Length of a Full Steiner Tree as a Function of Terminal Coordinates
LLength of a Full Steiner Tree as a Function of Terminal Coordinates
Alexei Yu. Uteshev, Elizaveta A. Semenova St. Petersburg State University, St. Petersburg, Russia
Given the coordinates of the terminals { ( x j , y j ) } nj =1 of the full Euclidean Steiner tree, its lengthequals (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 z j U j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where { z j := x j + i y j } nj =1 and { U j } nj =1 are suitably chosen 6th roots of unity. We also extend thisresult for the cost of the optimal Weber networks which are topologically equivalent to some fullSteiner trees. The problem of construction of the shortest possible network interconnecting the set of points (furtherreferred to as terminals ) P = { P j } nj =1 ⊂ R d , n ≥ , d ≥ Euclidean Steinerminimal tree problem ( SMT problem). We will treat here only the planar version of the problem,i.e. d = 2. For this case, some specializations of the set P guarantee the solution to the problem inthe form of a minimum spanning tree. However, for other configurations of P , the optimal networkcontains some extra vertices (junctions) known as the Fermat-Torricelli or Steiner points . Everysuch a point in the optimal network possesses the degree 3, and has the adjacent edges meeting at2 π/ P is the subjectof numerous and inventive research efforts. In the variety of potential solutions, it is possible todistinguish a type of subtrees with a relatively ordered structure. A Steiner tree with n terminals iscalled a full Steiner tree if it contains n − P . Any pair of these subtrees might have at most one commonterminal.The first aim of the present paper is express the length of a full Steiner tree as an explicit functionof the terminal coordinates. In [1] this length is computed via the lengths of the segments | P j P j +1 | (for the appropriate numeration of terminals). In [12] the expression for the length was obtainedin the so-called hexagonal coordinates of terminals which are claimed to be more natural than thecommon Cartesian coordinate system . We intend to dispute this claim by presenting the formula for { alexeiuteshev,semenova.elissaveta } @gmail.com a r X i v : . [ m a t h . C O ] F e b he length in terms of Cartesian coordinates. For the aim of deducing the formula from the Abstract,we will place the problem into the complex plane. In Section 3 we present the proof of the lengthformula and discuss its relation to one Maxwell’s result [6].A natural generalization of the SMT problem is the (multifacility) Weber problem . It is statedas that of location of the given number (cid:96) ≥ facilities ) { W r } (cid:96)r =1 ⊂ R d connectedto the terminals of the set P that solve the optimization problemmin { W ,...,W (cid:96) }⊂ R d (cid:40) n (cid:88) j =1 (cid:96) (cid:88) r =1 m rj | W r P j | + (cid:96) (cid:88) k =1 (cid:96) − (cid:88) r = k +1 (cid:101) m rk | W r W k | (cid:41) ; (1.1)here some of the weights m ij and (cid:101) m ik might be zero. The value (1.1) will be referred to as the minimal cost of the network . It can be expected that this problem should be more complicatedthan the SMT one. It really is even for the planar case, i.e. for d = 2. In contrast with theSMT, the optimal Weber network might contain facilities of degree greater than 3. However, weconjecture that in a restricted version of the planar Weber problem, one may expect the existenceof the counterpart for the Steiner tree length formula expressing the minimal cost by radicals withrespect to the problem parameters of (terminal coordinates and weights). This relates to the networkstopologically equivalent to some full Steiner trees, i.e. the networks where each terminal has degreeequal to 1 while the degree of each facility equal to 3. We discuss this issue in Section 5. Notation.
We set z j := x j + i y j and denote the 6th roots of unity υ k := cos πk i sin πk k ∈ { , , . . . , } , (1.2)while the third roots of unity as ε = 1 , ε = −
12 + i √ , ε = − − i √
32 ; (1.3)Evidently, υ = 1 = − υ , υ = − υ = − ε , υ = − υ = ε . L and C stand for respectively minimal length of the Steiner network and minimal cost of theWeber network.For the convenience of the reader, we place here the Classical Geometry result which we oftenrefer to in what follows: Theorem 1.1 (Ptolemy)
A quadrilateral is inscribable in a circle if and only if the product of thelengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.
