tarjan and E.L. Lawler, \The recognition of series parallel digraphs, " Siam J. Comput., vol. 11, pp. 298-313, May 1982.

.... by joining s 1 , t 1 with, respectively, s 2 , t 2 (a parallel join) is also series parallel with terminals s 1 , t 1 (see [Duf65] Series parallel graphs have received much attention because hard problems on networks tend to become easy when restricted to them (see [Hof88] TNS82] [VTL82], Win86] The fastest known algorithm for solving the linear cost network flow problem on series parallel graphs is that due to Bein, Brucker and Tamir [BBT85] with a time bound of O(nm m log m) where n is the number of nodes and m is the number of arcs. Below, we will apply Algorithm 2 to ....

Valdes, J., Tarjan, R. E. and Lawler, E. L., "The recognition of series parallel digraphs," SIAM J. Comput. 11 (1982), 298--313.

.... O(n 2 ) and even with volume O(n) 1 2 Preliminaries For the basic graph theoretical concepts we refer to [13] and for algorithms and their notations to [6] Notations and de nitions for the three dimensional graph drawing are mostly taken from [4, 2] and for series parallel digraphs from [1, 3, 14]. The Fary grid drawing of a graph is a three dimensional drawing where vertices are placed at integer coordinates, edges are straight lines and crossings of edges are not allowed. A drawing of an acyclic digraph is upward, if each edge is drawn as a curve monotonically nondecreasing in the prede ....

....digraph obtained by identifying s 1 ; s k into a single vertex s and identifying t 1 ; t k into a single vertex t. Throughout this paper, we assume that there is no parallel edges. A seriesparallel digraph G is associated with a rooted tree T , called SPQ tree or decomposition tree [14, 1, 3]. There are three types of nodes (S , P and Q nodes) in a decomposition tree: 1. If G is single edge, then T consists of a single Q node. 2. If G is created by the parallel composition of series parallel digraphs G 1 ; G k with decompositions trees T 1 ; T k , then the root of ....

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J. Valdes, R. Tarjan, and E. Lawlers. The recognition of series parallel digraphs. SIAM J. Comput., 11(2):298-313, 1982. 13

....tree in which the leaves represent distinct instances of the base graph (BG) and the internal nodes denote either a series (S) or a parallel (P) connection. Both sequential and parallel algorithms for recognizing a series parallel graph and obtaining its decomposition tree exist in the literature [13, 23]. 3 The minimum vertex cover problem Let G = V; E) be a graph under the usual notation convention, where V represents the set of vertices and E the set of edges. A covering set of G is a subset C V such that for any edge (i; j) 2 E, fi; jg C 6= The minimum vertex cover problem, also ....

....our variant, a binary tree, representing the decomposition tree of a TTSP graph, is the input to the problem. Obtaining the decomposition tree from the original TTSP graph is not the task of the real time algorithm. This is obtained by a preprocessing step, using any one of the existing algorithms [13, 23]. Let G = V; E) be a TTSP graph with terminals T 1 and T 2 . The cardinality of the minimum covering set of G is computed at the root of the parse tree, after processing the tree in a bottom up fashion. The result of this computation is a four element integer vector X, whose components are ....

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J. Valdes, R.E. Tarjan and E.L. Lawler, "The recognition of series parallel digraphs", SIAM J. Comput. Vol.11, No. 2, May 1982, pp. 298-313 17

....the left child precedes the right one. Thus, the decomposition tree is ordered. However the order between children of a parallel composition has no importance. Clearly, decomposition trees are not unique, as it is possible to have ties between successive compositions of the same type. In [12] a linear time algorithm was presented for recognizing the general class of series parallel graphs and constructing a binary decomposition tree. However, their algorithm breaks ties between successive compositions of the same type in an arbitrary way. Another algorithm was presented in [10] for ....

J. Valdes, R. E. Tarjan, E. L. Lawler, "The Recognition of Series Parallel Digraphs ", SIAM J. of Computing, 11, (1982) 298-313.

....use of a regular prefix free language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree. The advantage to represent graphs by trees is that properties of graphs can be verified by induction on the tree, often leading to efficient algorithms [4, 8, 10, 13]. In section 2 we give some preliminary definitions. In section 3 the graph representation is introduced, some properties of this representation are shown and the relationships between graph substitution and language concatenation is stated. The main result of section 4 is the proof that the ....

