The Length of a Minimal Tree With a Given Topology: generalization of Maxwell Formula

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by  A. O. Ivanov

The Length of a Minimal Tree With a Given
1
Topology: generalization of Maxwell Formula
1
0
2
A.O. Ivanov, A.A. Tuzhilin

n
a
January 12, 2011
J

1
1
Abstract

]
G The classic Maxwell formula calculates the length of a planar locally
minimal binary tree in terms of coordinates of its boundary vertices and
M
directions of incoming edges. However, if an extreme tree with a given
. topology and a boundary has degenerate edges, then the classic Maxwell
h
formulacannotbeapplieddirectly,tocalculatethelengthoftheextreme
t
a tree in this case it is necessary to know which edges are degenerate. In
m
this paper we generalize the Maxwell formula to arbitrary extreme trees
[ in a Euclidean space of arbitrary dimension. Now to calculate thelength

ofsuchatree,thereisnoneedtoknoweitherwhatedgesaredegenerate,
1
or the directions of nondegenerate boundary edges. The answer is the
v
maximum of some special linear function on the corresponding compact
7
convex subset of the Euclidean space coinciding with the intersection of
1
1 some cylinders.
2
.
1 Introduction
0
1
1 The present paper is devoted to investigation of extreme trees in Euclidean
: spaces. These trees attract interest since the class of such trees is a natural
v
extensionofthe setoflocallyminimaltreesandshortesttrees(Steiner minimal
i
X trees). ThelatteronescanbeconsideredassolutionstoTransportationproblem
r andsohavegrateimportanceforapplications. Itisknown,thatthesearchingof
a
ashortesttreespanningagivenboundarysetisaverytime-consumingalgorith-
mic problem (an NP-hard problem), that gives reason for active investigation
of heuristic solutions. As such solutions, one can choose as extreme trees, so
as minimal spanning trees. The latter ones are used often, because there is a
polynomialalgorithmoftheirconstruction,andquickrealizationsofitarewell-
knownandwidespread. Therelativeerrorofthis heuristicinthe worstpossible
situation is called the Steiner ratio of the ambient metric space, see [1].
Atpresent,the Steiner ratioisnotknownforanyEuclideanspaces,starting
with the two-dimensional plane. Notice that in 60th of the previous century,
E. N. Gilbert and H. O. Pollak [1] conjectured that the Steiner ratio of the
Euclideanplaneisattainedatthevertexsetofaregulartriangleandisequalto
1
Preliminaries 2
√3/2. But in spite of many attempts of different authors (see a review in [2]),
this conjecture is not proved yet. The most known attempt was taken in 90th
by D. Z. Du and F. K. Hwang [3]. But it turns out that their proof contains
seriousgapswhichwerepointedoutasbytheauthors,soasbyotherspecialists.
As a result, currently the conjecture is considered as open.
For the Euclidean spaces of dimension three and more we know even less.
It is proved in [4] that in these spaces the Steiner ratio is not achieved at the
vertex set of a regular simplex. Moreover, a fast growing low estimate for a
possible number of points in boundary set, the Steiner ratio could be achieved
at is found. Therefore, there is no any reasonable conjecture concerning the
Steiner ratio value for these spaces.
In the present paper a new formula is obtained, which gives an opportunity
to calculate the length of an extreme network spanning a given boundary set
without the network construction. It turns out that the length of such network
can be found as a maximal value of some linear function ρ on an appropriate
convexcompactsubsetS oftheconfigurationspaceRN. Thefunctionρdepends
on the coordinates of the boundary points only, and the subset S is completely
defined in terms of the structure of the parameterizing tree of the extreme
network. It seems to us, that the formula obtained gives a new view onto
relationsbetweenthelengthsofdistinctextremetrees,inparticular,thelengths
of shortest trees and minimal spanning trees.
The authors like to use the opportunity to express their gratitude to aca-
demician A. T. Fomenko for his permanent attention to their work.
The workis partly supportedby RFBR (projectN 10–01–00748),President
Program “Leading Scientific Schools of RF” (project NSh–3224.2010.1), Euler
programofDAAD,andalsobyFederalProgramsRNP2.1.1.3704,FCP02.740.11.5213
and FCP 14.740.11.0794.