Geometric conditions for the existence of a full Steiner tree for the case n = 3 and n = 4 terminalsare well known, and we assume them to be fulfilled for the terminal sets treated in the present2ection. In [8] these conditions are converted into analytical form, aside with formulas for Steinerpoint coordinates and the tree length. This provides one with an opportunity to verify directly thetest examples of this and the next sections.For the case of 3 terminals, the Torricelli-Simpson construction of the Steiner (Fermat-Torricelli)point is based on finding an extra point Q in the plane yielding the equilateral triangle P P Q (Fig.1). (a) (b)Figure 1. Steiner tree construction for three terminalsThe point of intersection of the segment Q P with the circle circumscribing this triangle is theSteiner point S . Due to Theorem 1.1, one gets | Q S | = | P S | + | P S | , and therefore the length of the SMT connecting the terminals P , P , P equals | Q P | . The coordi-nates of the point Q can be easily determined. There are two possibilities for its location (on eachside of the line P P ), and we need the variant opposite to P . For a configuration similar to thatdisplayed in Fig. 1 (i.e. the triangle vertices are numbered counterclockwise), one gets: Q := ( q x , q y ) = (cid:32) x + 12 x − √ y + √ y , √ x − √ x + 12 y + 12 y (cid:33) . (2.1)or, alternatively, in the complex plane q := q x + i q y = (cid:32)
12 + i √ (cid:33) x + (cid:32) − i √ (cid:33) x + (cid:32) − √
32 + i (cid:33) y + (cid:32) √
32 + i (cid:33) y (cid:32)
12 + i √ (cid:33) z + (cid:32) − i √ (cid:33) z = υ z + υ z = − ε z − ε z . For this case, the length of the (full) Steiner tree equals L = | Q P | = | z − q | = | υ z + υ z + z | = | z + ε z + ε z | . (2.2) Example 2.1
For the terminals P = (4 , , P = (2 , , P = (7 , , (Fig. 1), formula (2.2) yields L = | i + ε (4 + 4 i ) + ε (2 + i ) | = | √ / − i (3 / √ | = (cid:113)
28 + 15 √ ≈ . . For the case of 4 terminals { P j } j =1 , the geometrical algorithm for Steiner tree construction [3]with two Steiner points in the topology P P S S P P , is illuminated in Fig. 2. Coordinates of Q are computed similar to Q : Q := ( q x , q y ) (cid:55)→ q = q x + i q y = υ z + υ z . The length of the tree equals L = | Q Q | = | q − q | = | υ z + υ z + υ z + υ z | (1 . = | ε ( z − z ) + ε ( z − z ) | From this immediately follows equivalent equalities L = L · | − ε | = | ( z − z ) + ( z − z ) ε | = | ( z − z ) + ( z − z ) ε | . (2.3)In the case of existence a Steiner tree in the alternative topology P P ˜ S ˜ S P P , its length is givenby the formula ˜ L = | ( z − z ) + ( z − z ) ε | . Example 2.2
For the terminals P = (2 , , P = (1 , , P = (9 , , P = (6 , , and the Steiner tree topology P P S S P P (Fig. 2), formula (2.3) yields L = | − i + ( − − i ) ε | = | − / − √ i (7 + 5 √ / | = (cid:113)
115 + 62 √ ≈ . . P P P P Q Q S S P P P P S S Figure 2. Steiner tree construction for four terminals5 heck.
The Steiner points: S = (cid:18) √ , √ (cid:19) ≈ (2 . , . ,S = (cid:18) − √ , √ (cid:19) ≈ (5 . , . . Our next aim is to discover the correlation between the sequences of terminals { z j } and the rootsof unity (1.2) in the general formula for the tree length. Consider a full Steiner tree T with n ≥ P j be incident to the edge SP j linking it to the adjacent Steiner point S of the tree. We treat the edge SP j to be oriented from S to P j and denote by −→ (cid:96) j := −−→ SP j / | SP j | (3.1)the direction of the terminal P j in the tree T , i.e. a unit vector with its starting point attachedto the origin. It is known that for the full Steiner tree, the angle between −→ (cid:96) j and −→ (cid:96) k is an integermultiple of π/
3. Therefore, all the terminals can be sorted according to the 6 possible directions. Forour purpose, we do not need the exact values of these directions, but only their relative positions.