J. Valdez, R. E. Tarjan and E. Lawler, "The recognition of series parallel digraphs", SIAM Journal of Computing, 11 (1982) 298--313.

....in the case of 2 structures has been known as the prime tree family [2] but we will use the term modular decomposition for consistency with prior terminology on graphs. The cotree decomposition of cographs [14] and the series parallel decomposition of General Series Parallel partial orders [15] are the special cases of the decomposition on these graph classes. A definition of a module that satisfies the requirements of the algebraic model of [5] exists for Boolean functions, set systems, k ary relations [5] and k structures [16] so a modular decomposition exists for these structures as ....

J. Valdes, R.E. Tarjan, , and E.L. Lawler, The recognition of series-parallel digraphs, Siam J. Comput., 11 (1982), 299--313.

....operations. 5. 3 Series parallel languages The study of series parallel languages, or sp languages, from the point of view of rationality and regularity was initiated in [8] Series parallel languages play an important role in theoretical computer science, and we refer the reader to [10] and [15] for more details on this connection. In this section, we tie together the results of this paper with those in [8, 9, 10] As in Section 5.2, we consider = fkg and a theory T = E) but E now asserts the associativity and commutativity of the parallel product. The free T algebra over A is ....

J. Valdes, R.E. Tarjan and E.L. Lawler. The recognition of series-parallel digraphs, SIAM J. Comput. 11 (1981) 298-313. 32

....operations. 5. 3 Series parallel languages The study of series parallel languages, or sp languages, from the point of view of rationality and regularity was initiated in [8] Series parallel languages play an important role in theoretical computer science, and we refer the reader to [10] and [15] for more details on this connection. In this section, we tie together the results of this paper with those in [8, 9, 10] As in Section 5.2, we consider Sigma = fkg and a theory T = Sigma; E) but E now asserts the associativity and commutativity of the parallel product. The free T algebra ....

J. Valdes, R.E. Tarjan and E.L. Lawler. The recognition of series-parallel digraphs, SIAM J. Comput. 11 (1981) 298--313. 33

....only compatible triples may be combined. For compatible triples, l , c , Y) is obtained as below: l : max l 1 , l 2 l : c 1 c 2 u Y1 S w(u) Y : Y 1 Next, l , c , Y) is added to Z. 3.4. Complexity The series parallel decomposition of an SPDAG can be determined in O(n) time [VALD79]. By keeping each F(G i ) as four separate lists of triples, one for each of the four possible values for the third coordinate of the triples, F(G 1 G 2 ) and F(G 1 G 2 ) can be obtained in O( #F(G 1 )# #F(G 2 )# ) time from F(G 1 ) and F(G 2 ) Since F(G 1 ) F(G 2 ) contains only non ....

....u V(G) S w(u) For SPDAGs with unit delay or unit weight, this is O(n 2 ) 12 4. General Series Parallel Dags General series parallel dags (GSPDAGs) were introduced in [LAWL78, MONM77, SIDN76] A linear time algorithm to determine whether or not a given dag is a GSPDAG was developed in [VALD79]. This paper also contains a linear time algorithm to obtain a series parallel decomposition of a GSPDAG. The definitions and terminology used in this section are derived from [VALD79] A transitive dag is a dag G = V , E) such that i , j E whenever there is a path from i to j . The ....

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J. Valders, R. E. Tarjan, and E. L. Lawler, "The recognition of Series Parallel digraphs", SIAM J. Comput., 11 (1982), pp. 298-313.

.... 30 75 60 44 85 Table 5: Arc weights for the 29 nodes 37 arcs graph 15 Unfortunately there is not a polynomial algorithm able to delete the smallest number of arcs in a graph in such a way to obtain a series parallel graph, even if we can recognize series parallel graphs in polynomial time (see [22]) A possible heuristic to improve the algorithm can therefore start from a spanning rooted tree of G and then add to G as many arcs as possible by preserving the property of being series parallel. A further improvement to the algorithm can be obtained by using the results of Sidney [21] In fact, ....

J. Valdes, R. E. Tarjan and E. L. Lawler, The Recognition of Series Parallel Digraphs, SIAM J. Comput., 11, 298--313, 1982.

