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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

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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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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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