Finding networks with minimal cost to connect points is a key problem in VLSI design, which can be described as obstacle-avoiding shortest path and minimum Steiner tree problem according to whether the number of points is greater than 2. Connection graphs, such as track based on graphs GC and GT and free area based graphs GF and GG, are effective tools for the shortest path problem, which is the foundation of the Steiner tree problem. The contribution of this paper includes three points: The dynamic algorithms for querying the shortest path between two points on each connection graph are designed and analyzed for the first time; secondly, all algorithms for Steiner problem on each connection graph are analyzed; The number of candidate Steiner points is reduced from O((e+p)2) to O((t+p)2) in the 3-Steiner algorithm on GC, where e, t, p presents the number of edges, extreme edges of obstacles and terminals; An average Q(t) algorithm for 3-Steiner problem are designed on GG.