1 Preliminaries
Consider an arbitrary tree G=(V,E) with the vertex set V = v containing
k
{ }
a fixed subset B = v ,...,v , and with the edge set E. We assume that all
1 n
{ }
the vertices of the tree G having degree 1 or 2 lie in B. Such a set B is called
a boundary of the tree G and is denoted by ∂G. Vertices from ∂G and edges
incident to such vertices are said to be boundary, and the remaining vertices
v ,...,v and the remaining edges are said to be interior. A network of
n+1 n+s
type G in the Euclidean space Rm is an arbitrary mapping ΓV Rm. The
→
restrictionof Γ onto ∂G is called the boundary of the network Γ and is denoted
by ∂Γ.
EachnetworkΓisusefultorepresentasalinear graph,associatingeachedge
v v ofthe tree G with the segment Γ(v ),Γ(v ) (which could be degenerate).
k l k l
This segment is called by edge of th(cid:2)e network Γ(cid:3). Thereby, an angle between
adjacentnon-degenerateedgesofthenetworkisnaturallydefined. Thelengthof
thenetwork isalsonaturallydefinedasthesumofthelengthsofallitsedges. A
network,all whose edges are non-degenerate,is called non-degenerate. Besides,
Preliminaries 3
an edge of a network is called boundary (interior), if such is the corresponding
edge of the tree G. An edge of the tree G is said to be Γ-degenerate (Γ-non-
degenerate), if such is the corresponding edge of the network Γ.
Let α = G be a family of non-intersecting subtrees of G. Define the
k
{ }
reduced treeG/αasfollows. ItsverticesarealltheG togetherwiththevertices
k
from G, which are not in G . Join the vertices from G/α by an edge, if and
k
only if they are joined in G by an edge. The boundary of the tree G/α consists
of all its vertices containing elements from ∂G.
For each network Γ, define the Γ-reduction of the tree G taking the G to
k
be equal to the connected components of the subgraph of G generated by Γ-
degenerate edges of the tree G. Notice that the network Γ generates naturally
the non-degenerate network of the type G/α, which is denoted by Γˆ and called
reduced.
Let ϕB Rm be a mapping, which is one-to-one with its image. Consider
→
all possible networks with the boundary ϕ. Then a network having the least
possible length among all such networks is called a Steiner minimal tree or a
shortest network with the given boundary.
AnetworkΓissaidtobe locally minimal, if∂Γisone-to-onewithits image,
and the angles between adjacent edges of the reduced network Γˆ are at least
120◦. NoticethatthevertexdegreesofthenetworkΓˆ canbeequalto1,2,or3,
and all its vertices of degree 1 and 2 are boundary. A locally minimal network
whose set of boundary vertices coincides with the set of its vertices of degree 1
is called binary.
Each shortest network is locally minimal. Locally minimal networks, in
turn,areshortest“insmall”,i.e., anysufficiently smallpartofsuchnetworkisa
shortestnetworkwiththecorrespondingboundary. Fullinformationconcerning
locally minimal networks can be found in [2] and [5].
For an arbitrary network Γ of type G we put z = Γ(v ) Rm. Let us
k k
identify the boundary of the network Γ with the point z = (z ,∈...,z ) Rmn
1 n
of the configuration space Rmn. ∈
By , we denote the Euclidean scalar product in Rmn. To start with,
h· ·i
let n = 2, z = z , and ν be the direction of the segment [z ,z ] oriented to
1 2 1 2
6
z (correspondent, ν is the direction of this segment oriented to z ). In the
2 1
−
configuration space we define the vector θ = ( ν,ν). That is, the vector θ is
−
composed from the directions of the segment [z ,z ] oriented to its consecutive
1 2
vertices. Then
z z
2 1
z,θ = z , ν + z ,ν = z z , − = z z ,
1 2 2 1 2 1
h i h − i h i D − z2 z1 E k − k
k − k
i.e. z,θ is equal to the length of the segment [z ,z ].
