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# Finding a minimal cost tree for the network

Bob
Guest
Posts: n/a

 04-30-2005
All,

Any one from this group can help me to solve this problem please!!!

Thanks
Bob.

The cost for a communication line between two stations is proportional
to the length of the line. The cost for conventional minimal spanning
trees of a set of stations can often be cut by introducing "phantom"
stations and then constructing a new Steiner tree. This device allows
costs to be cut by up to 13.4% (= 1- sqrt(3/4)). Moreover, a network
with n stations never requires more than n-2 points to construct the
cheapest Steiner tree. Two simple cases are shown in Figure 1.
For local networks, it often is necessary to use rectilinear or
"checker-board" distances, instead of straight Euclidean lines.
Distances in this metric are computed as shown in Figure 2.

Suppose you wish to design a minimum costs spanning tree for a local
network with 9 stations. Their rectangular coordinates are: a(0,15),
b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0)
i(10,3). You are restricted to using rectilinear lines. Moreover, all
"phantom" stations must be located at lattice points (i.e., the
coordinates must be integers). The cost for each line is its length.

Find a minimal cost tree for the network.

Patricia Shanahan
Guest
Posts: n/a

 04-30-2005
Bob wrote:
> All,
>
> Any one from this group can help me to solve this problem please!!!
>
> Thanks
> Bob.
>
> The cost for a communication line between two stations is proportional
> to the length of the line. The cost for conventional minimal spanning
> trees of a set of stations can often be cut by introducing "phantom"
> stations and then constructing a new Steiner tree. This device allows
> costs to be cut by up to 13.4% (= 1- sqrt(3/4)). Moreover, a network
> with n stations never requires more than n-2 points to construct the
> cheapest Steiner tree. Two simple cases are shown in Figure 1.
> For local networks, it often is necessary to use rectilinear or
> "checker-board" distances, instead of straight Euclidean lines.
> Distances in this metric are computed as shown in Figure 2.
>
> Suppose you wish to design a minimum costs spanning tree for a local
> network with 9 stations. Their rectangular coordinates are: a(0,15),
> b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0)
> i(10,3). You are restricted to using rectilinear lines. Moreover, all
> "phantom" stations must be located at lattice points (i.e., the
> coordinates must be integers). The cost for each line is its length.
>
> Find a minimal cost tree for the network.
>

rectilinear distance minimum steiner tree

It produced several useful-looking hits.

In particular:

http://portal.acm.org/citation.cfm?id=238997.239033

@article{239033,
author = {Joseph L. Ganley and James P. Cohoon},
title = {Rectilinear Steiner trees on a checkerboard},
journal = {ACM Trans. Des. Autom. Electron. Syst.},
volume = {1},
number = {4},
year = {1996},
issn = {1084-4309},
pages = {512--522},
doi = {http://doi.acm.org/10.1145/238997.239033},
publisher = {ACM Press},
address = {New York, NY, USA},
}

Patricia