algorithm - Place connected boxes on a 2D plane -

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given n rectangular boxes , m connections between them, want place them on plane efficiently such total sum of length of connections remains minimum.

the throught process have split plane in grid n or more spaces, , place boxes maximum number of connections farther apart in grid starting diagonally opposite corner spaces.

this might not efficient when there 1 box connected n-1 boxes , connections. expect 1 box in center , other boxes around it.

is there standard solution such problem? can pointer how approach such problem?

it non-linear optimization problem , can solved, instance, using simulated annealing or general objective function minimization such gradient descent methods.

given layout of boxes, let l denote sum of lengths of connections given layout. want minimize l. simple simulated annealing scheme works this:

layout = random_layout() t = 1.0 while(true)   l = sum_of_lengths(layout)   layout' = move_one_box(layout)   l' = sum_of_lengths(layout)   if (l' < l  or  random(0..1) < t)     layout = layout'   t = t * 0.999

initially, algorithm moves boxes randomly around, when t decreases, algorithm gradually changes greedy optimizer. can run multiple runs of algorithm , pick best result. simulated annealing scheme.


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