# How to linearize your cluster

**URL:** <https://delvingbitcoin.org/t/how-to-linearize-your-cluster/303>\
**Category:** Implementation\
**Tags:** wg-cluster-mempool, mempool\
**Created:** [December 18, 2023, 8:55pm UTC](https://delvingbitcoin.org/t/how-to-linearize-your-cluster/303 "2023-12-18T20:55:14Z")\
**Posts on this page:** 1\
**Showing post:** 8

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**Author:** ![lorbax](https://delvingbitcoin.org/user_avatar/delvingbitcoin.org/lorbax/32/797_2.png) [@lorbax](https://delvingbitcoin.org/u/lorbax)\
**Post date:** [January 7, 2025, 12:33pm UTC](https://delvingbitcoin.org/t/how-to-linearize-your-cluster/303/8 "2025-01-07T12:33:49Z")

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Thank you, now I understand. All of this started in my head while trying to understand the proof of the following.

> [@sipa](#):
>
> every highest-feerate topologically-valid set among those that include inc and exclude exc _must_ contain every topologically valid subset of pot

I tried to expand the proof of it, using the additional material you provided. From the comment above we can deduce the following

_Lemma._

1. pot is _the_ (i.e. unique) highest amongst all the sets including inc and excluding exc.
2. t\in pot if and only if feerate(t)\> feerate(pot).

_Proposition._ Let B an element that maximize the feerate in the following collection.\{U \text{ topological} \mid inc \subseteq U \subseteq exc^c \}. If C \subseteq pot is topological, then C\subseteq B.

_Proof._ By contraddiction, suppose that there is B like in the hypothesis and C topological, C\subseteq pot and C \not \subseteq B. Since C is topological, possibly replacing t with some of its ancestors, it is no loss to assume anc(t) \subseteq B. Since t\in pot, by 2. of Lemma above, feerate(t) \> feerate(pot). By characterization of pot in 1. of Lemma above, we have that feerate(pot) \ge feerate(B). So, feerate(t)\> feerate(B). Consider B' = \{t\} \cup B. Clearly, feerate(B')\> feerate(B); moreover, since anc(t) \subseteq (B), B' is topological. This is against the definition of B.

Let me know what do you think.

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