Gilbert Yale Steiner

Pseudo-Gilbert-Steiner trees

The Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. In this paper, we define a special class of minimum Gilbert networks, called pseudo-Gilbert–Steiner trees, and we show that it can be constructed by Gilbert’s generalization of Melzak’s method. Besides, a counterexample, a pseudo-Gilbert–Steiner tree, is con...

The Steiner ratio conjecture of Gilbert and Pollak is true.

Let P be a set of n points on the euclidean plane. Let Ls(P) and Lm(P) denote the lengths of the Steiner minimum tree and the minimum spanning tree on P, respectively. In 1968, Gilbert and Pollak conjectured that for any P, Ls(P) >/= (radical3/2)Lm(P). We provide an abridged proof for their conjecture in this paper.

Algebra Notes: Gilbert&Gilbert

Sets and Intervals • An interval with round brackets such as (a, b) means all the numbers between a and b not including a and b. We call this interval an open interval. • An interval with square brackets such as [a, b] means all the numbers between a and b including a and b. We call this interval a closed interval. • An interval with one square bracket and one closed bracket is called a clopen ...

An Approach for Proving Lower Bounds: Solution of Gilbert-Pollak's Conjecture on Steiner Ratio

'Let {gi(z)}iEr be a family of finitely many continuous functions on a polytope X. Consider the problem of minimizing the function f(z) = maxiGrgi(2) on X. Many problems about lower bounds can be reduced to this general form. In this paper, we show that if every gi(z) is a concave function, then the minimum value of f(z) is achieved at finitely many special points in X. As an application, we so...

Steiner Trees and Steiner Forests