1 2
h i
Now,letΓbeanon-degeneratelocallyminimalbinarynetworkwithabound-
ary ϕ. By ρ(Γ) we denote the length of the network Γ. Let ν be the direction
k
ofthe boundaryedgeofthe networkΓenteringinto the pointz ,k n. Letus
k
≤
define the vector θ of the boundary edges directions of the network Γ as follows:
θ = (ν ,...,ν ). Summing the above expressions for the length of a segment
1 n
and taking into account the additivity of the scalar product and the fact, that
Preliminaries 4
the sum of the direction vectors of the edges entering into a vertex of degree 3
is equal to zero, we have:
ρ(Γ)= z,θ .
h i
The latter equality is referred as Maxwell formula (see, for example, [1], [2],
[6]). Maxwell formula can be naturally generalizedto locally minimal networks
of general form. In this case the corresponding reduced network can contain
boundary vertices of degree 2 or 3, where the edges meet by angles of at least
120◦. In Maxwell formula for such networks, at each boundary vertex we need
to take the sum of vectors of all the entering edges.
Noticethatfornon-degeneratelocallyminimalbinarynetworksintheplane,
the vector θ can be calculated by the direction of a single edge and by the
planar structure of the network. It is convenient to do this in terms of so-
called twisting numbers. Let us recall the corresponding definition. Let the
standard orientation of the plane be fixed, that gives us a possibility to define
the positive (left) and negative (right) rotations. Let e and e be two edges of
1 2
aplanarimmersedbinarytreeG′,andγ betheuniquepathinG′ joiningthem.
Then the twisting number tw(e ,e ) from the edge e to the edge e is defined
1 2 1 2
as the difference between the numbers of left (positive) and right (negative)
turns in the interior vertices of the path γ during the motion from e to e in
1 2
G. Notice that this function is skew-symmetric and additive along the paths
(see [2] or [5]).
Let us identify the plane R2 with the complex field C: the points z are
k
considered as complex numbers, and the directions ν are considered as unit
complex numbers eiψ. Then the configuration space can be identified with Cn.
Instead of the Euclidean scalar product, here we consider the Hermitian scalar
product which is denoted in the same way.
Let e be the unique edge of the network Γ entering into the point z ,
k k
k n, and θ = (eiψ1,...,eiψn) be the directions vector of the edges ek. Then
≤
the number z,θ is real, and it is equal to the length ρ(Γ) of the network Γ.
h i
Indeed, it suffices to verify this equality for a network having a single edge
[z ,z ], z = z . Let eiψ be the direction of the segment [z ,z ] oriented from
1 2 1 2 1 2
6
z to z (in this case eiψ is the direction of this segmentoriented to the point
1 2
−
z ). Then θ =( eiψ,eiψ), and hence
1
−
z,θ =z ( e−iψ)+z e−iψ =(z z )e−iψ =(z z )(z z )/z z = z z .
1 2 2 1 2 1 2 1 2 1 2 1
h i − − − − | − | | − |
Lett =tw(e ,e )be the twisting number fromthe edgee to the edge e .
pq p q p q
Then
eiψq = eiψpeiπ3tpq.
−
Put tk = eiπ3tk1,...,eiπ3tkn . Notice, that the kth component of the vector
tk is equal(cid:0)to 1. Then θ = (cid:1)eiψktk. Thus, starting with the direction of one
−
boundary edge only, one can find the directions of all the remaining boundary
edges(andalltheotheredgesalso)ofthetreeΓusingjustthetwistingnumbers.
As an application of the above formula, let us calculate the length of the
network Γ and the directions of all its edges without an explicit construction
Preliminaries 5
of the tree and using only the information concerning the boundary mapping
ϕ and the planar structure of the network Γ (the twisting numbers for all the
pairs of edges of the corresponding immersed planar tree G′). Since
ρ(Γ)= z,θ = eiψk z,t ,
k
h i − h i
thenρ(Γ)= z,t andψ isequaltotheargumentofthenumber z,θ / z,t ,
k k k
h i −h i h i
andsince z,(cid:12)θ isa(cid:12)positivereal,thenψ isequaltotheargumentofthenumber
h (cid:12) i (cid:12) k
t ,z . Ifweknowψ ,thenwecanfindouttheremainingψ bymeansofthe
k k p
−h i
above formulas.