....with a 1 and b 2 as its terminals. This graph is the series composition of G 1 and G 2 . 2. The graph obtained by identifying a 1 and a 2 and also b 1 and b 2 is a series parallel graph, the parallel composition of G 1 and G 2 . This graph has a 1 ( a 2 ) and b 1 ( b 2 ) as its terminals. In [VTL82] the authors present a linear time algorithm to decide whether a given digraph is seriesparallel, and if this is true, produce a parse tree (or decomposition tree) specifying how G is constructed using the above rules. The size of the parse tree is linear in the size of the input graph. The ....

....as guess value for the optimum cost in the final algorithm. Notice that the optimum fixed cost is an integer between 0 and mC. For 0 b B we define CG (b) to be the maximum flow that can be achieved by using edges of total cost no more than b. In our algorithm we first use the algorithm from [VTL82] to obtain a decomposition tree for the input graph G in time O(n m) We then use dynamic programming and the decomposition tree to compute all the values CG (b) f = 0; B in O(mB 2 ) time. Clearly, if G consists of just the two vertices s and t joined by an edge (s; t) we can ....

J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series-parallel digraphs, SIAM Journal on Computing 11 (1982), no. 2, 298--313.

....directed graphs. Cohen [5] gave an O(log 4 n) time, O(n 2 log n) work CREW PRAM algorithm for all pairs shortest paths in planar directed graphs. An O(log n) time, O(n 2 ) work EREW PRAM algorithm for all pairs shortest paths in series parallel digraphs can be obtained from the results of [33] (by using the parallel tree contraction technique [14] Note that a seriesparallel digraph is a directed acyclic graph with exactly one source and exactly one sink, such that the graph can be constructed by series and parallel compositions. Series parallel digraphs are a special case of planar ....

....in their polylogarithmic running time is rather 2 (a) b) s t s t Figure 1: a) A planar directed grid graph. b) A corresponding planar layered digraph. large. An O(log n) time, O(n) work EREW PRAM algorithm for single source shortest paths in series parallel digraphs was presented in [33]. There are a few parallel algorithms for computing a shortest path between one pair of vertices in certain graphs. Aggarwal and Park [1] and Apostolico et al. 2] obtained an O(log 2 n) time, O(nlog n) work CREW PRAM algorithm for finding a source to sink shortest path in planar directed ....

J. Valdes, R. E. Tarjan, and E. L. Lawler, "The recognition of series parallel digraphs," SIAM J. Comput., 11 (1982), pp. 298--313.

....1 2 3 (a) 4 8 5 6 7 (b) S 3 2 1 4 P S P S S 6 P 8 5 7 Figure 1: a) Example of an SP1 graph and (b) the corresponding decomposition tree. and G 0 2 such that G 0 1 is a proper subgraph of G 1 . Such a decomposition tree for the SP1 graph of Figure 1(a) is illustrated in Figure 1(b) In [12], Valdes, Tarjan and Lawler presented an algorithm of linear time (in the number of vertices and arcs) for recognizing the general class of series parallel graphs and for constructing a binary decomposition tree and a canonical decomposition tree where successive compositions of the same type are ....

....be used for the construction of our minimal binary decomposition tree. 5 8 3 2 1 4 P S P S 5 (a) S S 6 P 8 7 3 2 1 4 P S P S 6 P 7 (b) Figure 2: Decomposition trees by the algorithm of Valdes Tarjan Lawler. a) binary decomposition tree. b) canonical decomposition tree. The algorithm of [12] also determines whether a graph is series parallel . If it is not, clearly it is not series parallel 1 either. If yes, one obtains the canonical decomposition tree. In order to determine whether the graph is SP1, it suffices to check the children of S ( series operation) nodes. Indeed, for a ....

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J. Valdes, R. E. Tarjan, E. L. Lawler, "The Recognition of Series Parallel Digraphs", SIAM J. of Computing, 11, (1982) 298-313.

.... The dag shown in Figure 1 is called the interdictive graph (IG) It is well known that an st dag G is series parallel if and only if there are no IGs in G [5] Figure 1: The interdictive graph (IG) When we say an st dag is series parallel we mean that it is two terminal edge series parallel (see [10] for a discussion of the relationship between vertex and edge series parallelity) The series parallel st dags are recursively defined as follows: ffl An st dag having a single edge e is two terminal series parallel (with s(e) s and t(e) t) ffl If G 1 = V 1 ; E 1 ) and G 2 = V 2 ; E 2 ) ....

J. Valdes, R. Tarjan, and E. Lawler, The recognition of series parallel digraphs, SIAM J. Comput., 11 (1982), pp. 298--313.

