g-harel / superpermutations

calculate and check superpermutations

superpermutations

A superpermutation of the string n is another string that contains all the permutations of the characters in n. For example, 1221 is a superpermutation of 12. However, 121 is a shorter superpermutation of 12 because it overlaps characters.

This repository contains code to produce superpermutations that are close to minimal (proving a superpermutation is minimal for any n is still not a solved problem).

The algorithm starts with the original string n and progressively appends more character(s). The number of appended/shifted characters each iteration is taken from a sequence of integers which is generated from the input string's length.

Shift sequences

len(n)	sequence
1	-
2	1
3	112
4	111211121113
5	111121111211112111131111211112111121111311112111121111211114
...	...

When the shift is of 1, the first character of the most recent permutation can be appended to the end of the string. This adds a unique permutation of the string n to the end of the result string.

123     shift(1)
 231    shift(1)
  312   shift(1)
12312   ...

When the shift (s) is larger than one, the first s characters of the most recent permutation are mirrored and appended to the end of the result string. This process repeats until the sequence is exhausted, which means half + 1 of the result has been computed. The rest of the string is produced by mirroring this first part.

1234         shift(1)
 2341        shift(1)
  3412       shift(1)
     2143    shift(3) 341 -> 143
      1432   shift(1)
1234121432   ...

more information about superpermutations

more information about the integer sequences

Usage

$ go get -u github.com/g-harel/superpermutations/superpermutations

CLI

Usage:
  superpermutations [flags]

Flags:
  -c, --check          check correctness of result (big performance hit)
  -h, --help           help for superpermutations
  -l, --length int     set input string length (max 13) (default 5)
  -o, --out string     write result to a file
  -p, --print          print the result (may be very large)
  -r, --runes string   custom list of chars (looped if < length)
  -s, --silent         silence all output (except --print)
      --version        version for superpermutations

Package

import "github.com/g-harel/superpermutations"

func main() {
  // generate superpermutation
  s := superpermutations.Find("01234")

  // confirm that it contains all permutations
  fmt.Println(superpermutations.Check("01234", s))
}