N. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54:169--213, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1, 18, 32] branchwidth, for k 3 (see [8] node search number, for k 3 (see [10, 20]) and mixed search number, for k 2 (see [34] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma 1, there are at most k vertices incident to ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Disc. Appl. Math., 54:169--213, 1994. 29

....tracks, characterizations and recognition of graphs with path width at most two J anos Bar at # , P eter Hajnal Bolyai Institute University of Szeged, Hungary Abstract Nancy G. Kinnersley and Michael A. Langston has determined [3] the excluded minors for the class of graphs with path width at most two. Here we give a simpler presentation of their result. This also leads us to a new characterization, and a linear time recognition algorithm for graphs width path width at most two. 1 History and introduction Based on the ....

....OTKA Grants F.026049 and F.030737 The second author s research was supported by OTKA Grant T.030074 1 In this paper we give two characterizations of graphs with path width at most two. The first one is the forbidden minor characterization that is already published by Kinnersley and Langston [3]. The list of forbidden graphs has 110 members. The second characterization is a much more useful one, it is based on some special reductions preserving the property having path width at most two. This characterization gives a short list of graphs, with the property that any graph with path width ....

N.G. Kinnersley and M.A. Langston, Obstruction set isolation for the gate matrix layout problem, Discrete Appl. Math., 54 (1994), pp. 169-213.

....implication will be shown in the next section. The other (much more technical) direction is presented in [2] 8 Remark 4.2 One can extend the above characterization by listing all excluded minors of the class of graphs with path width at most two. The complete list consists of 110 graphs, see [4]. This type of characterization is also given implicitly in [2] One major point of our paper is that the appropriate theorem is not the excluded minor theorem but the one we presented. 5 Path width of the non reducible graphs The excluded minors can be reduced to the following ten fundamental ....

N.G. Kinnersley, M.A. Langston, Obstruction set isolation for the gate matrix layout problem, Discrete Applied Math. 54 (1994), 169-213.

....decompositions. Much research has been done towards the construction of linear time algorithms solving Pi d k (T ) and Pi c k (T ) In [5] a linear (on the size of the input) time algorithm for treewidth was constructed. For further results concerning on related graph theoretic parameters see [3, 7, 8, 13, 16, 14, 15, 19, 21, 19, 22, 23, 24]. In this paper, we find analogous results to those of [5] for the parameter of branchwidth. Namely, we prove that, for any fixed k, one can construct a linear 2 time algorithm that solves Pi d k (B) and Pi c k (B) An immediate consequence of this result is that, for any fixed k, one can ....

N. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54:169--213, 1994.

....(siddhart cs.utk.edu) 146 THE STRUCTURE AND NUMBER OF OBSTRUCTIONS TO TREEWIDTH 147 rithm for w = 4. Although both these algorithms are practical, no general techniques are known for finding a complete set of reductions for w 4. The pathwidth of a graph is a concept akin to treewidth (see [9, 17], for example) There has been considerable interest in the obstructions for treewidth and for pathwidth [6, 14, 17, 19, 27] Obstruction based algorithms have been used for integrated circuit design and other applications [12, 15, 18] The reasons for studying the obstructions are two fold. ....

....both these algorithms are practical, no general techniques are known for finding a complete set of reductions for w 4. The pathwidth of a graph is a concept akin to treewidth (see [9, 17] for example) There has been considerable interest in the obstructions for treewidth and for pathwidth [6, 14, 17, 19, 27]. Obstruction based algorithms have been used for integrated circuit design and other applications [12, 15, 18] The reasons for studying the obstructions are two fold. First, a better comprehension of their structure and number can help one design better algorithms for the fixed parameter ....

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N. G. Kinnersley and M. A. Langston, Obstruction set isolation for the gate matrix layout problem, Discrete Appl. Math., 54 (1994) pp. 169--213.

....is the tendency of the number of obstructions for natural parameterized families of lower ideals to grow explosively as a function of the parameter k. For example, the number of minor order obstructions for k Pathwidth is 2 for k = 1, 110 for k = 2, and provably more than 60 million for k = 3 [5]. We favor the following as a working hypothesis: Natural forbidden substructure theorems of feasible size are feasibly computable. The remaining sections of this paper are organized as follows. First, we formally define minor order obstructions and the cycle cover graph families that we ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Technical Report TR-91-5, Dept. of Computer Science, University of Kansas, January 1991, to appear Annals of Discrete Math.

.... that is closed under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1,20,35] branchwidth, for k 3 (see [6] pathwidth, for k 2 (see [22,23]) and mixed search number, for k 2 (see [38] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma 1, there are at most k vertices incident to ....

....therefore L 2 is the obstruction set for the class of graphs with linear width 2. 2.3 Reducing graphs to simpler ones We will first prove a series of lemmata, enabling us to restrict our study to more simple graphs. Analogous lemmata for pathwidth and the mixed search number have been proved in [14,22,23] and [38] respectively. Lemma 4 Let H be a graph with linear width k. The following hold. i. Let v; v 0 be vertices such that v 2 V (H) dH (v) 2, and v 0 62 V (H) If H 0 = V (H) fv 0 g; E(H) ffv; v 0 gg) then linear width(H 0 ) k (notice that v 0 is a simply pendant ....