To calculate similarly the length and the edges directions of a locally mini-
mal planar network of general form, we need to partition the reduced network
Γˆ into the binarycomponents,cutting itby the verticesof degree2 andbound-
ary vertices of degree 3, and to proceed the calculation for each component
separately.
Let us return to the networks in Rm. By [G,ϕ] we denote the set of all
the networks in Rm having the type G and the boundary ϕ. Each network Γ
from [G,ϕ] is uniquely defined by the images z = Γ(v ), k > n, of all the
k k
interior vertices of G. Consequently writing down the vectors z , k >n, as the
k
components of the vector (z ,...,z ), we identify the set [G,ϕ] with the
n+1 n+s
spaceRms. Notice,thatthelengthofthenetworkisareal-valuedfunctionρ
G,ϕ
on [G,ϕ]. It is easy to see, that this function is convex and tends to infinity
as the arguments unlimitedly increase. Therefore, the set of minima of this
function is non-empty and convex. Each network corresponding to a minimum
ofthisfunctionissaidtobeanextremenetworkof the typeG with theboundary
ϕ.
Recall that if among the extreme networks of a type G with a boundary ϕ
a locally minimal network exists, then the extreme network in [G,ϕ] is unique,
see[2]. Thus, [G,ϕ] containsatmostonelocallyminimalnetwork. Notice,that
an extreme network need not always be locally minimal. For example, it can
contain vertices of degree more than 3.
Bydefinition,thelengthofanextremenetworkcanbecalculatedastheleast
valueofthe function ρ onRms. Butthis functionis asumofsquarerootsof
G,ϕ
square polynomials on the coordinates of the interior vertices,that complicates
the investigationof the extreme networks in its terms. In the present paper we
show,howthe lengthofanextremenetworkcanbecalculatedbymaximization
of a linear function. But this maximization need to be proceeded on a more
complicatedsubsetoftheconfigurationspace,namely,onanintersectionofsome
cylindersandalinearsubspace. WegeneralizetheMaxwellformulaandmakeit
uniformforalltypes ofnetworksobtainedfroma giventype G by degeneration
ofsomeedgesofthegraphG. Besides,inthecaseofplanarnetworks,weescape
the necessity to go over all possible non-equivalent immersions G′ of a binary
tree G.
Generalized Maxwell formula 6
2 Generalized Maxwell formula
Let G = (V,E) be an arbitrary tree with a boundary B = v ,...,v V
1 n
{ } ⊂
consistingofnelements. UsingthestructureofthetreeG,letusformasystem
S consistingofequationsandinequalitiesonvariablesθj whichareconsidered
G k
as the standard coordinates (θ1,...,θm,...,θ1,...,θm) in the space Rmn. We
1 1 n n
put θ =(θ1,...,θm) and θ =(θ ,...,θ ).
k k k 1 n
Let e be some edge of the tree G. By G = (V ,E ), r = 1, 2, we denote
r r r
the connected components of the graph G e. Thus, G e=G G , and put
1 2
\ \ ⊔
B = B V . By (e) we denote the resulting partition B ,B of the set
r r G 1 2
∩ P { }
B. Let us choose one of B (e), and let B = v ,...,v . By σ we
r ∈ PG r { k1 kp} e
2
denotetheinequality p θ 1,andbyσ wedenotethevectorequation
(cid:13) q=1 kq(cid:13) ≤
n θ = 0. The sy(cid:13)sPtem S i(cid:13)s formed from σ and σ over al edges e of the
k=1 k (cid:13) G (cid:13) e
tPree G.
Let S Rmn be the set of all solutions to the system S . Notice that
G G
| | ⊂
S does not depend on the choice of the components B , since equality σ is
G k
|valid|. Besides, each inequality σ defines a convex subset in Rmn, bounded by
e
an elliptic cylinder, i.e. it is the product of an m-dimensional elliptic disk and
Rm(n−1). The origin 0 together with the points of the subspace Π defined by
equation σ, which are close to 0, are solutions to the system S . Therefore,
G
S is a convex body in the subspace Π.
