delvingbitcoin

Merging incomparable linearizations

Merging incomparable linearizations

Original Postby ajtowns

Posted on: November 30, 2023 19:01 UTC

The provided text appears to delve into the conceptual understanding of feerate diagrams and their construction, particularly with respect to linearizations in a transaction selection scenario.

The discussion illustrates that a certain diagram $N$ is entirely positioned on or above another diagram $D$, suggesting that $N$ represents a more favorable or efficient scenario compared to $D$. This conclusion is based on the analysis of subsets of good chunks combined with bad parts, denoted as $d_i$, where the cumulative feerate is equal to or less than $f$.

The argument progresses by considering the impact of adding successive chunks to the diagram, which effectively 'kinks' it upwards, indicating an improved feerate diagram over previous sections. This concept is clarified with reference to a specific comment on the Delving Bitcoin forum, providing an example of how one might compare different linearizations to determine their optimality.

Further refining the idea, the author suggests employing rays from the origin to points $P(\gamma_i)$, representing optimal chunking within the fee rate diagram. By plotting these points for specific transaction prefixes $\gamma_i$ and their corresponding returns $R(\gamma_i)$, one can visualize and compare the efficacy of different chunkings. In an optimal scenario, this would result in a monotonic decreasing line rather than a concave one.

Lastly, the method for determining the most beneficial chunking involves creating a diagram for each transaction chunked individually. The optimal feerate function $f_{opt}(x)$ is then defined as the maximum feerate for all transactions at or beyond a certain point, represented mathematically as $\max(f(x^\prime), x^\prime \ge x)$. This approach streamlines the process of identifying the most advantageous transaction organization, providing a clear and systematic method for maximizing returns.