How to linearize your cluster

The discussion focuses on the concept of linearization in the context of Bitcoin transactions, particularly how transactions within a cluster can be ordered optimally. Linearization refers to a topological ordering process for transactions within a cluster, ensuring that parent transactions precede their children. This concept is pivotal for understanding the dynamics of transaction handling and optimization in blockchain technology.

A significant aspect of linearization is depicted through the fee-size diagram, which illustrates the convex hull connecting points representing the size and fee of all prefixes of a linearization. The segments of this diagram highlight "chunks" of the linearization, or groups of transactions that financially support each other. This can be seen as an extension of the Child Pays for Parent (CPFP) mechanism, showcasing how certain groupings within a transaction set can provide mutual benefit in terms of transaction fees.

The comparison of linearizations is based on their respective fee-size diagrams. A linearization is deemed at least as good as another if its diagram is never below that of the other linearization. This comparative analysis is crucial because it is often unclear how much of a cluster will be included in a block when evaluated independently. Therefore, a linearization that maintains superiority across all possible prefixes signifies its optimal nature for inclusion in a block.

Furthermore, it has been proven that for every cluster, there exists at least one optimal linearization, which stands superior or equal to all other possible linearizations for that cluster. This assertion simplifies the complex problem by using the convex-hull approximation, facilitating the identification of a globally optimal linearization despite uncertainties regarding the extent of a cluster's inclusion in a block. Optimal linearizations can be constructed by identifying the highest feerate topologically closed subset within the cluster and positioning it at the forefront of the linearization. Alternatively, the Greedy Granular Transformation (GGT) method divides the cluster into increasingly smaller segments, which eventually form the chunks of the linearization.

For those interested in delving deeper into the theoretical underpinnings and practical applications of this topic, additional resources and discussions can be found at Introduction to Cluster Linearization for a more accessible overview, and Cluster Mempool Definitions Theory for a comprehensive theoretical exploration.