Content Delivery Networks (CDNs) are increasingly being used to deliver live streaming on today's Internet. The new application type of live streaming exposes unique characteristics and challenges that require more advanced design of CDN infrastructure. Unlike traditional web-object delivering, which allows CDN content servers to cache contents and thus typically only involves certain CDN servers (e.g. edge content servers) for delivering contents, live streaming requires a real-time full CDN streaming path that spans across the streaming source, the CDN ingest server, the CDN content server, and end-viewers. Though the ingest server is typically fixed for a particular live stream, appropriate content servers need to be selected for delivering the stream to end viewers. Though today's CDNs typically employ layered infrastructure for delivering live streaming, in this work, we propose a flat-layered design which is referred to as Constrained Server Chaining (CSC) for selecting optimal content servers to deliver live streams. Rather than employing a strictly layered infrastructure, CSC allows CDN streaming servers to dynamically choose upstream servers, thus saves transit cost for CDN providers.