Let G = ( V , E ) be a graph with n nodes in which each pair of nodes is joined

Question

Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satisfy the triangle inequality wik wij + wjk . For a subset V V, we will use G[V] to denote the subgraph (with edge weights) induced on the nodes in V.

We are given a set X V of k terminals that must be connected by edges. We say that a Steiner tree on X is a set Z so that X Z V, together with a spanning subtree T of G[Z]. The weight of the Steiner tree is the weight of the tree T.

Show that the problem of finding a minimum-weight Steiner tree on X can be solved in time O(nO(k)).

Explanation / Answer

Consider a Steiner tree represented by T in which X U Z, V|X| = k.

The X represents the terminals and Z represents the extra nodes.

Therefore, |Z| <= k.

Thus, the complexity would be O(nO(k)).

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