....within the subroutine (Algorithm 2) see also Figure 1. Assuming that the activity on arc diagram is stored by adjacency lists the worst case complexity is O(jAj (conv(sp) prod(sp) jAj 2 ) Although it takes only linear time to decide whether a given network is series parallel [20], Dodin s algorithm requires more effort. The problem is that once a reduction step has been performed one has to check if new parallel arcs have been created (if statement in Algorithm 2) However, the average performance turns out to be much better, in particular, the bounds of Kleindorfer and ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM Journal on Computing, 11:298--314, 1982.

....notation, we give the definitions for arbitrary communication delays. Since we suppose that the precedence constraints are given by a series parallel order, we also introduce the necessary definitions and notations for series parallel orders here. A comprehensive overview can be found in [12] and [17]. 2.1 Notations in Scheduling Theory An instance I = m; V; p; Theta; c) for the problem P jseries parallel; p j ; c ij jCmax consists of the number m of machines, a set V of jobs, a series parallel precedence order Theta on the set V of jobs, a processing time p(v) 1 for every job v 2 V , ....

....(V; OE) consisting of a set V and a strict order relation OE on V , i.e. a transitive and asymmetric binary relation, denoted by u OE v. There exist several slightly different definitions of series parallel orders, such as single source single target parallel and edge seriesparallel orders (see [17, 12]) These are special cases of the following order theoretic definition. A partial order Theta is called series parallel if it can be obtained recursively from singletons by two operations, the series composition and the parallel composition of two (series parallel) sub orders. ffl The smallest ....

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Jacobo Valdes, Robert E. Tarjan, and Eugene L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11(2):298--313, 1982.

....20, 22, 26, 33] some of them for special cases or generalizations of the problem. The cotree decomposition of cographs and the series parallel decomposition of series parallel partial orders are special cases on graphs and digraphs, respectively, for which linear time solutions have been given [7, 34]. O(n m log n) 8] and O(n mff(m; n) 31] bounds for arbitrary undirected graphs have recently been given. Here, we give a modification of the algorithm of [31] that eliminates the ff(m; n) factor in the time bound for modular decomposition, giving a linear time bound for the problem. A ....

J. Valdes, R.E. Tarjan, , and E.L. Lawler. The recognition of series-parallel digraphs. Siam J. Comput., 11:299--313, 1982.

....alphabets (fag; f(a; a)g) that is closed under taking finite direct sums and complex products. The trace monoids generated by dependence alphabets from D form the class of cograph (trace) monoids, denoted by M. There is a well known graph theoretical characterization of cographs, see [10] [11] or [12] They are precisely undirected (loop less) graphs ( Sigma ; D Gamma f(a; a) j a 2 Sigma g) containing no induced subgraph isomorphic to L 3 = a Gamma b Gamma c Gamma d] A description for a dependence alphabet ( Sigma ; D) is a well formed linear term built from the constants a 2 ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM Journal of Computing, 11(2):298--313, 1981.

....with relatively free word order, while it is so diOEcult, in the Lmabek calculus to obtain the same syntactical analysis for two constructions, where two contiguous constituents are permuted. In fact we are able to do so at least for each such phenomena involving serie parallel orders, see e.g. [LTV82] which precisely correspond to orders which may be described with and , see [Ret93, Ret95] 2.3.3 Incremental strategy Modules can be plug in any order, but it is natural to rst try the order in which they appear, preserving this nice property of categorial grammar. Nevertheless it is ....

E.L. Lawler, R.E. Tarjan, and J. Valdes. The recognition of Series-Parallel digraphs. SIAM Journal of Computing, 11(2):298313, May 1982.

....to compute the resistance of an (electrical) network of resistors assumes that the underlying graph is in fact a series parallel graph. A well studied problem is the problem to recognise series parallel graphs. A linear time algorithm for this problem has been given by Valdes, Tarjan, and Lawler [11]. Also, it is known that when a decomposition tree for a series parallel graph is given, then many problems can be solved in linear time, including many problems that are NP hard for arbitrary graphs [2, 5, 9, 10] Valdes et al. also show how to obtain such a decomposition tree in linear time. ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11:298--313, 1982.

....specified circumstances. It is also shown that Timing Models correlate better with speed values from experiments than the popular model proposed by Amdahl. This dissertation formally defines an SCTF graph, and utilizes a Forbidden Subgraph (see section 3.1.1. 3) to determine if a graph is SCTF [99]. The translation strategy from SCTF graph to CDT is outlined. A set of template rules are developed for component speed, availability, and performability. These templates are responsible for integrating Microanalysis results into application dependent structures, resulting in a performability ....