Theorem 3.1
For the full Steiner tree T with n ≥ terminals, assign to each P j the value U j of th root of unity (1.2) according to the following rules:(a) P → U := υ = 1 ;(b) P j → U j := υ k if the angle between −→ (cid:96) and −→ (cid:96) j counted clockwise equals kπ/ .Then L ( T ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 z j U j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.2) Proof is by induction on n . For n = 3 the statement holds as it is demonstrated in Section2. Assume the result to be true for an arbitrary full Steiner tree with n − ≥ T n with n terminals, and number these terminals and Steiner pointssomehow with the only requirement that the terminals P n − and P n should be adjacent to the sameSteiner point S n − which, in turn, should be adjacent to the Steiner point S n − .6eplace the pair P n − , P n by a point defined by either of alternatives Q (cid:55)→ q = (cid:26) υ z n − + υ z n ,υ z n − + υ z n (3.3)in accordance with its location on the side of the line P n − P n other that of S n − (Fig. 3).(a) (b)Figure 3. Location of the point Q .The new tree with terminals P , . . . , P n − , Q and Steiner points S , . . . , S n − is the full Steinertree (the points S n − , S n − and Q are collinear). Due to Theorem 1.1, one has: | S n − S n − | + | P n − S n − | + | P n S n − | = | S n − S n − | + | S n − Q | = | S n − Q | . Therefore, the new ( n − T n − has its length equal to L ( T n ). The direction of theterminal Q in the tree T n − (i.e. −→ (cid:96) = −−−−→ S n − Q/ | S n − Q | ) obeys the assignment rule from the statementof the theorem, i.e. it differs from −→ (cid:96) by an integer multiple of π/
3. Based on the inductive hypothesis,one may conclude that L ( T n − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) j =1 z j U j + q U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where the sequence U , . . . , U n − , U is computed in accordance with the assignment rule. Assume,for definiteness, that • −→ (cid:96) coincides with −→ (cid:96) , i.e. U = υ = 1, • and q be defined by the first alternative from (3.3).7hen L ( T n ) = L ( T n − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) j =1 z j U j + υ z n − + υ z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In the obtained formula, the assignment rule for terminal directions is kept fulfilled. The direction −→ (cid:96) n − is π/ −→ (cid:96) n is π/ −→ (cid:96) (Fig. 3 (a)). (cid:3) Corrolary 3.1
In the notation of Theorem 3.1, one has n (cid:88) j =1 U j = 0 . The proof can be extracted from that of the theorem. From this statement and from the homo-geneity of (3.2) with respect to the terminal coordinates, the following assertion holds
Corrolary 3.2
The value (3.2) is invariant under the affine transformation of the complex plane z (cid:55)→ w z + z , ∀{ w , z } ⊂ C , | w | = 1 . In particular, it is possible to choose such a rotation of the plane R around the origin that,in the new coordinates, the expression (cid:80) nj =1 z j U j in (3.2) becomes a positive real number. In thisversion, formula (3.2) transforms into the interpretation by Gilbert and Pollak of Maxwell’s theoremon the equilibrium condition of the pin-jointed rigid rods holding a prescribed system of externalforces [3, 6]. However, generically, to find this rotation for an arbitrary given set P is not a trivialtask. For its evaluation, direction of at least one terminal P j in the tree T should be established, i.e.the coordinates of the adjacent Steiner point in (3.1). For Example 2.2, this “good” rotation is givenby w = cos ϕ + i sin ϕ with ϕ := π + arcsin (cid:18) (cid:113) − √ (cid:19) . To complete the present section, we convert (3.2) into real numbers.
Corrolary 3.3
Let { U jx := Re ( U j ) , U jy := Im ( U j ) } nj =1 . Then { U jx } nj =1 ⊂ { , ± / } , { U jy } nj =1 ⊂ { , ±√ / } , { U jx + U jy = 1 } nj =1 . Set X := ( x , . . . , x n ) , Y := ( y , . . . , y n ) , U X := ( U x , . . . , U nx ) , U Y := ( U y , . . . , U ny ) . Then L ( T ) = (cid:113) ( (cid:104) U X , X (cid:105) − (cid:104) U Y , Y (cid:105) ) + ( (cid:104) U Y , X (cid:105) + (cid:104) U X , Y (cid:105) ) where (cid:104)·(cid:105) is the standard inner product in R n . Examples
We present here two examples illuminating the usage of formula (3.2).
