K. Takamizawa, T. Nishizeki, and N. Saito, "Linear Time Computability of Combinatorial Problems on Series Parallel Graphs," J. ACM 29 (1982), 623--641.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a single top down traversal over an SP Term. 2. 1 SP Terms for Control Flow Graphs of Flowchart Programs Algebraic Construction of Series Parallel Graphs Control flow graphs of flowchart programs are graphs with tree width at most 2, which are known as series parallel directed graphs (digraphs) [32]. Such graphs can be specified in terms of SP Term. Note that the following definition is somewhat simplified compared to that in Section 4.1 for general cases. Constants (e (e , l 1 , l 2 ) # 2 Series composition 2 1 l 1 # 2 l # 1 # # S l 1 l 2 # # # # # ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. Journal of the Association for Computing Machinery, 29:623--641, 1982.

....we will show that the catamorphic framework in Section 2 can be applied on control flow graphs by the use of the algebraic construction, called SP Term. Before entering to the general and formal study in Section 4, this section introduces the simple case study, series parallel graph [35]. This class of graphs corresponds to the control flow graphs of structured programs (in strict sense) i.e. programs consist of single entry and single exit blocks. 3.1 SP Terms for Representing Structured Control Flow Graphs Algebraic Construction of Series Parallel Graphs A control flow graphs ....

....flow graphs of structured programs (in strict sense) i.e. programs consist of single entry and single exit blocks. 3. 1 SP Terms for Representing Structured Control Flow Graphs Algebraic Construction of Series Parallel Graphs A control flow graphs of flowchart scheme are series parallel graphs [35], which are graphs with tree width at most 2. We give the transformation from the control flow graph to a SP Term. Note that this definition is somewhat simplified compared to the definition in Section 4.3 for general cases. Definition 1. An SP Term is a pair of a ground term t and a tuple (l 1 ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on seires-parallel graphs. Journal of the Association for Computing Machinery, 29:623--641, 1982.

.... H, K, L, M A, F, H, M A, F, G, H Figure 1: Tree decomposition of a planar graph. 4 Bounded Tree Width Subgraph Isomorphism As a subroutine, we need to perform subgraph isomorphism testing in graphs of bounded tree width. This can be done by a standard dynamic programming technique [9, 46]. The exact statement of the problem we solve is complicated by the requirement that we count or list each subgraph isomorph exactly once. For simplicity, we state the bounds for this problem with one parameter measuring both the tree width of the text and the size of the pattern. De nition 1 A ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach. 29:623-641, 1982.

....called data constructors applying to an element of type and a bounded number of recursive components. Though seemly restricted, these polynomial data types are powerful enough to cover our commonly used data types, such as lists, binary trees, rooted trees [BLW87] and series parallel graphs [TNS82] Moreover, other data types like the rose trees, a kind of regular data type de ned by RTree = Node [RTree ] can be encoded into one of these polynomial data types. This will be demonstrated in Chapter 4. 14 For each data constructor C i , we de ne F i by F i f (e; x 1 ; x n i ....

....graphs. 5.10.2 Generality Our approach is general (polytypic) enough to deal with maximum marking problems on data structures that are not lists or trees. To illustrate this, we derive a linear time algorithm solving the maximum two disjoint paths problem on series parallel graphs [TNS82] The series parallel graph is de ned as follows: SPG : Base (Vert; Vert; Edge) j Series SPG SPG j Parallel SPG SPG Here Vert represents the type of vertices and Edge represents the type of edges. Every graph should have a single source and a single sink. The two data constructors are ....

[Article contains additional citation context not shown here]

Kazuhiko Takamizawa, Takao Nishizeki, and Nobuji Saito. Linear-time computability of combinatorial problems on seriesparallel graphs. Journal of the Association for Computing Machinery, 29:623-641, 1982.

