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=} that approximately preserves its pairwise distances. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured 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.