delvingbitcoin

## Cluster mempool definitions & theory

### Posted on: December 11, 2023 01:16 UTC

The conversation delves into the intricacies of determining an optimal transaction ordering in a block, particularly when dealing with transactions of equal fee and size but differing fee rates.

The discussion suggests that while there may not be an objective reason to prioritize smaller transactions first, it might be beneficial to consider them last for potentially improved performance in greedy knapsack selection algorithms. There is an acknowledgment that achieving a strictly unique and perfect linearization of transactions is implausible due to scenarios where multiple equally efficient orderings exist, as illustrated by the example involving transactions A, B, and C.

Furthermore, the email touches on the concept of canonical ordering to reduce uncertainty and ambiguity in transaction processing. However, it critiques the use of arbitrary orders to break ties between transactions, emphasizing that such choices remain arbitrary even when guided by predefined rules. The author also questions the definition of the optimization function provided, suggesting an alternative formulation that considers the highest-feerate subset of a given set of transactions and builds upon valid linearizations.

A non-constructive proof approach is proposed to demonstrate the uniqueness claim of an optimal linearization. The approach involves defining a function based on the number of possible linearizations greater than or equal to a given linearization, and selecting one which maximizes this count. If no such linearization exists that is greater than all others, a merge operation is suggested to construct a new linearization that would satisfy the criteria, leading to a contradiction and thereby supporting the claim of optimality.

The mathematical expressions within the email, such as the redefined optimization function and the inequalities used in the proof strategy, are central to the argument presented. The correspondence seems to aim at refining the methodology for transaction selection and ordering within the context of blockchain technologies, with implications for both efficiency and consistency in block creation.