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Proof

If q were such a vertex, the Steiner minimal tree would contain either a edge from q to v i not containing v j , or vice versa. If the SMT contains a edge from q to v i , for example, the SMT can be shortened by deleting [ v i , v ] and adding [ q , v j ] , a contradiction.

Similar statements can be made to further specify where a Steiner hull may be.

Remark

It is easy to see that a full Steiner topology with n terminals has 2 n - 3 edges. Let f ( n ) , n 2 , denote the number of full Steiner topologies with n - 2 Steiner points. Adding a Steiner point to the middle of any of the edges and connecting it to a new terminal results in a full Steiner tree for n + 1 terminals. Geometrically, this causes the edge to become two separate edges, connected by the new Steiner point. In order to be consistent with the fact that edges must meet at 120 angles at Steiner points, the two new edges will rotate to meet at an angle of 120 with each other and with the edge connecting the new Steiner point to the new terminal. Thus there are 2 n - 3 ways to create a new full Steiner tree, so

f ( n + 1 ) = ( 2 n - 3 ) f ( n )

which has the solution

f ( n ) = ( 2 n - 4 ) ! 2 n - 2 ( n - 2 ) !

Let F ( n , k ) denote the number of Steiner topologies with n terminals and k Steiner points such that no terminal is of degree 3. Then F ( n , k ) can be obtained from f(k) by first selecting k + 2 terminals and a full Steiner topology on it, and then adding the remaining n - k - 2 terminals one at a time at interior points of some edges, creating a "kink" in each edge. The first terminal can go to one of the 2 k + 1 edges, the second to one of 2 k + 2 edges, and so on, until the ( n - k - 2 ) th point is added to one of the 2 k + ( n - k - 2 ) = n + k - 2 edges. Thus

F ( n , k ) = n k + 2 f ( k ) ( n + k - 2 ) ! ( 2 k ) !

Letting n 3 denote the number of terminals of degree 3

F ( n ) = k = 0 n - 2 n 3 = 0 n - k - 2 2 n n 3 F ( n - n 3 , k + n 3 ) ( n + n 3 ) ! k !

A table containing the values of f ( n ) and F ( n ) for n = 2 , ... , 8 is given.

n 2 3 4 5 6 7 8
f ( n ) 1 1 3 15 105 945 10395
F ( n ) 1 4 31 360 5625 110880 2643795

Computational complexity

The optimization problem:

Given : A set N of terminals in the Euclidean plane

Find : A Steiner tree of shortest length spanning N

can be recast as a decision problem:

Given : A set N of terminals in the Euclidean plane and an integer B

Decide : Is there a Steiner tree T that spans N such that | T | B

The Steiner tree problem is NP-complete. This means that it is at least as hard as any problem in NP, but the solution is still verifiable in polynomial time. However, the problem cannot be solved in polynomial time. The discrete Euclidean Steiner problem, however, is not known to be NP-complete.

Given : A set N of terminals with integer coordinates in the Euclidean plane and an integer B .

Decide : Is there a Steiner tree T that spans N such that all Steiner points have integer coordinates and the discrete length of T is less than or equal to B , where the discrete length of each edge of T is the smallest integer not less than the length of that edge.

There is a version of this problem that is in P: Given a finite set in a Banach-Minkowski plane, a minimal Steiner tree for this set can be found in polynomially bounded time with an algorithm that executes only graph theoretic functions.

The steiner ratio

The Steiner ratio is the smallest possible ratio between the total length of a minimum Steiner tree and the total length of a minimum spanning tree. It is conjectured that, in the Euclidean Steiner problem, the Steiner ratio is 3 2 . This is called the Gilbert-Pollak conjecture, and was though to be proven in 1990, but some people do not accept this proof. Gilbert and Pollak also conjectured the ratio for higher dimensional spaces, but this was disproven.

It's easy to see that this is true in the four point case. Alternatively, the Steiner ratio is sometimes given as the total length of the minimum spanning tree over the length of the minimum Steiner tree. The Steiner ratio for the rectilinear Steiner tree, which is when all edges of the Steiner tree are perpendicular, is 3 2 .

Other applications

Steiner trees have many biological applications. They can be used to describe the ways proteins connect and fold. Steiner trees are also used in phylogeny. Organic chemistry diagrams often include Steiner trees. For example, CH 6 N 3 is a regular hexagon Steiner tree.

Steiner trees have many computer science applications as well, such as in network design and circuit layout.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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