G
| L|etϕB Rmbeanarbitrarymapping,andz =ϕ(v ),andz =(z ,...,z ).
k k 1 n
→
Put ρ (θ)= θ,z .
ϕ
h i
Theorem 1 Under the above notations, the length of each extreme network
from [G,ϕ] is equal to the largest value of the linear function ρ on the convex
ϕ
set S .
G
| |
Proof. Let θ S be an arbitrary point of the set S . To each pair (v,e),
G G
∈ | | | |
where e E is the edge of the tree G incident to the vertex v V, we assign
the vecto∈r θ(v,e) Rm using the vector θ as follows. Let (∈e) = B ,B ,
G 1 2
∈ P { }
and v B and B = v ,...,v . Then we put
∈ 1 1 { k1 kp}
p
θ(v,e)= θ .
kq
Xq=1
Lemma 1 For any edge e=vw of the tree G we have θ(v,e)= θ(w,e).
−
Proof. The statement of Lemma follows immediately from equality σ.
Lemma 2 For any boundary vertex v , 1 k n, we have θ = θ(v ,e).
k k k
≤ ≤
e:vPk∈e
Proof. Lete =v w ,q =1,...,p,be allthe edgesfromG,whichareincident
q k q
to v . We cut the tree G by the vertex v , and let G = (V ,E ) be the
k k q q q
component containing w . Put B =V B, then
q q q
∩
B =B B v ,
1 p k
⊔···⊔ ⊔{ }
Generalized Maxwell formula 7
hence
p p
0= θ(w ,e )+θ = θ(v ,e )+θ ,
q q k k q k
−
Xq=1 Xq=1
where the last equality follows from Lemma 1, which was to be proved.
Lemma 3 Lete ,j =1, ..., r,bealltheedgesofthetreeG,whichareincident
j
to its interior vertex v. Then the equality
r
θ(v,e )=0
j
Xj=1
holds.
Proof. Indeed, let e = vw , then, due to Lemma 1, we have: θ(v,e ) =
j j j
θ(w ,e ). Let (e ) = Bj,Bj , and v Bj for each j. Then B = Bj,
− j j PG j { 1 2} ∈ 1 ⊔j 2
and therefore,
n
θ(v,e )= θ(w ,e )= θ =0,
j j j k
− −
Xj Xj Xk=1
which was to be proved.
Lemma 4 For each vertex v of the tree G and each its edge e incident to v, we
have θ(v,e) 1.
k k≤
Proof. Takingintoaccountequalityσ(θ),thestatementofLemmaisequivalent
to the validity of inequality σ (θ).
e
Let Γ [G,ϕ] e an extreme network,and v , ..., v be the set of all
n+1 n+s
∈ { }
interiorverticesofthetreeG. Forn+1 k n+swealsodefinez asfollows:
k
≤ ≤
z =Γ(v ).
k k
Lemma 5 Under the above notations, we have
z,θ = z ,θ(v ,e) .
k k
h i
(vXk,e)(cid:10) (cid:11)
vk∈e
Proof. Letus partitionthe sumin the righthandpartofthe equalityinto two
sums: the first one is over all interior vertices v , and the second one is overall
k
the boundary vertices. The first sum vanishes due to Lemma 3. In the second
sumwe groupthe terms correspondingto the same vertexandapply Lemma2.
Lemma is proved.
Thus, due to Lemma 5,
z,θ = z ,θ(v ,e) = z z ,θ(v ,e) ,
k k k l k
h i h − i
(Xvk,e)(cid:10) (cid:11) e=Xvkvl
vk∈e
Generalized Maxwell formula 8
therefore, due to Lemma 4, we have
z,θ = z z ,θ(v ,e) z z =ρ(Γ).
k l k k l
h i − ≤ k − k
e=Xvkvl(cid:10) (cid:11) e=Xvkvl
Since, as we remember, θ is an arbitrary point from S , we conclude that the
G
| |
maximal value of the function ρ (θ) on S does not exceed ρ(Γ).
