Some of the routing protocols used in telecommunication networks route traffic on a shortest path tree according to configurable integral link weights. One crucial issue for network operators is finding a weight function that ensures a stable routing: when some link fails, traffic whose path does not use that link should not be rerouted. In this paper we improve on several previously best results for finding small stable weights. As a conceptual contribution, we draw a connection between the stable weights problem and the seemingly unrelated unique-max coloring problem. In unique-max coloring, one is given a set of points and a family of subsets of those points called regions. The task is to assign to each region a color represented as an integer such that, for every point, one region containing it has a color strictly larger than the color of any other region containing this point. In our setting, points and regions become edges and paths of the shortest path tree, respectively, and based on this connection, we provide stable weight functions with a maximum weight of O(n log n) in the case of single link failure, where n is the number of vertices in the network. Furthermore, if the root of the shortest path tree is known, we present an algorithm for determining stable weights bounded by 4n, which is optimal up to constant factors. For the case of an arbitrary number of failures, we show how stable weights bounded by 3(n)n can be obtained. All the results improve on the previously best known bounds.