Some upper bounds for minimal trees

Information Processing North-Holland

38 (1991) 209-213

31 May 1991

asart Departament d’Inforr+tica, Facultat de Cihcies, Universitat Autbnoma de Barcelona, 08193 Bellaterra, Catalonia. Spain Communicated by L. Boasson Received 21 February 1990 Revised 5 June 1990

Abstract Basart, J.M., Some upper bounds for minimal trees, Information Processing Letters 38 (1991) 209-213. This paper introduces two upper bounds for the length of two kinds of minimal trees. The first upper bound, a(n) < T([(( n I)/IT)*/~] + l/2) + l/2, is related to the Euclidean Steiner Minimal Tree obtained from any set of n points enclosed inside a circumference of radius 1. The second upper bound, G(n) 6 2( n - 1)/k, is related to the Rectilinear Minimal Tree obtained from any set of n = !( k( k + 2) + 1)/21 points (k = 1, 2, 3,. . . ), enclosed in a square with sides equal to 1. Moreover, we also present the only case where the well-known bound u(n) = 1 + fi is attained for the length of the Rectilinear Steiner Minimal Tree for any set of n = t2 points (t = 2, 3,. . _) enclosed in a square with sides equal to 1.

Keywork

Computational geometry, combinatorial problems

UCtiOll

The Euclidean Steiner Minimal Tree (ESMT) problem is an old problem known since at least the XVIIth century, when it was considered by the famous mathematician Pierre Fermat. The terms of the problem are: given a set P of n points on the euclidean plane, find the shortest set of lines which links them making a tree. What makes this problem so difficult-that is to say NP-hard [4]-is the fact that as many as n - 2 points can be added to P in order to reduce the length of the final tree. Of course, how many new points (Steiner points in the literature) are needed, and where they have to be placed, is currently unknown in the general case. The work collecby Gilbert and Pollack [S] contains a TS. tion of geometrical properties for the e of the most interesting aspects about the : what is can have 0020-0190/91/$03.50

when its n points are located inside some closed region? The first approach was obtained by Fete [ 31; considet ing a unit square its bound was D( n ) < fi + 7/4. As far as we know, the last result for this problem was given by Chung and Graham in [I]. They proved that, considering the same region that Few considered, there exists an integer t which verifies that a(n) K 0.99&h for every integer n 2 t. The rectilinear distance between two points p =(P~, p2)andq=(ql, q2)is@venby Ipl-qll + 1pz - q2 I. The Rectilinear Steiner Minimal Tree (RSMT) problem [6] can be described as the ES was, the difference is that in this case the rectilinear distance is used instead of the euclidean one, so, all the lines in the tree will be horizontal lines rtical ones. or [I], the authors pr ose the conjecture a( n; o(n) being now t \( 1 + fi for any value the oint

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square of one unit side. This conjecture is based on the fact that they showed (in [l] using 121) that a(n)=l+fi when n=t’, t=2,3,4 ,... . In this paper we present three theorems related to these topics. The first one gives an upper bound a(n) for the length of the ESMT when the region considered is a circumference of radius 1. The second one identifies the only distribution of points where, in the case of the RSMT, a(n) is attained when n = t2, t = 2, 3, 4,. . . . Finally, the last result gives an upper bound @r(n) for the length of any Rectilinear Minimal Tree (RMT) in a unit square. The RMT has the same purpose as the SMT but it does not allow Steiner points, only the n initially given points will be considered. Fig. 1.

2.

44

The method that will be used to obtain this bound is based on Few’s idea. Briefly, Few’s process takes three steps: (1) Draw a suitable skeleton S of homogeneous lines covering the closed region where the points are located. (2) Link each point in P to its nearest skeleton-line by means of a straight segment. (3) Remove from the configuration obtained in (2) the longest set of lines without loosing the connectivity among the points in P. The result is a tree, and its length can now be easily computed and it is taken as the upper bound for the ESMT. . Given any distribution of n points enclosed in a circumference radius equal to 1, the maximum length of its E is

where [ ] represents the ceiling function. . A natural way of constructing a skeleton S in a circumference Co is to take m - 1 inner concentric circumferences Ci, C2,. . . , Cm_ i with radii equal to 1 - (l/m ), 1 - (2/m ), . . . ,I 210