Example 4.1
For terminals P = (3 , , P = (1 , , P = (6 , , P = (10 , , P = (8 , the Steiner tree is displayed in Fig. 4 (a). The assignment scheme from Theorem 3.1 is given inFig. 4 (b). L = | z + z + υ z + υ z + υ z | = | ( z − z + z ) + ε z + ε z | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − √
32 + i
19 + 9 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:113)
152 + 86 √ ≈ . . Check.
The Steiner points S S S = (cid:0) + √ , + √ (cid:1) = (cid:0) − √ , − √ (cid:1) = (cid:0) + √ , + √ (cid:1) ≈ (3 . , . ≈ (5 . , . ≈ (8 . , . xample 4.2 For terminals P = (2 , − √ , P = (8 , , P = (4 , √ , P = ( − / √ , , P = ( − , , P = ( − , − √ the Steiner tree is displayed in Fig. 5 (a). The assignment scheme is given in Fig. 5 (b) L = | z + υ z + υ z + υ z + υ z + υ z | = | ( z − z ) + ε ( z − z ) + ε ( z − z ) | = | ( − − / √ − i (8 + 15 √ | = 30 + 15 / √ ≈ . . Check.
The Steiner points: S = (4 , , S = (2 , √ , S = (2 , − √ , S = ( − , . (a) (b)Figure 5. (a) The Steiner tree for 6 terminals and (b) its assignment scheme (Example 4.2). We now turn to the Weber problem mentioned in Introduction. In the present section we assume thefulfillment of the conditions for the existence of the networks in the prescribed topologies (some ofthem are presented in [9, 10]). In particular, we suppose all the involved geometric constructions tobe feasible. Hereinafter all the weights m j , m, . . . are positive real numbers and the angles denotedby Greek letters α, β, γ are assumed to be within the interval [0 , π ].We start with the three terminal problem (also known as the generalized Fermat-Torricelli prob-lem) as that of finding the facility W providingmin W ∈ R (cid:88) j =1 m j | W P j | . (5.1)10 geometric solution to the problem is given in the paper by Launhardt [4, 5]; it is just a counterpartof the Torricelli-Simpson construction for the Steiner problem from Section 2. Example 5.1
Find the optimal position of the facility W to the problem (5.1) where (cid:26) P = (2 , P = (1 , P = (5 , m = 2 m = 3 m = 4 (cid:27) . Solution.
First find the point Q lying on the opposite side of the line P P with respect to thepoint P and such that | P Q | = m m | P P | , | P Q | = m m | P P | . (5.2)These relations mean that the triangle P P Q is similar to the so-called weight triangle of theproblem, i.e. the triangle containing the edges formally coinciding with the values of the weights m , m , m . We will further denote the angles of the weight triangle by α , α , α as shown in Fig. 6(a). Next, draw the circle C circumscribing P P Q . Finally draw the line through Q and P .The intersection point of this line with C is the position of the optimal facility W . It possessesthe property that directions of the terminals, i.e. −→ (cid:96) j := −−→ W P j / | W P j | , make the following angles(Fig. 6 (b)) (cid:92) −→ (cid:96) , −→ (cid:96) = π − α , (cid:92) −→ (cid:96) , −→ (cid:96) = π − α , (cid:92) −→ (cid:96) , −→ (cid:96) = π − α . The corresponding (minimal) cost equals m | P Q | (consequence of (5.2) and Theorem 1.1). (cid:3) (a) (b)Figure 6. (a) Weight triangle and (b) Weber point location (Example 5.1).11e now rewrite the analytics similar to that from Section 2. The coordinates of the point Q areas follows [9] Q = (cid:18)
12 ( x + x ) + ( m − m )( x − x ) − √ k ( y − y )2 m ,
