Computation Costs of Filters
it's in textbooks.  you'll find that, for a sharp filter of given sharpness, that IIR filters will cost less than FIR.

given it's IIR, you'll find that the Direct Form I and Direct Form II will have less computational cost than the State-Variable Filter, Lattice, Normalized Ladder, Gold-Rader, or Zölzer, or Harris-Brooking forms (and there are a zillion other forms, i s'pose, but i can't see them being cheaper to implement than the Direct Forms).  and then you can implement many forms (like DF1, DF2, or Lattice) as one-big direct filter section or as a cascade of second-order sections.  but there are numerical tradeoffs doing it the cheap (from a computation time POV) way.

given it's FIR, there are at least 3 different ways of implementing an FIR, all with different cost issues.  if the FIR filter is totally general (but still causal), then the choice is between the simple direct method (the "transversal" FIR form) and what we like to call Fast Convolution (implementing a long FIR using the FFT).  for some FIRs of a greatly restricted class (that is FIRs that look like IIRs), there is a method that is called Truncated IIR (TIIR) that is about as cheap as an IIR, but it's mathematically FIR.

good question, no short answer that is also comprehensive.  need to get O&amp;S and read it.