....of the resulting algorithms and also at the expense of increased algorithm time complexity as a function of k. In contrast, results giving explicit practical algorithms in this setting are usually limited to a few selected problems on either (full) k trees [9] partial 1 trees or partial 2 trees [25]. We intend to cover the middle ground between these two extremes, by investigating the time complexity as a function of both input size and the treewidth k. We assume that the input graph is given with a width k tree decomposition, computable in linear time for fixed k [6] Our algorithms employ ....

K.Takamizawa, T.Nishizeki and N.Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. ACM 29(1982) 623-641.

.... graph obtained 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 ....

Takamizawa, K., Nishizeki, T. and Saito, N., "Linear--time computability of combinatorial problems on series--parallel graphs," Journal of A. C. M. 29 (1982), 623--641.

....problem : given a graph G and a fixed pattern graph H, one wants to determine the maximum number of vertex disjoint copies of H in G. The problem is known to be NP complete if H contains more than three vertices [10] but for some restricted class of graph, it can be computed in linear time [14]. It is known that such e#ciency can be achieved by decomposing the graph G into a tree, followed by a dynamic programming [4] Chemical structures do not confine themselves in a well behaving class of graphs, but the number of cycles is limited, and the degree of nodes is bounded. Moreover, we ....

Takamizawa, K., Nishizeki, T., and Saito, N., Linear-time computability of combinatorial problems on series-parallel graphs, J. Assoc. Comput. Mach., 29(3):623--641, 1982.

....Usually, if the defined algebra does not have a finite carrier, then some conditions must be fulfilled in order to evaluate the parse tree by applying a parallel tree contraction algorithm. For more details about these conditions, see [1] 1. 2 Contributions of this paper Takamizawa et al. [21] showed in a unified manner that for many combinatorial problems on graphs, including minimum vertex cover, there exist algorithms whose running time is linear in the size of the input, if the graphs under consideration are restricted to the class of series parallel graphs. In this paper we show ....

....is an NP complete problem [9] In our case, however, the input to the problem is a TTSP graph and we are only interested to find the cardinality of the minimum covering set, or the best possible approximation of it. Under these restrictions the problem can be solved in sequential linear time (see [21] for example) In 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 ....

K. Takamizawa, T. Nishizeki and N. Saito, "Linear-time computability of combinatorial problems on series-parallel graphs", J. Assoc. Comput. Mach., 29 (1982), pp. 623-641.

....4, we conclude with some remarks on future research. 2 Series Parallel Graphs All graphs considered in the paper, unless otherwise stated, are directed graphs (digraphs) The class of series parallel graphs is known as a class for which several scheduling problems are polynomially solvable [1, 6, 11], while the same problems are NP complete for a general graph. Here, we are interested in a subclass of seriesparallel graphs which we call series parallel 1 graphs. We define the class and subclasses of series parallel graphs using a quadruple notation, G = V; E; I; T ) for a graph G, where V; ....

K. Takamizawa, T. Nishizeki, N. Saito, "Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs", JACM, 18, (1982) 623-641. RR n2566 24 L. Finta, Z. Liu, I. Milis, E. Bampis

.... s are called data constructors applying to an element of type and a bounded number of recursive components. Though seemly restricted, these polynomial data types are powerful enough to cover our commonly used data types, such as lists, binary trees, rooted trees [6] and series parallel graphs [23]. Moreover, other data types like the rose trees, a kind of regular datatype de ned by RTree = Node [RTree ] can be encoded into one of these polynomial data types. This will be demonstrated in Section 6. For each data constructor C i , we de ne F i by F i f (e; x1 ; xn i ) e; ....

....chaining problem. 7.2 Generality Our approach is general (polytypic) enough to deal with maximum weightsum problems on data structures that are not lists or trees. To illustrate this, we derive a linear time algorithm solving the maximum two disjoint paths problem on series parallel graphs [23]. The series parallel graph is de ned as follows. SPG : Base (Vert; Vert; Edge) j Series SPG SPG j Parallel SPG SPG Here Vert represents the type of vertices and Edge represents the type of edges. Every graph should have a single source and a single sink. The two data constructors are ....

