Center of a graph

Question

Given an unoriented tree with weightless edges with N vertices and N-1 edges and a number K find K nodes so that every node from a tree is within S distance of at least one

Answer 1

For me it looks like a clustering problem. Try it with the k-Means (wikipedia) algorithm where k equals to your K. Since you have a tree and all vertices are connected, you can use as distance measurement the distance/number of edges between your vertices. When the algorithm converts you get the K nodes which should be found. Then you can determine S by iterating through all k clusters. There you calculate the maximum distance for every node in the cluster to the center node. And the overall max should be S.

Update: But actually I see that the k-means algorithm does not produce a global optimum, so this algorithm wouldn't also produce the best result ...