ϕ G
| |
Now,letusshowthatthismaximalvalueisreached. ForeachΓ-nondegenerate
edge e = vw, by ξ(v,e) we denote the unit vector from Rm having the same
directionastheedgeeofthenetworkΓenteringthepointΓ(v)has. Noticethat
ξ(v,e)= ξ(w,e). Further,duetoExtremeNetworksLocalStructureTheorem
−
(see [2], Theorem 4.1, or [5], Theorem 4.1), for any Γ-degenerate edge e = vw
the pair (v,e) can be assigned with some vector ξ(v,e) Rm, ξ(v,e) 1, in
∈ ≤
suchaway,thattheequalityξ(v,e)= ξ(w,e)holdsandalsot(cid:13)hevecto(cid:13)r-valued
− (cid:13) (cid:13)
function ξ (now ξ is defined on all the pairs (v,e) where vertex v is incident to
edge e) meets relation
ξ(v,e)=0 (1)
eX:v∈e
atanyinteriorvertexv ofthe treeG. We putξ = ξ(v ,e),1 k n
and show that the vector ξ =(ξ1,...,ξn) Rmnkis aPsoel:uvkt∈ioen tokthe sys≤tem≤SG
∈
and ρ (ξ)= z,ξ =ρ(Γ).
ϕ
h i
Indeed, consider an arbitrary edge e = vw of the tree G, and let G e =
\
G G , G =(V ,E ), and v V and B =B V = v ,...,v .
1⊔ 2 r r r ∈ 1 1 ∩ 1 { k1 kp}
Lemma 6 Under the above notations, we have ξ(v,e)= p ξ .
q=1 kq
P
Proof. At first, assume that v B and v =v . Since the equality ξ(v′,e′)=
∈ ks
ξ(w′,e′) holds for any edge e′ =v′w′, we conclude that
−
0= ξ(v′,e′)+ξ(w′,e′) =
e′=vX′w′∈E1(cid:0) (cid:1)
= ξ(v ,e′)+ ξ(v,e′)+
kq
q∈{1,X...,p}\{s} e′∈EX1:vkq∈e′ e′∈EX1:v∈e′
+ ξ(v′,e′)=
v′∈XV1\B1e′∈EX1:v′∈e′
= ξ + ξ ξ(v,e) + ξ(v′,e′).
kq ks −
q∈{1,X...,p}\{s} (cid:0) (cid:1) v′∈XV1\B1 e′∈EX1:v′∈e′
Sincethelattersumvanishesinaccordancetoequality(1), weobtainthe state-
ment of Lemma for the boundary vertex v.
Generalized Maxwell formula 9
Now, let v be an interior vertex of the tree G. Then
0= ξ(v′,e′)+ξ(w′,e′) =
e′=vX′w′∈E1(cid:0) (cid:1)
p
= ξ(v ,e′)+ ξ(v,e′)+
kq
Xq=1 e′:vXkq∈e′ e′∈EX1:v∈e′
+ ξ(v′,e′).
v′∈V1\XB1∪{v} e′∈EX1:v′∈e′
(cid:0) (cid:1)
Thefirstsuminthelatterformulacoincideswith p ξ duetodefinitions,
q=1 kq
thesecondoneisequalto ξ(v,e)inaccordancewithPequality(1),andthethird
−
one vanishes due to equality (1) also, thus, we get the statement of Lemma for
the interior vertex v. Lemma is proved.
Lemma 7 The equality n ξ =0 holds.
k=1 k
P
Proof. Again,taking into accountthe fact thatthe equality ξ(v,e)= ξ(w,e)
−
is valid for any edge e=vw, we get
n
0= ξ(v,e)+ξ(w,e) = ξ(v ,e)+ ξ(v,e).
k
e=Xvw∈E(cid:0) (cid:1) Xk=1 e:Xvk∈e v∈X(V\B) e∈XE:v∈e
The first sum in the latter equality coincides with n ξ due to definitions,
k=1 k
and the second one vanishes due to equality (1). LePmma is proved.