[Article contains additional citation context not shown here]

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. Journal of the Association for Computing Machinery, 29:623-641, 1982.

....data constructors which may have a bounded number of elements of type ff and a bounded number of recursive components. Though being seemly restricted, they are powerful enough to cover our commonly using data types such as lists, binary trees, rooted trees [BLW87] and series parallel graphs [TNS82]. Moreover, for other data types like the rose trees defined by RTree ff = Node ff [RTree ff] we can easily convert it into that in the above form, as will be demonstrated in Section 5. 3.2 Catamorphism Catamorphisms, one of the most important concepts in program calculation [MFP91, SF93, ....

....is general (polytypic) enough to deal with maximum weightsum problems on generic data structures not being restricted to lists or trees. As an example, we shall derive a linear time algorithm to solve the maximum two disjoint paths problem on series parallel graphs. This problem was discussed in [TNS82], but as far as we know no practical linear algorithm has been given so far. The definition of the series parallel graph is defined as follows. SPG : Base Vert Vert Edge j Series SPG SPG j Parallel SPG SPG 9 Here, Vert represents type of vertices, and Edge represents type of edges. Every ....

[Article contains additional citation context not shown here]

K. Takamizawa, T. Nishizeki, and N. Saito. Lineartime computability of combinatorial problems on series-parallel graphs. Journal of the Association for Computing Machinery, 29:623--641, 1982.

....we show a key structural property of planar graphs, that if they have low diameter they also have low tree width. Such a result was implicit already in the work of Baker [5] With a bound on tree width we can use dynamic programming techniques to compute many graph properties in linear time [8, 40]. A result similar to the one in this section follows easily from the Robertson Seymour wall lemma [36] Lemma 5 below) However we give the following direct proof to make explicit the dependence on the diameter, and to show that the result does not introduce any of the scary constants ....

.... The maximal cliques of a chordal graph can be arranged in a tree in such a way that the intersection of any two cliques is a subset of the cliques occurring along the corresponding path in the tree; this tree can be used for many e#cient dynamic programming algorithms in treewidth bounded graphs [8, 40]. The following lemma is the main result of this section. Lemma 1. Let planar graph G have diameter D. Then G has tree width O(D) and a tree decomposition of G with width O(D) can be found in time O(Dn) Proof: We assume without loss of generality that G is maximal planar. We fix an embedding ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Computing Machinery, 29:623--641, 1982.

.... graph according to the above operations, one can perform many other computations on the graph in linear time; these computations include problems such as maximum matchings, maximum independent sets, minimum dominating sets, and other problems including many that for general graphs are NP complete [3, 9, 12]. Such a decomposition can be constructed in linear time. Recently, a parallel algorithm was given by He and Yesha for recognizing directed series parallel graphs and providing a decomposition into the series parallel composition operations [7] This algorithm takes time O(log 2 n) and uses ....

K. Takamizawa, T. Nishizeki, and N. Saito, Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs, J. ACM 29 (1982) 623--641.

.... H, K, L, M A, F, H, M A, F, G, H Figure 1: Tree decomposition of a planar graph. 4 Bounded Tree Width Subgraph Isomorphism As a subroutine, we need to perform subgraph isomorphism testing in graphs of bounded tree width. This can be done by a standard dynamic programming technique [9, 46]. The exact statement of the problem we solve is complicated by the requirement that we count or list each subgraph isomorph exactly once. For simplicity, we state the bounds for this problem with one parameter measuring both the tree width of the text and the size of the pattern. De nition 1 A ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach. 29:623-641, 1982.

....Section 4, we conclude with some remarks on future research. 2 Series Parallel Graphs All graphs considered in the paper, unless otherwise stated, are directed graphs (digraphs) The class of series parallel graphs is known as a class for which several scheduling problems are polynomially solvable [1, 6, 11], while the same problems are NP complete for a general graph. Here, we are interested in a subclass of series parallel graphs which we call series parallel 1 graphs. We define the class and subclasses of series parallel graphs using a quadruple notation, G = V; E; I; T ) for a graph G, where V; ....