12 ( y + y ) + ( m − m )( y − y ) + √ k ( x − x )2 m (cid:19) , (5.3)or, equivalently, in the complex plane: Q (cid:55)→ q = 12 m (cid:104) ( m + m − m + i √ k ) z + ( m − m + m − i √ k ) z (cid:105) (5.4)Here k := ( m + m + m )( − m + m + m )( m − m + m )( m + m − m ) , (5.5)and, due to Heron’s formula, √ k / m + m − m m m = cos α , m − m + m m m = cos α , (5.6)while √ k m m = sin α , √ k m m = sin α . (5.7)Finally, one has q = 1 m [ m (cos α + i sin α ) z + m (cos α − i sin α ) z ]and C = m | P Q | = | m z − m q | = | m z + m z { cos( π − α ) + i sin( π − α ) } + m z { cos(2 π − α − α ) + i sin(2 π − α − α ) } | . The obtained formula is the true counterpart of the formula (2.2) for the length of the full Steinertree for 3 terminals.Let us now turn to the bifacility Weber problem for 4 terminals { P j } j =1 as that of finding thepoints W and W which yield min { W ,W }⊂ R F ( W , W ) where F ( W , W ) = m | W P | + m | W P | + m | W P | + m | W P | + m | W W | . (5.8)We first recall the geometric solution outlined by Georg Pick in the Mathematical Appendix ofWeber’s book [11]. We illustrate this algorithm with the following example. We recall the assumption made at the beginning of the present section: all the angles denoted by Greek lettersare assumed to lie within [0 , π ]. Therefore all the sine values for such angles are nonnegative. xample 5.2 Find the optimal position for the facilities W and W for the problem (5.8) where (cid:26) P = (1 , P = (2 , P = (7 , P = (6 , m = 3 m = 2 m = 3 m = 4 m = 4 (cid:27) . Solution.
First find the point Q lying on the opposite side of the line P P with respect to thepoint P and such that | P Q | = m m | P P | , | P Q | = m m | P P | . (5.9)The exact coordinates of this point are given by (5.3) where the substitution m → m is made. Findthen the second point Q with the similar property with respect to the points P and P (Fig. 7): | P Q | = m m | P P | , | P Q | = m m | P P | . Figure 7. Construction of the points Q and Q .It is evident that triangle P P Q is similar to the weight triangle with the edges m, m , m , while P P Q is similar to the weight triangle with the edges m, m , m (Fig. 8) Next, draw the circle C circumscribing P P Q and C circumscribing P P Q .13igure 8. Weight triangles for the bifacility Weber problemFinally draw the line through Q and Q (Fig. 9). P P P P Q Q W W C C P P P P W W Figure 9. Pick’s construction for the Weber network (Example 5.2).The intersection points of this line with C and C are the position of the optimal facilities W and W for the network with the corresponding (minimal) cost equal to C = m | Q Q | . Remark.
Pick’s solution can be interpreted as a counterpart of the algorithm worked out byGergonne in 1810 (and rediscovered by Melzak in 1961) for solution of the SMT problem for fourterminals. It should be noted that Pick did not provide any reasons of validity for his construction(we also failed to find any references to Pick’s solution in subsequent papers on the subject). In [9, 10]we have verified the accuracy of Pick’s treatment via an analytical representation of the coordinates14f the optimal facilities W , W and substitution them into the gradient of the function (5.8). Someother qualitative conclusions have been deduced, such as, for instance, that, in the case of existence,the bifacility network is less costly than any unifacility one. All the computations for the examplesbelow can be found in the cited sources.Thus, the coordinates for the facilities from the previous example are as follows W = (cid:18) √ √ √ ( √ √ √ ) , √ √ √ ( √ √ √ ) (cid:19) ≈ (3 . , . W = (cid:18) √ √ √ ( √ √ √ ) , √ √ √ ( √ √ √ ) (cid:19) ≈ (4 . , . C = 18 (cid:113) √
15 + 5118 √
55 + 1890 √ ≈ . . (5.10)Let us deduce the general formula for C = m | Q Q | for the networks which are topologicallyequivalent to the one dealt with in Example 5.2. The coordinates of the point Q are determined by(5.4) where substitution m → m is made. The coordinates of Q are also obtained from (5.4) byreplacement ( m , m , m ) → ( m , m , m ). Thus C = m | Q Q | = m | q − q | = | m z (cos α + i sin α ) + m z (cos α − i sin α ) − m z (cos α + i sin α ) − m z (cos α − i sin α ) | . (5.11)Here α , α are the angles of the first weight triangle of the problem, while α , α are those of thesecond (Fig. 8). Cosine and sine functions of these angles are computed similar to (5.6) and (5.7).For the network of Example 5.2 one gets the cost value C = (cid:12)(cid:12)(cid:12)(cid:12) i (cid:18)
