K. Skodinis. Computing optimal linear layouts of trees in linear time. In M. Paterson, editor, Proc. ESA 2000, number 1879, pages 403--414. Springer-Verlag, Lectures Notes in Computer Science, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....separation of trees can be computed in linear time, and they gave an O(n log n) time algorithm for computing the corresponding layout. It yields another O(n) time algorithm returning the search number of trees, and an O(n log n) time algorithm returning a minimal search strategy in trees. In [29], Skodinis recently showed that an optimal layout can also be computed in linear time, and hence a minimal search strategy can be computed in linear time. Beside network security [14] the graph searching problem has many other applications, including pursuit evasion problems in a labyrinth [26] ....

K. Skodinis. Computing optimal linear layouts of trees in linear time. In 8th European Symp. on Algorithms (ESA '00), Springer, LNCS 1879, pages 403-414, 2000.

....B i such that both u; v 2 B i ; and 3. for all 1 i j k p, B i B k B j . The width of B is maxfjB i j j 1 i pg 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16]. The next two results about trees and pathwidth are given in [15, 7] Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T n v have pathwidth h and the remainder have pathwidth at most h 1. As de ned in [7] we say that a vertex v ....

....pathwidth at most h 1. Lemma 5 [7] Let T be a tree rooted at r. If at most two subtrees rooted at children of r have pathwidth h, neither has an h critical vertex, and every other subtree rooted at a child of r has pathwidth at most h 1, then pw(T ) h. Finally, the following lemma is given in [7, 16]: Lemma 6 [7, 16] A tree T has at most one pw(T ) critical vertex. In the next section we prove optimal upper and lower bounds on the number of layers required by short layered drawings of trees. We follow that with optimal upper and lower bounds for proper, upright and unconstrained layered ....

[Article contains additional citation context not shown here]

Konstantin Skodinis. Computing optimal linear layouts of trees in linear time. In Mike Paterson, editor, Algorithms, 8th European Symposium (ESA 2000.

....such that both u, v B i ; and 3. for all 1 i j k p, B i B k B j . The width of B is max B i p 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16]. The next two results about trees and pathwidth are given in [15, 7] Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T v have pathwidth h and the remainder have pathwidth at most 1. As defined in [7] we say that a vertex ....

....at most h 1. Lemma 5 [7] Let T be a tree rooted at r. If at most two subtrees rooted at children of r have pathwidth h, neither has an h critical vertex, and every other subtree rooted at a child of r has pathwidth at most h 1, then pw(T ) h. Finally, the following lemma is given in [7, 16]: Lemma 6 [7, 16] A tree T has at most one pw(T ) critical vertex. In the next section we prove optimal upper and lower bounds on the number of layers required by short layered drawings of trees. We follow that with optimal upper and lower bounds for proper, upright and unconstrained layered ....

[Article contains additional citation context not shown here]

Konstantin Skodinis. Computing optimal linear layouts of trees in linear time. In Mike Paterson, editor, Algorithms, 8th European Symposium (ESA 2000.

....a path decomposition (P, X) that is not more than 3pw(H) in O(n) time. The bound stated in [7] is only O(nlogn) because the algorithm described there relies on obtaining optimal path decompositions of trees. It has more recently been shown that such decompositions can be computed in linear time [12]. Thus the algorithm of [7] runs in linear time as well. Adding the two vertices to every element of X gives a path decomposition of G of width 3pw(H) 2 3pw(G) 2 because H G. If H were biconnected, we could use the algorithm of [2] to obtain a path decomposition of width 2pw(H) ....

K. Skodinis. Computing optimal linear layouts of trees in linear time. Technical Report MIP-9904, Department of Computer Science, University of Passau, Passau, Germany, 1999.

....research is to try to solve the problem for specific (typically small) values of the treewidth w. No algorithm of this type exists for cutwidth when w 1, while, for pathwidth, the best, so far, result is an approximation algorithm in [10] for outerplanar graphs that have treewidth 2 (see also [28, 45]) ....

K. Skodinis. Computing optimal linear layouts of trees in linear time. In i. Paterson, editor, Pro. ESA 2000, number 1879, pages 403 414. Springer-Verlag, Lectures iNotes in Computer Science, 2000.

.... on graph searching, we refer the reader to, e.g. 9, 12, 14, 15] Graph searching is a non trivial interesting and challenging problem; even determining whether s(G) k for arbitrary G and k, is NP complete [26] Not surprisingly, the research has focused on restricted classes of graphs (e.g. [19, 25, 27, 33, 34]) and on bounded search numbers (e.g. Departament de Matem atica Aplicada IV, Universitat Polit ecnica de Catalunya, Spain. lali mat.upc.es. CNRS, Laboratoire de Recherche en Informatique, Universit e Paris Sud, France. http: www.lri.fr pierre. School of Computer Science, Carleton ....

K. Skodinis. Computing optimal linear layouts of trees in linear time. In 8th European Symp. on Algorithms (ESA '00), Springer, LNCS 1879, pages 403-414, 2000. (To appear in SIAM J. Computing.)

....to solve the problem for speci c (typically small) values of the treewidth w. No algorithm of this type exists for cutwidth when w 1, while, for pathwidth, the best, so far, result is an approximation algorithm in [10] for outerplanar graphs (outerplanar graphs that have treewidth 2 see also [26, 43]) ....

K. Skodinis. Computing optimal linear layouts of trees in linear time. In M. Paterson, editor, Proc. ESA 2000, number 1879, pages 403-414. Springer-Verlag, Lectures Notes in Computer Science, 2000.

....for writing the output. However, more compact representations of path decompositions exists, e.g. mark for each vertex the first and last bag it belongs to, or one can use the equivalent notion of vertex separations. These representations have size linear in the number of vertices. As Skodinis [9] has shown that (with such representations) one can find an optimal path decomposition of a given tree in linear time, we conjecture that the algorithm of Section 4 can be made to run in linear time, but there are several unresolved matters in this, and we leave this as an open problem. As a side ....

K. Skodinis, Computing optimal linear layouts of trees in linear time, Technical Report MIP-9904, Dept. of Computer Science, University of Passau, Passau, Germany, 1999.

....for writing the output. However, more compact representations of path decompositions exists, e.g. mark for each vertex the rst and last bag it belongs to, or one can use the equivalent notion of vertex separations. These representations have size linear in the number of vertices. As Skodinis [9] has shown that (with such representations) one can nd an optimal path decomposition of a given tree in linear time, we conjecture that the algorithm of Section 4 can be made to run in linear time, but there are several unresolved matters in this, and we leave this as an open problem. As a side ....

K. Skodinis, Computing optimal linear layouts of trees in linear time, Technical Report MIP-9904, Dept. of Computer Science, University of Passau, Passau, Germany, 1999.

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K. Skodinis. Computing optimal linear layouts of trees in linear time. In M. Paterson, editor, Proc. ESA 2000, number 1879, pages 403--414. Springer-Verlag, Lectures Notes in Computer Science, 2000.

No context found.

K. Skodinis. Computing optimal linear layouts of trees in linear time. In 8th European Symp. on Algorithms (ESA '00), Springer, LNCS 1879, pages 403-414, 2000. (To appear in SIAM J. Computing.)

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