Lemma6andthecondition ξ(v,e) 1imply thatξ meetseachinequality
k k≤
σ . Lemma 7 implies that ξ also meets condition σ. Thus, ξ S . Further,
e G
∈| |
n n
ρ (ξ)= z,ξ = z ,ξ = z ,ξ(x ,e) =
ϕ k k k k
h i h i
kX=1 Xk=1e:Xvk∈e(cid:10) (cid:11)
n+s
= z ,ξ(x ,e) = z z ,ξ(v ,e) =
k k k l k
−
kX=1e:Xvk∈e(cid:10) (cid:11) e=vXkvl∈E(cid:10) (cid:11)
z z
k l
= z z ,ξ(v ,e) = z z , − =
k l k k l
− (cid:28) − z z (cid:29)
e=vXkvl∈E(cid:10) (cid:11) e=vXkvl∈E k k− lk
zk6=zl zk6=zl
= z z =ρ(Γ).
k l
k − k
e=vXkvl∈E
zk6=zl
Theorem 1 is proved.
Remark. Notice that, due to Lemmas 2 and 4, each component of vector θ
S is bounded, therefore S is a compact subset of Rmn. Thus, taking th∈e
G G
| | | |
above into account, we conclude that S is a convex compact.
G
| |
Generalized Maxwell formula 10
As above, let G = (V,E) be a tree with a boundary B = v ,...,v V
1 n
{ } ⊂
and E E be some family of edges of the tree G. By α = G we denote
d k
⊂ { }
thefamilyofnonintersectingsubtreeofG,suchthattheunionofalltheiredges
coincides with E . Assume that each G intersects B by at most one vertex.
d k
Such families E are said to be admissible. Consider the reduced tree G/α and
d
choose its boundary to be the set of all the vertices which intersect the set B.
Due to the restrictions imposed on the subtree G from α, the boundary set
k
obtained consists of the same number of points as the set B does, that gives us
anopportunitytoidentifyitwithB. Thus,foranarbitrarymappingϕB Rm,
→
the next two spaces are defined: [G,ϕ] and [G/α,ϕ]. If Γ [G,ϕ] degenerates
∈
all the edges from E , then Γ generates naturally the network from [G/α,ϕ],
d
which we denote by Γ/α.
As above, consider the system S consisting of equation σ and inequali-
G
ties σ , and through out of it all the inequalities corresponding to the edges
e
e E . By S E we denote the resulting system.
d G d
∈ \
Theorem 2 Under the above notations, the length of extreme network Γ/α
∈
[G/α,ϕ]is equaltothelargestvalueofthelinearfunctionρ on theset S E
ϕ G d
| \ |
of all the solutions to the system S E .
G d
\
Proof. It is sufficient to notice that S E =S , since for any edge e from
G d G/α
\
G/α we have: (e)= (e). Theorem 2 is proved.
G/α G
P P
Thus, throwing out of inequalities of the form σ from the system S is
e G
equivalent to factorization of the tree G by the corresponding edges e. The
authorshope thatthe formulaobtainedforthe calculationofextreme networks
lengths can be useful in investigation of Steiner ratio of Euclidean spaces.
Remark. The results of this paper could be generalized in the following di-
rections. First, Extreme Networks Local Structure Theorem from [2] and [5]
is proved for the case of extreme weighted networks. The classical Maxwell
formula also can be easily transformed to this case (namely, the length of the
direction vector of a weighted edge should be equal to the weight of the edge).
Therefore, the above results can be easily generalized to the case of weighted
trees.
Second, the classical Maxwell formula remains valid for extreme networks
with cycles. Therefore it seems not very difficult to generalize Theorems 1
and 2 to the case of arbitrary extreme networks in Rm (not necessary the trees
as in the present paper), and also to the case of arbitrary weighted extreme
networks in Rm.
At last, in [2] we obtained theorems on the local structure of extreme net-
worksinnormedspaces. Itwouldbe interestingtoworkoutananalogueto the
Maxwell formula for this case in terms of so-called ρ-impulse, see [2], and after
thatto generalizethe results ofthe presentpaper to the caseofnormedspaces.