N.G. Kinnersley, M.A. Langston, Obstruction set isolation for the gate matrix layout problem, Disc. Appl. Math. 54 (1994) 169--213.

....pathwidth at most 4. If G 0 has pathwidth more than 2 then it must contain at least one of the pathwidth 2 obstructions as a minor. In particular, any such obstruction for 2 Pathwidth must also be a member of 1 Feedback Edge Set. All of the 20 possible forbidden minors with one cycle, given in [11], have at least three pendant paths of length 2, i.e. three legs of the spider graph S(K 1;3 ) attached to the single cycle. Property 2 is applied as follows. By considering incident edges from vertices x and y to G 0 , we know that G must have: a) three disjoint cycles or (b) one cycle and ....

Nancy G. Kinnersley and Michael A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Discrete Applied Mathematics, 54:169--213, 1994.

....in [Kin92] has shown that the pathwidth of a graph is identical to the vertex separation of a graph. The concept of pathwidth has been popularized by the theories of Robertson and Seymour (see for example, RS85] Thus, since the gate matrix layout cost, another well studied VLSI layout problem [KL94, Moh90], equals the pathwidth plus one [FL89] it also equals the vertex separation plus one. This paper shows that vertex separation is also related to another area besides computer science, namely computational biology. 2 Main Result In this section, we formally define our fixed parameter problems ....

Nancy G. Kinnersley and Michael A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Discrete Applied Mathematices, 54:169--213, 1994.

....is the tendency of the number of obstructions for natural parameterized families of lower ideals to grow explosively as a function of the parameter k. For example, the number of minor order obstructions for k Pathwidth is 2 for k = 1, 110 for k = 2, and provably more than 60 million for k = 3 [6]. We favor the following as a working hypothesis: Natural forbidden substructure theorems of feasible size are feasibly computable. 2 Preliminaries and The Basic Theory Let m be the minor order on graphs, that is, for two graphs G and H, H m G if and only if a graph isomorphic to H can be ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Technical Report TR-91-5, Dept. of Computer Science, University of Kansas, January 1991, to appear Annals of Discrete Math.

.... the possibility of an absolute approximation algorithm (unless P=NP, obviously) 20] Synthesizing a Predatory Search Strategy for VLSI Layouts 07 05 98 5 This problem is specially important on the theory of NP Completeness because of the surprising nonconstructive results obtained recently [21 23]. For instance, it has been proved that there exists a decision algorithm that verifies, in polynomial time, the existence of a k tracks layout, for any integer positive k. However, this existence proof is nonconstructive, such that, although it is known that the algorithm must in fact exist, it ....

N.G. Kinnersley, and M.A. Langston, "Obstruction set isolation for the gate matrix layout problem," Discrete Applied Math., vol. 54, pp. 169-213, 1994.

....and only if G does not contain K 4 as a minor. iii) Arnborg, Proskurowski, and Corneil [6] A graph G = V; E) has treewidth at most 3, if and only if it does not contain any of the four graphs, shown in Figure 1 as a minor. The obstruction sets of graphs with pathwidth 1 and 2 are also known [52]. The size of the obstruction sets can grow very fast: for instance, the obstruction set of the graphs with pathwidth at most k contains at least k 2 trees, each containing 5 Delta3 k Gamma1 2 vertices [98] Ramachandramurthi [68] investigated the graphs with k 1, k 2 and k 3 ....

N. G. Kinnersley and M. A. Langston, Obstruction set isolation for the gate matrix layout problem, Tech. Rep. CS-91-126, Computer Science Department, University of Tennessee, Knoxville, USA, 1991.

....) and O(n 4 ) heuristic algorithms based on longest paths on the associated disjoint graph generated from the personality matrix. Through experimental testing, they compared their work with that of Hachtel et al. 6] See [10] for a brief history of the development of PLA folding heuristics. See [9] and [13] for some recent works related to gate matrix layout. 2 The Optimal PLA Folding Problem In a PLA the input signals and their complements run, say vertically, through a matrix of circuit elements called the AND plane. Combinations of the inputs and their complements are selected in this ....

Kinnersley, N. G. and Langston, M. A., "Obstruction set isolation for the gate matrix layout problem", Technical Report, Department of Computer Science, University of Tennessee, Knoxville, January, 1991.

.... under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1, 18, 32] branchwidth, for k 3 (see [8] node search number, for k 3 (see [10, 20]) and mixed search number, for k 2 (see [34] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma 1, there are at most k vertices incident to ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Disc. Appl. Math., 54:169--213, 1994.

