Merging incomparable linearizations

[sipa]

@ajtowns

I think the intuition for this is that you’re constructing a (possibly non-optimal) feerate diagram for L_G+L_B by taking the subsets L_G + \sum_{i=1}^j d_i and showing this is a better diagram than that of L. Because you can’t say anything about d_i (the bad parts of each chunk), you’re instead taking the first j chunks as a whole, which have a feerate of f or lower, and then the suffix of L_G that was missed, which has a feerate of f or high, which kinks the diagram up. Kinking the diagram up means this subset has a better diagram than the chunks up to and including c_j.

Exactly. I’ve added a drawing to illustrate this, I think.

@ajtowns

Ah! Actually, isn’t a better “diagram” to consider the ray from the origin to these P(\gamma_i) points in general, ie, for x=0...c_1 plot the y value as R(\gamma_1), for x=c_1..c_2 plot R(\gamma_2) etc. Then (I think) you still compare two diagrams by seeing if all the points on one are below all the points on the other; but have the result that for an optimal chunking, the line is monotonic decreasing, rather than concave.

Do you perhaps mean plotting F(\gamma_i) (fee) instead of R(\gamma_i) (feerate)?

If so, that’s always monotonically increasing (just because fees are never negative).