OFFSET
1,2
COMMENTS
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.
REFERENCES
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).
FORMULA
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).
EXAMPLE
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Dmitry Kamenetsky, Sep 21 2011
STATUS
approved