....families to grow explosively as a function of the parameter k. For example, the number of minor order obstructions for k Pathwidth (i.e. graphs with 1200 Dinneen M.J. Minor Order Obstructions . pathwidth at most k) is 2 for k = 1, 110 for k = 2, and provably more than 60 million for k = 3 [KL94]. It is known that there are at least k 2 obstructions that are trees for each k [TUK91] In this section we study common families of parameterized minor order lower ideals and provide (1) a practical constructive result that allows one to compute the disconnected obstructions of k F from the ....

Nancy G. Kinnersley and Michael A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Discrete Applied Mathematics, 54:169-- 213, 1994.

.... have been successfully computed [CD94, CDF95] Since (i) and (iii) can be effectively derived from an MSO description of F [Co90a] our Theorem 1 shows that (ii) is essential in the earlier positive result of [FL89b] Other work on the systematic computation of obstruction sets has appeared in [APS90, CDDFL97, GI91, Kin94, KL91, LA91, Lag93, Pr93]. 2 Preliminaries All of our discussion concerns finite simple graphs. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by a sequence of operations chosen from the list: i) delete a vertex, ii) delete an edge, iii) contract an edge. When applying the edge ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Appl. Math. 54 (1994), 169--213.

.... is closed under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1, 21, 36] branchwidth, for k 3 (see [7] pathwidth, for k 2 (see [10, 22, 23]) and mixed search number, for k 2 (see [38] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma 1, there are at most k vertices incident to ....

....therefore L 2 is the obstruction set for the class of graphs with linear width 2. 2.3 Reducing graphs to simpler ones We will first prove a series of lemmata, enabling us to restrict our study to more simple graphs. Analogous lemmata for pathwidth and the mixed search number have been proved in [14, 22, 23] and [38] Lemma 4 Let H be a graph with linear width k. The following hold. i. Let v; v 0 be vertices such that v 2 V (H) dH (v) 2, and v 0 62 V (H) If H 0 = V (H) fv 0 g; E(H) ffv; v 0 gg) then linear width(H 0 ) k (notice that v 0 is a simply pendant vertex of H 0 ....

N. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54:169--213, 1994.

....4. A Fast ecision Algorithm We have adopted the standard mapping of a circuit into a graph representing the interconnection relationships between the nets [DKL] Each vertex in the graph corresponds to a net in the circuit. We test only for six out of the 110 obstructions to a three track layout [KL]. The six obstructions are the graphs ; and , shown in Figure 1. These six graphs are the smallest obstructions to a three track gate matrix layout. Given the density of these six obstructions relative to the other 104 possibilities [KL] it is reasonable to expect that a circuit that cannot ....

....six out of the 110 obstructions to a three track layout [KL] The six obstructions are the graphs ; and , shown in Figure 1. These six graphs are the smallest obstructions to a three track gate matrix layout. Given the density of these six obstructions relative to the other 104 possibilities [KL], it is reasonable to expect that a circuit that cannot be laid out in three tracks will contain one of them. Our algorithm operates on this hypothesis. 4. Algorithm escription Excluding 4 implies that an input graph must be series parallel. The well known fact that every series parallel graph ....

N. G. Kinnersley and M. A. Langston, "Obstruction Set Isolation for the Gate Matrix Layout Problem," University of Tennessee Computer Science Technical Report CS--91--126, 1991.

....if we are given the information: v) An MSO expression OE that describes the graphs of the lower ideal F . Then from this we can effectively derive (i) iii) and (iv) This result is mainly due to Courcelle [Co90] 5) Other work on the systematic computation of obstruction sets has appeared in [Pr93, APS90, CD94, CDF95, Kin94, KL91]. Some of these results support practical implementations that have led to some significant mechanical or partly mechanical proofs of new and nontrivial forbidden substructure theorems. There has been a considerable amount of overlapping work in this area which is sometimes confusing to sort ....

N. G. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Appl. Math. 54 (1994), 169--213.

....although the algorithm is not practical. 5 Three track Gate Matrix Layout GML(1) is trivially solved. GML(2) can be solved in O(nm) time using [BL] The first genuinely difficult case is GML(3) 5. 1 The Six Smallest Obstructions It is known that there are exactly 110 obstructions to GML(3) [KL]. The number of vertices in these obstructions ranges from 4 to 22. Consider the six smallest obstructions, which we call A, B, C, D, E and F (see Figure 2) Graph A is K 4 . Graph B is known as the Haj os graph. We are able to use the structure and relative density of these six obstructions to ....

N. G. Kinnersley and M. A. Langston, "Obstruction set isolation for the Gate Matrix Layout problem," Discrete Applied Mathematics 54 (1994), 169--213.

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N. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54:169--213, 1994.

No context found.

N. Kinnersley and M. A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54:169--213, 1994.

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N. G. Kinnersley and M. A. Langston, Obstruction set isolation for the gate matrix layout problem, Discrete Appl. Math. 54 (1992) 169-213.

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N.G.Kinnersley, M.A.Langston, Obstruction set isolation for the Gate Matrix Layout problem, Discrete Applied Mathematics 54 (1994), pp 169-213.

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