....simulation. The goal in this dissertation is an analytical model that is complete as well as unified, scalable and verifiable. 2.3. 3 Series Parallel Graphs and Performability Speed and availability were modeled using a series parallel graph [98] which is converted into a binary decomposition tree [99] for probabilistic analysis in [79] This latter work inspired Macroanalysis, which is presented in section 3.3. 31 2.3.3.1 Terminology and Definitions The following terms and definitions are used throughout the dissertation, and collected here for convenience. A graph G = V, E consists of a ....

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Valdes, J., Tarjan, R.E., and Lawler, E.L. "The recognition of series parallel digraphs." SIAM Journal of Computing, Vol. 11, No 2, May 1982, pp. 298 - 313.

....function. The output of the function is true if and only if the corresponding arc is on a shortest path from the source to the destination (see [4, 16] In this paper we consider Series Parallel Graphs (SPGs) a particular topology widely studied in literature for several types of problems (see [3, 12, 13, 14, 25]) We show how to apply Interval Routing Schemes to directed SPGs, and we introduce a new more efficient technique for directed and undirected SPGs, the Distance Routing (DR) This technique is similar to Boolean Routing, and Prefix Routing, however, the PRS with their classical definition cannot ....

....to the DSPG, so that it is possible to cycle back and w.l.o.g. we also remove all the multiple arcs. We now show an interesting property of SPGs: v z u v u u v ii) i) v u e e e e e e 2 1 2 1 Figure 1: Directed Series and Parallel Composition. Figure 2: The forbidden graph for DSPGs. Theorem 1 [3, 25] Let G be a directed acyclic graph with one source and one sink. G is a DSPG iff G does not contain a subgraph homeomorphic to the one in figure 2. In a DSPG, a node is an opening node iff it has out degree greater than one; a node is a closing node iff it has in degree greater than one. An arc ....

J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series-parallel digraphs. SIAM Journal of Computing, 11(2):298--313, 1982.

....been published in preliminary form in [7, 5] The idea of reduction algorithms originates from Duffin s [12] characterization of seriesparallel graphs: a multigraph is series parallel if and only if it can be reduced to a single edge by applying a sequence of series and parallel reductions. In [18] it was shown how a reduction algorithm based on this set of reduction rules can be implemented in linear time, and hence series parallel graphs can be recognized in linear time. Arnborg and Proskurowski [4] extended these ideas, and obtained reduction rules that characterize the graphs of ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11:298--313, 1982.

.... T 9 T 10 T 13 9,10 15,11 19 26 13 3 T 15 and predecessors T i , i 8 28 35 28 35 14 3 T 15 and all predecessors 87 87 We now consider the time complexity of this scheduling algorithm. The generalized seriesparallel graph decomposition can be computed in linear time [Valdes78] [Valdes79]. Lemma 4.13 is applied at most n times and it takes O (1) time to compute the costs for this lemma. Lemmas 4.14 and 4.15 are together applied exactly n 1 times and they each require O (n) time to update the cost vectors. The step of scheduling the AND only graph takes O( A n log n) time. ....

Valdes, Jacobo, Robert E. Tarjan and Eugene L. Lawler. The Recognition of Series Parallel Digraphs. Proceedings of the 11th Annual ACM Symposium on Theory of Computing (1979) pp. 1-12.

....G 1 jjG 2 , the parallel composition of G 1 and G 2 , is the union of G 1 and G 2 , where s 1 is identified with s 2 and t 1 is identified with t 2 . G is a serial parallel (sp) network, if it can be constructed using only the operations = and jj from networks containing a single link. See also [7, 16]) Lemma 3: The update procedure in Section 2.2, when applied to an sp network G of jEj links, produces an sp network of at most jEj Gamma 1 links. For any e 2 E, we denote by G Delta e and G Gamma e the networks resulting from the contraction and deletion of e respectively. The proof of the ....

