Minimum covering polyplet

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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. 1.0 1.1 1.2 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