# Science: Mathematicians learn how to join the dots

Joining up a set of numbered dots to make a picture is child’s play.

But joining the dots so that the resulting network is as short as possible

has defeated the best mathematicians. Now, however, two mathematicians have

solved a similar ‘optimisation’ problem, known as the Steiner ratio conjecture.

In order to understand the problem, consider a set P of n points on

the plane. The shortest network that connects the points in P is called

a Steiner minimum tree, or SMT, after Jakob Steiner, the 19th-century Swiss

mathematician.

To obtain such a network, Steiner introduced new points – so-called

Steiner points – and used them as vertices. Consider the simplest non-trivial

example, when P consists of three points only. If these points are the vertices

of an equilateral triangle, it is possible to obtain an SMT by adding a

Steiner point at the centre of the triangle, then joining it to all three

vertices.

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The problem of finding an SMT arises in areas such as the design of

computer circuits, the study of factory locations and the design of telephone

networks. But finding an algorithm that can compute an SMT if n is large

is hard. Computer scientists have instead tried to find approximate solutions.

One of these is the ‘minimum spanning tree’, or MST, of the set P. This

is the shortest network whose vertices are precisely the points in P (no

extra vertices are allowed). In the example of three points forming an equilateral

triangle, an MST is the ‘V’ formed by joining two of the vertices to the

third one.

In general, the MST is longer than the SMT. On the other hand, very

efficient algorithms already exist for computing it.

Finding a minimum spanning tree may be fast, but it is not very useful

to mathematicians unless they know how good an approximation it is. A measure

of its ‘goodness’ is the ratio ‘length of SMT/length of MST’, called the

Steiner ratio. In the example of the equilateral triangle, this ratio is

the square root of 3/2 (or approximately 0.866).

In 1968, Edgar Gibert and Henry Pollak at Bell Laboratories in New Jersey,

speculated that the Steiner ratio never falls below the square root of 3/2.

If true, this would mean that finding the SMT will result in a network that

is only 13 per cent shorter than its approximation in the best case.

In any practical application, the length of a network is directly related

to cost. So knowing this 13 per cent limit in potential savings has economic

implications.

By 1989, mathematicians had proved that the conjecture was true for

any set P with less than seven points. They had also managed to push up

the lower bound of the Steiner ratio from 0.57 (a value found in 1976) to

0.824. But further progress towards a complete solution was hindered because

the amount of computation required was growing rapidly.

Ding-Zhu Du, a visiting Chinese mathematician at Princeton University,

and Frank Hwang, of AT&T Bell Laboratories, came up with a different

approach, which bypassed the computations altogether (Proceedings of the

National Academy of Sciences, vol 87, p 9464). They transformed the Steiner

ratio conjecture into a ‘minimax’ problem, similar to those in game theory.

First, they showed some important properties of the class of maximum

points. Then they translated these properties back to the original geometrical

problem. This implied that the conjecture would be true in general if it

was true in a particular special case. The case was when the points of P

sit at the vertices of equilateral triangles placed side by side to form

a tiling of the plane. Du and Hwang settled this special case, and so proved

the conjecture.

After their success Du and Hwang expect that their method will help

solve other optimisation problems involving lower bounds.