J. Valdes, R.E. Tarjan and E.L. Lawler, "The Recognition of Series Parallel Digraphs", In Proc. of the 11th Annual ACM Symposium on Theory of Computing, 1979. Itai, Shachnai: Adaptive Source Routing.. 25

....nodes are the leaves of SPQ(G) Fig. 4.11(b) shows a SPQ tree that is associated with the series parallel graph shown in Fig. 4.11(a) Notice that if G has no multiple edges, at most one child of a P node is a Q node. Given G, SPQ(G) can be constructed in linear time using the algorithm given in [108]. s=s 1 (b) e) c) s t (a) d) G 1 G 2 Q G 1 G 2 G 1 G G G G SPQ(G 1 ) SPQ(G 1 ) S G G 2 (f) G 1 SPQ(G 1 ) SPQ(G 1 ) P G G 2 t 1 =s 2 t=t 2 t=t 1 =t 2 s=s 1 =s 2 Figure 4.10: A series parallel graph G when it is: a) a single edge (s; t) b) a seriescomposition ....

....drawing with angular resolution at least 1= 48 d 2 ) that can be constructed in O(n) time. Proof: Let G 0 be a series parallel graph with degree d and n vertices. Construct a compact SPQ tree T (G) associated with G 0 , by first constructing SPQ(G 0 ) using the O(n) time algorithm of [108], and then compacting SPQ(G 0 ) in O(n) time using the compacting procedure described earlier. Let Delta E be an equilateral triangle. Construct Drawing D(G 0 ; Delta E ) using the recursive algorithm given above. It is easy to see that the algorithm executes in O(n) time, giving a total ....

J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM J. Comput., 11(2):298--313, 1982.

....) t 2 t 3 c c c # # # b b b ae ae t 5 P( 4 ) t 4 S( 3 ) Figure 2: Binary decomposition tree time function for any such graph (with n nodes) can be computed in O(np 2 ) time. A number of different but equivalent definitions of series parallel graphs exist. The one we will use is taken from [34], in which a series parallel DAG can be parsed as a binary decomposition tree (BDT) in time proportional to the number of edges. The leaves of such a tree correspond to the DAG nodes themselves and internal tree nodes describe either parallel (P) or series (S) compositions. Figure 2 illustrates ....

J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11(2):298--313, May 1982.

.... decompose is based on the observation that any s; t numbering will suffice [He] Efficient methods for finding biconnected components and computing s; t numberings are known from [Ta,ET] Techniques for determining whether directed graphs are 2TSP and finding decomposition trees can be found in [VTL]. All these algorithms are linear in n and the number of edges; thus decompose runs in O(n) time. 3.2 Algorithm components Algorithm components finds the three edge connected components of a series parallel multigraph in linear time. The input to components is a series parallel graph and a ....

J. Valdes, R.E. Tarjan, and E. Lawler, "The Recognition of Series-Parallel Digraphs," SIAM Journal on Computing 11 (1982), 298--313.

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J. Valdes, R.E. Tarjan, and E.L. Lawler, "The Recognition of Series Parallel Digraphs," SIAM J. on Computing 11 (1982), 298--313.

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Valdes, J., T. R. & Lawler, E. (1982), `The recognition of series parallel digraphs', SIAM Journal of Computing 11(2), 298--313.

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J. Valdes, R.E. Tarjan, and E.L. Lawler, "The recognition of series-parallel digraphs", SIAM Jour. Comp. 11 (1982) pp. 298-313.

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J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series parallel digraphs, Proc. 11 th ACM Symp. on Theory of Computing (STOC),Atlanta, ACM, 1979, pp. 1---12; SIAM J. Comput. 11 (1982) 298---313; MR0564616 (81d:68088) and MR0652904 (84d:68073)

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J. Valdes, R.E. Tarjan, and E.L. Lawler. The recognition of series parallel digraphs. SIAM Journal of Computing, 2(11):298--313, 1982.

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J. Valdes, R.E. Tarjan, and E. Lawler, "The Recognition of Series-Parallel Digraphs," SIAM Journal on Computing 11 (1982), 298--313. 37

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E. L. Lawler J. Valdes, R. E. Tarjan. The recognition of series parallel digraphs. SIAM J. Comput., 11(2):298--313, 1982.

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Valdes, J.R., Tarjan, E. and Lawler E.L. (1982). The recognition of series-parallel digraphs. SIAM Journal on Computing, 11, 361-370.

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J. Valdez, R. E. Tarjan and E. Lawler, \The recognition of series parallel digraphs", SIAM Journal of Computing, 11 (1982) 298-313. .

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Valdes, J., Tarjan, R. E., and Lawler, E. L. (1982). The recognition of series parallel digraphs. SIAM J. Comput., 11:298--313.

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