((m - 1)/m), respectively. The value of m is a function of n and will be fixed later. Once S has been built, the points in P are linked to one of the Ci’S as follows: (a) points located inside the smallest circumference Cm_ 1 are radially linked to it and to its center; (b) points located on a circumference Ci are radially linked to Ci+ 1; and (c) points located between q and q+i are radially linked to both. So, in all cases, the length of the segments used for each point is l/m. Let us assume that at least one point in P is linked to the circumference Ck, k = 0, 1,. . . , rz - 1. Now, for every inner circumference Ci, C2,. . . , Cm-i, one point pi lying on Ci, i = 1, 2 . . , m - 1 is chosen supposed that it exists. For each of these m - 1 points located in the m - 1 circumference inner to Co, a new segment of length l/m is radially drawn linking the point on Cj to Ci_,. The situation at this moment is shown-for a particular case-in Fig. 1. f we consider ihe following: the even(odd)numbered circumferences, both segments affecting the point pi located on every odd(even)-numbered circumference Ci and the segments from the other points to the even(odd)-numbered circumferences, then, in both cases an interconnection net for P is obtained. )

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Let us suppose that the assumption made before has failed. If some Ck has no points linked to it, select a point linked to Ck _ 1 and extend its link till Ck+i. Right away, Ck is eliminated from the constructions. In this way, the resulting construction keeps both the interconnection net and its desired length. Now, the sum L of the lengths of both interconnection nets is less than or equal to _;f 2

2n l+l-;+1 ( +n$+l-,

iDD +l_“?-

1 m

1

1

which after some direct calculations as

can be stated

L, [a*], and performing some direct calculations, we obtain 42[y3’*]

+,)

+I.

So, this result implies that it is possible to obtain one tree with a length shorter than L/2, or,

as claimed above.

0

A Steiner Equally Spaced Distribution (SESD) for a set of n = t* points (t =2, 3, Lg,...) in a square of one unit side with the rectilinear metric, is a distribution of t rows and t columns making a regular grid, where the distance between every pair of points belongs to the succession k/(t - I), k= 1, 2,..., 2( t - 1). In Fig. 2 the SESDs corresponding to the cases tg = 4 and n = 9 are sketche

Fig. 2.

GivenasetPofn=t*(t=2,3,4....) points distiibuted inside a square of one unit side, ng the rectilinear distance among them, the corresporlding to P attains its maximum length a(n) = 1 + fi if, and on/j if, P is a SESD. . The sufficiency is clearly true. corresponding to a SESD will never use Steiner points, that is to say, every segment will dire&v link two of the n points. This is due to the fact that the set of points candidates to be Steiner points are always located at the intersection points of the grid obtained by drawing all the horizontal and vertical lines through the set of n initial points [B]. So, the RSMT will take n - 1 segments of length equal to l/( t - l), the sum of all these segments being 1 + fi. To show the reverse, the grid of lines formed by the correspondent SESD with n points is represented over the one unit side squared region. Next, P, which is assumed different from the SESD, is placed over the r = (t - l)* squared cells k,, k *, a. . , k, given by the SESD. ‘We consider that the cell ki (i = 1, 2,. . . , r) contains n, points of P. Each point of P placed on the border between two or four cells will be considered as belonging to any one of them, so, the equality n, + n2 + - - - + n, = n holds. The RST linking the nj, points contained in kj (j= 1, 2,..., r) has a length less than or equal to nj/( t - 1); in fact, the equality is attained only in those cells containing exactly two points placed in opposite corners. Note also, that the length nj/( t - I) always allows the RST contained in k, tar h;~va,a line with length equal to l/(t - 1) on one of the sides of kj and this line m that lies in the cell b ST can be constructed j of RSTs through his common horizontal line and, at the end, including a straight vertical line of 211

length 1 to ensure the connectivity among the pairs of partial RSTs. Therefore, the length L of the RST corresponding to P will be less than or equal to the total length used for connectmg the set of RSTs distributed inside the r cells; that is to say, L

<

Fig. 4.