78 + 124 i √ (cid:19) + 2(2 + i ) (cid:18) − i √ (cid:19) − . . . (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − − √ − √ − i (cid:18)
638 + 18 √
135 + 18 √ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) which coincides with (5.10). 15a) (b)Figure 10. Angles in the bifacility Weber network.Let us rewrite (5.11) uniformizing representation of the arguments of trigonometric functions: C = | m (cos α + i sin α ) z + m z (cos( π − α ) + i sin( π − α ))+ m z (cos( π + α ) + i sin( π + α )) + m z (cos(2 π − α ) + i sin(2 π − α )) | . (5.12)Here one can watch the generation rule for these arguments: they correspond to the direction anglesof the terminals P j in the network if we count them clockwise starting from the line W W (Fig. 10). Theorem 5.1
Let the optimal Weber network T with n ≥ terminals { P j } nj =1 and n − facilities { W k } n − k =1 be topologically equivalent to a full Steiner tree, i.e. each terminal from T is of the degree while each facility is of the degree . Let −→ (cid:96) j = (cos β j , sin β j ) be the direction of the terminal P j inthe network counted clockwise starting from some particular vector. The cost of the network T in theprescribed topology is given by the formula C ( T ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 m j z j (cos β j + i sin β j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.13)The results of Corollaries 3.1 and 3.2 from Section 3 can evidently be upgraded to the “weighted”version. For instance, one has n (cid:88) j =1 m j (cos β j + i sin β j ) = 0 . (5.14)To justify the result of the theorem, we suggest here some plausible reasonings . Example 5.3
Find the cost of the optimal network that minimize the function m | P W | + m | P W | + m | P W | + m | P W | + m | P W | + (cid:101) m , | W W | + (cid:101) m , | W W | (5.15)16 or the following configuration: (cid:26) P = (1 , P = (5 , P = (11 , P = (15 , P = (7 , (cid:101) m , = 10 m = 10 m = 9 m = 8 m = 7 m = 13 (cid:101) m , = 12 (cid:27) . Solution.
Existence of the optimal network and coordinates of the corresponding facilities W , W , W are established in [9]. We will discuss here only the minimal cost calculation.Figure 11. Weber network construction for five terminals (Example 5.3).Replace a pair of the terminals P and P by the phantom terminal Q defined by the formula (5.4)where the substitution m → (cid:101) m , is made, and assign this weight to Q . The minimal cost of thenew { } -network minimizing the function (cid:101) m , | Q W | + m | P W | + m | P W | + m | P W | equals the minimal cost of the original network (and facilities W , W in both optimal networkscoincide). By (5.12), C = | (cid:101) m , q (cos(2 π − γ ) + i sin(2 π − γ )) + m z (cos γ + i sin γ ))+ m z (cos( π − γ ) + i sin( π − γ )) + m z (cos( π + γ ) + i sin( π + γ )) | (5.16)with the angles { γ j } j =1 displayed in Fig. 12 (a). 17a) (b)Figure 12. Angles of the trifacility Weber network (Example 5.3).One has cos γ = (cid:101) m , + (cid:101) m , − m (cid:101) m , (cid:101) m , , cos γ = (cid:101) m , + m − (cid:101) m , m (cid:101) m , , cos γ = m + (cid:101) m , − m (cid:101) m , m , cos γ = m + (cid:101) m , − m (cid:101) m , m and the expressions for the corresponding sine values are represented, using the involved weights,similarly to (5.7). Next we replace q by the original terminals z and z via (5.4) ( m → (cid:101) m , ): q = 1 (cid:101) m , [ m (cos α + i sin α ) + m (cos(2 π − α ) + i sin(2 π − α ))]with the angles α , α displayed in Fig. 12 (b) and withcos α = (cid:101) m , + m − m m (cid:101) m , , cos α = (cid:101) m , + m − m m (cid:101) m , . Finally, C = | m z (cos( α − γ ) + i sin( α − γ )) + m z (cos(2 π − α − γ ) + i sin(2 π − α − γ )) + . . . | with the rest of the terms coinciding with those from (5.16). The terminal direction assignment rulefrom Theorem 5.1 is fulfilled: these angles are counted starting from the line W W . Substitution18f the expressions for sine and cosine functions in terms of all the involved weights yields the costrepresentation by radicals. For our particular example, C = (cid:12)(cid:12)(cid:12) − − √ − √ − √ − √ √ i (cid:18) √ − √
