I have heard that the Toric ideals allow Buchberger's algorithm to be considerably accelerated (see Basics of Grobner's Toric ideals, Observation 2,3). My question is twofold:
What are the precise theoretical limits of complexity known for Buchberger's algorithm for toric ideals?
Are there other kinds of non-trivial ideals on which a grobner base can be calculated efficiently? Can you please provide a reference?
For context, I want to use a grobner base as a way to encode data flow analysis problems in the compiler's construction, so that it allows multiple analyzes to "share" information easily. Therefore, knowing special types of ideals and fast algorithms to calculate your Grobner base would help design specific data flow analysis.