-&+i-=-i.+...+& *2

--.- o-u2

1

25 points, are shown, but whereas the solution tree in the SESD has a cost of 6, the RMT for P has a cost of 8. An Equally Spaced Distribution (ESD) for a set of n points in a unit square with the rectilinear metric, is-when it exists-a distribution that verifies the following: (i) for any of its points, all the other n - 1 ones are located at a rectilinear distance equal to d or 2d or 3d . . . . d being the minimum distance between two points in the ESD; (ii) if one of its n points is displaced, then the length of the RMT decreases; (iii) if a new point is added anywhere in the ESD, then the length of the new RMT does not change. Figure 4 shows three ESDs with values n = 2, 5 and 8. In these examples, the simple alternation of points in the vertices of the underlying grid shows how the ESDs are built, and proves that they do not always exist. By their construction, it is easy to verify that an ESD with n points exists whenever n is obtained from

+,

t-l 2 *,-I-n,+ s-8 -In, t-l = --+1 2 t-l t2+2t-1 =

2(t-1)

_

3-l,

and this is less than 1 + t when t >, 4. In both cases t = 2 and t = 3, the final tree does not need the last straight vertical line of length 1. When t = 2 it is clear that the resulting tree has a length shorter than 3, while the case t = 3 verifies t2 + 2t - 1

2(t-1)

C1+t*

o

Considering the fact that a RSMT for any SESD is in fact a RMT, it is very natural to ask if the SESDs are, or not, the worst (costliest) distributions when a RMT is requested. The answer to this question is: sometimes, they are not. In Fig. 3 the SESD and another configuration D, both with

k(k+2)+1 2

Fig. 3. 212

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k=l,

2, 3 ,...

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Moreover, by means of direct calculations, the length of an ESD with n points as above becomes: k+2 k+2-x

if k is even, 1

if k is odd,

therefore, the minimum distance d takes the value 2/k. But, what makes the ESDs more interesting is the following result : 3.

less than or equal to the cost of the r RMTs plus the cost of their interconnections, that is to say: L < d(n,-l)+d(n,-l)+-

1

points, k = 1, 2, 2,. . . , then the length L of its RMT is equal to 2(n - 1)/k. Moreover, L is greater than the L’ corresportding to any other distribution with n points.

a..

+d(n,-l)+d(r-1) =d(n-

1) = G(n)

as required. Note that the equality holds if, and only if, P is an ESD. c]

ent

If P is an ESD with

k(k+2)+1 n= 2 i

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The author would like to acknowledge the comments and suggestions provided by referees on earlier versions of this paper.

eferences PI F.R.K. Chung and R.L. Graham, On Steiner Trees for bounded point sets, Germ. Dedicaru 11 (1981) 353-361.

Basically, this follows the same idea used in Theoiem 2, but with the ESD now playing the role previously performed by the SESD. So, given P with n points, let us consider the grid given by its respective ESD. Let kl, k,, . . . , k, and n,, n2,.. ., n, be as in Theorem 2, verifying C n, = n (i = 1, 2 ,..., r). The RMT lkking the ni points belonging to kj always has a length less than or equal to d(nj - l), so, the length L for the RMT linking P will be

PI F.R.K. Chung and F-K. Hwang, The largest Minimal Rectilinear Steiner Trees for a set of n points enclosed in a rectangle with a given perimeter, Nerwurk.r9 (1979) 19-36. [31 L. Few, The shortest path and shortest road through n points, Murhematiku 2 (1955) 141-144. [41 M.R. Gare;J, R.L. Graham and D.S. Johnson, The complexity of computing Steiner Minimal Trees, SIAM J. Appl. Math. 32 (1977) 835-859.

PI E.N. Gilbert and H.O. Pollak, Steiner Minimal Trees, SIAM

J. Appl. Math.

16 (1968) l-29.

Fl M. Hanan, On Steiner’s Problem with rectilinear distances, SIAM

J. App!. _&?& 14 (1966) 255-265.

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