# Minimum covering polyplet

(Redirected from MCPS)

A minimum covering polyplet (MCP) of a pattern is a polyplet (i.e. orthogonally/diagonally connected pattern) of minimal population covering said pattern.[1] The minimum covering polyplet size (MCPS) of a pattern is the size of a minimum covering polyplet[1]; unlike the minimum covering polyplet itself, this is a single, well-defined number.

There is one pattern with MCPS = 1, two patterns with MCPS = 2, 8 patterns with MCPS = 3, and 39 patterns with MCPS = 4 (not distinguishing between different orientations).[2][3]

## Computation

Finding a minimum covering polyplet for a given pattern is an instance of the Steiner tree problem[4],[1] which is NP-hard[5]; however, finding a minimum covering polyplet for a given small pattern is often easy in practice.

## Uses

Oscar Cunningham proposed using the minimum covering polyplet size to gauge the size of a methuselah, as it penalizes both population and bounding box.[6] The resulting metric is L/MCPS.

## References

1. Oscar Cunningham (January 20, 2018). Re: Largest and oldest methuselah ever found! (discussion thread) at the ConwayLife.com forums
2. Oscar Cunningham (October 26, 2022). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
3. confocaloid (March 8, 2023). Re: Search Requests Thread (discussion thread) at the ConwayLife.com forums
4. Steiner tree problem at Wikipedia
5. NP-hard problem
6. Oscar Cunningham (January 20, 2018). Re: Largest and oldest methuselah ever found! (discussion thread) at the ConwayLife.com forums