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A195647
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a(n) is the optimal wire-length for an n X n grid.
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0
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0, 6, 24, 60, 116, 200, 318, 472, 668, 914, 1214, 1568, 1988, 2480, 3040, 3680, 4408, 5224, 6130, 7140, 8260, 9478, 10816, 12280, 13864, 15576, 17430, 19428, 21560, 23850, 26304, 28908, 31680, 34632, 37760, 41060, 44556, 48254, 52130, 56216, 60520, 65030
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OFFSET
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1,2
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COMMENTS
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This problem is also known as the linear arrangement problem or the wire-length problem. The task is to label the vertices of a graph with distinct positive integers such that the sum of label differences over all the edges is minimal. More formally, given a finite simple graph G=(V,E) with vertex set V and edge set E, we need to find a map f from V onto {1,2, ..., |V|} that minimizes the sum |f(u) - f(v)| over all edges (u,v) in E. In general this problem is NP-hard, but exact solutions are known for rectangular grids. This sequence corresponds to optimal solutions for n X n square grids.
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REFERENCES
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T. Y. Berger-Wolf, Wirelength of a Grid Graph, 2001.
P. Fishburn and P. Tetali and P. Winkler, Optimal linear arrangement of a rectangular grid, Discrete Mathematics, 2000, pages 123-139.
D. O. Muradyan and T. E. Piliposjan, Minimal Numberings of Vertices of a Rectangular Lattice, Akad. Nauk. Armjan. SSR. Dokl. 70, 1980, pages 21-27 (in Russian).
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LINKS
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FORMULA
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a(n) = n*(n^2 + n - 2) - t*(2t^2 - 6nt + 3n^2 + 3n - 2)/3, where t = round((6n - sqrt(6*(2 - 3n + 3n^2)))/6).
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EXAMPLE
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For n=2 an optimal grid arrangement is
1 2
4 3
The value of this arrangement is |1-2| + |1-4| + |2-3| + |3-4|=6.
For n=8 an optimal grid looks like so:
64 63 60 40 25 07 02 01
62 61 59 39 26 08 04 03
58 57 56 38 27 09 06 05
55 54 53 37 28 12 11 10
52 51 50 36 29 15 14 13
49 48 43 35 30 18 17 16
47 46 42 34 31 23 20 19
45 44 41 33 32 24 22 21
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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