Cluster mempool definitions & theory

In the realm of block building within blockchain technology, a critical insight emerges regarding the optimization of transaction linearization. It is revealed that for any given cluster of transactions, any linearization that maximizes the unconvexified area under the fee-size diagram not only achieves this maximum but also aligns with what is defined as an optimal linearization. This notion of optimality is anchored in the convex diagram, where such a linearization ensures the convexified area under the diagram reaches its peak potential. Importantly, while the definition of optimality based on the convex diagram abstracts away from the specifics of transaction ordering within each chunk, the unconvexified diagram's area is sensitive to these details.

The significance of the unconvexified area extends beyond theoretical interest; it correlates directly with the average feerate achievable within a block, under the premise that block boundaries are randomly distributed across the cluster. This direct relationship prompts the proposal of defining sub-chunk-optimal linearizations. These are characterized by their ability to maximize the area under the unconvexified diagram, situating them as a particular subset of the broader category of optimal linearizations.

Despite the clear advantages of identifying and utilizing sub-chunk-optimal linearizations, the challenge lies in the computational aspect. Currently, there is no known efficient algorithm dedicated to determining these sub-chunk-optimal sequences. Nevertheless, an existing methodology, referred to as the post-linearization algorithm, exhibits capabilities that contribute to this endeavor. While primarily serving other functions, this algorithm inherently acts as a sub-chunk optimizer by ensuring that the unconvexified area under the diagram is not diminished. For those interested in exploring the technical depths of these concepts, further reading and resources can be found through detailed discussions at Delving into Bitcoin and insights into the post-linearization process at Post-Linearization Processing.