Exact Steiner Trees in Graphs and Grid Graphs

AN ALGORITHM FOR STEINER TREES IN GRID GRAPHS AND ITS APPLICATION TO HOMOTOPIC ROUTING

Journal of Circuits System and Computers ◽

10.1142/s0218126696000029 ◽

1996 ◽

Vol 06 (01) ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

MICHAEL KAUFMANN ◽

SHAODI GAO ◽

K. THULASIRAMAN

Keyword(s):

Steiner Tree ◽

Steiner Trees ◽

Minimum Length ◽

Routing Problem ◽

Tree Algorithm ◽

Grid Graphs ◽

Steiner Minimal Trees ◽

Homotopic Routing

In this paper we present an algorithm for Steiner minimal trees in grid graphs with all terminals located on the boundary of the graph. The algorithm runs in O(min{k4, k2n}) time, where k and n are the numbers of terminals and vertices of the graph, respectively. It can handle non-convex boundaries and is the fastest known for this case. We also consider the homotopic routing problem and apply our Steiner tree algorithm to construct minimum-length wires for multi-terminal nets.

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Optimal Steiner trees under node and edge privacy conflicts

Journal of Combinatorial Optimization ◽

10.1007/s10878-020-00690-1 ◽

2021 ◽

Author(s):

Alessandro Hill ◽

Roberto Baldacci ◽

Stefan Voß

Keyword(s):

Steiner Trees

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Connected perimeter of planar sets

Advances in Calculus of Variations ◽

10.1515/acv-2019-0050 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

François Dayrens ◽

Simon Masnou ◽

Matteo Novaga ◽

Marco Pozzetta

Keyword(s):

Minimization Problem ◽

Existence Of Solutions ◽

Lower Semicontinuous ◽

Representation Formula ◽

Steiner Trees ◽

Lower Semicontinuous Envelope ◽

Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

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Peg Solitaire on Cartesian Products of Graphs

Graphs and Combinatorics ◽

10.1007/s00373-021-02289-7 ◽

2021 ◽

Vol 37 (3) ◽

pp. 907-917

Author(s):

Martin Kreh ◽

Jan-Hendrik de Wiljes

Keyword(s):

Cartesian Product ◽

Cartesian Products ◽

Connected Graphs ◽

Grid Graphs ◽

Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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