319 + 32 √
143 + 964 √ √ (cid:19) (cid:12)(cid:12)(cid:12) ≈ . . (cid:3) The result of the theorem remains valid even for the networks without the imposed restriction ontheir topologies. For instance, the unifacility Weber problem (generalized Fermat-Torricelli problem)for n ≥ W ∈ R n (cid:88) j =1 m j | P j W | , (5.17)in the case of existence of solution W ∗ = ( x ∗ , y ∗ ) (cid:54)∈ { P j } nj =1 , has the value (5.13) equal to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 m j z j (cos β j + i sin β j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5 . = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 m j ( z j − z ∗ )(cos β j + i sin β j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , z ∗ := x ∗ + i y ∗ . If the direction (cos β j , sin β j ) of the terminal P j is counted starting from the x -axis, then 2 π − β j isjust the argument of the number z j − z ∗ .However, the posed restriction on the network topology is significant for the constructive compu-tation of the directions of the terminals in the network. Indeed, in the case of networks possessingthe claimed property, formulas for the direction of any terminal can be represented by radicals viathe weights of the problem provided that these directions are counted starting from any edge ofthe optimal network. Note that the coordinates of facilities are not required for this aim, the onlyassurance of the existence of the network in the prescribed topology matters.This is not the case for the networks with facilities of the degree higher than 3. Even the problem(5.17) cannot be resolved by radicals for general case of n = 4 terminals. For instance, when theweight m from Example 5.2 increases continuously from the initial value m = 4, the facilities W and W of the corresponding optimal bifacility networks tend to a collision point W ∗ at the value m = m ∗ .The point W ∗ is the solution to the unifacility Weber problem (5.17) with n = 4. Coordinates of W ∗ satisfy the 10th degree algebraic equations over Z ; so do the critical value m ∗ ≈ . Though the deduced formulas for the length or cost of optimal networks do not require the coor-dinates of any network facility, it is possible to generate them in the above exploited complexification19deology. For instance, in the complex plane, the Steiner point of the triangle P P P (in the case ofits existence and provided that the triangle vertices are numbered counterclockwise) is given as i √ (cid:0) ε z − ε z + ( z − z ) L / L (cid:1) where L := z + ε z + ε z is the expression from formula (2.2), i.e. |L| is the length of the fullSteiner tree. For the terminals of Example 2.1, this formula yields454 + 250 √ i (262 + 150 √ √ √ i
293 + 135 √ ≈ . i . . It looks challenging to find the minimum value for (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 z j (cid:101) U j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in the set of all vectors ( (cid:101) U , . . . , (cid:101) U n ) ∈ C n such that (cid:110) (cid:101) U j = 1 (cid:111) nj =1 , n (cid:88) j =1 (cid:102) U j = 0 . It resembles a knapsack problem. 20 eferences [1] Booth R.S. Analytic formulas for full Steiner trees.
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LNCS , Springer,2020, V.12251, pp. 395–411.[11] Weber A. ¨Uber den Standort der Industrien. Bd. 1: Reine Theorie des Standorts. 1909. T¨ubin-gen. English translation: Friedrich C.J. (Ed.). Alfred Weber’s theory of location of industries.Chicago: The University of Chicago Press, 1929.[12] Weng J.F. Steiner trees, coordinate systems and NP-hardness. In