Before revision:  We consider \alpha=}   2W , 4W , 6W, where W is the maximum edge weight in G. We first show that every n-node graph has a subsetwise 2W-spanner on O(n |S|^{1/2) edges where S \subseteq V and W is a constant.
After revision:  We consider \alpha=} 2W and + 4W , 6W, where W is the maximum edge weight in G. We first show that every n-node graph has a subsetwise 2W-spanner on O(n |S|^{1/2) edges where S \subseteq V and W is a constant.