K. Takamizawa, T. Nishizeki, N. Saito, "Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs", JACM, 18, (1982) 623-641.

....Many combinatorial problems defined on graphs are NP complete, and hence there is probably no polynomial time algorithm for any of them. But it is shown that there exist efficient algorithms for many combinatorial problems if an input graph is restricted to the class of series parallel graphs [9], 2] 6] 7] Two examples are the generalized matching problem and the decision problem. In a generalized matching problem we would like to find a maximum number of vertex disjoint copies of a fixed graph contained in an input graph [6] In a decision (i.e. yes no) problem we would like to ....

K. Takamizawa, T. Nishizeki, and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. Assoc. Comput. Mach., 29 (1982), pp. 623--641.

.... 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. In this paper, we assume a specific form of the decomposition tree, and use the term sp trees for these trees. He and Yesha gave a parallel algorithm for recognising directed series parallel graphs [8] Their ....

K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM, 29:623--641, 1982.

....of the resulting algorithms, and also at the expense of increased algorithm time complexity as a function of k. In contrast, results giving explicit practical algorithms in this setting are usually limited to a few selected problems on either (full) k trees [9] partial 1 trees, or partial 2 trees [25]. We intend to cover the middle ground between these two extremes by investigating time complexity as a function of both input size and treewidth k. We assume that the input graph is given with a width k tree decomposition, computable in linear time for fixed k [6] Our algorithms employ a binary ....

<F3.719e+05> K. Takamizawa, T. Nishizeki, and N.<F3.91e+05> Saito,<F3.539e+05> Linear-time computability of combinatorial problems on series-parallel<F3.91e+05> graphs, J. ACM, 29 (1982), pp. 623--641.

....for series parallel graphs can be reduced in the case when the graph is 3 planar. Series parallel graphs arise in a variety of problems such as scheduling, electrical networks, data flow analysis, database logic programs, and circuit layout. Also, they play a central role in planarity problems [20, 28, 5, 6]. In Fig. 2 we show two di#erent planar embeddings of the same series parallel graph. Besides each planar embedding we show the orthogonal drawing with the minimum number of bends that can be constructed preserving that embedding. In the drawing of Fig. 2a we have almost twice the number of bends ....

....in the number of vertices of degree 4. The emphasis of this paper is on the existence of polynomial time algorithms. However, the time complexity of the algorithms presented in the paper could perhaps be improved. For example, one can think of applying techniques similar to those of [20]. Several of the algorithms presented in the paper can be easily parallelized by using standard techniques. However, the bound that can be achieved with such techniques is not optimal. The problem of finding e#cient parallel algorithms still exists. Fig. 24. A problem in extending Theorem 6.2 ....

<F3.755e+05> K. Takamizawa, T. Nishizeki, and N.<F3.8e+05> Saito,<F3.504e+05> Linear-time computability of combinatorial problems on series-parallel<F3.8e+05> graphs, J. Assoc. Comput. Mach., 29 (1982), pp. 623--641.

No context found.

K. Takamizawa, T. Nishizeki, and N. Saito, "Linear Time Computability of Combinatorial Problems on Series Parallel Graphs," J. ACM 29 (1982), 623--641.

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K. Takamizawa, T. Nishezeki, and N. Sato, "Linear-time computability of combinatorial problems on series-parallel graphs", Jour. Assoc. Comput. Mach., 29 (1982) pp. 623-641.

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K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. Journal of the Association for Computing Machinery, 29:623-641, 1982.

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K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Math, 29, 623--641, 1982.

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K. Takamizawa, T. Nishlzeki, N. Saito, 1982, Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs, J. ACM 29,623-641.

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K. Takamizawa, T. Nishizeki, and N. Saito (1982), "Linear-time computability of combinatorial problems on series-parallel graphs", Journal of the Association for Computing Machinery 29, (3) 623-641. 77

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