deltam / perm

Permutation generator based on group theory written in Go

perm

Permutation generator based on group theory¹.

• Fast
• Stateless
  • generate next permutation from current permutation only.
• Permutation order is NOT lexical.

Installation

go get github.com/deltam/perm

Usage

Index only:

func main() {
	g, err := perm.New(3)
	if err != nil {
		log.Fatal(err)
	}

	for ; !g.Done(); g.Next() {
		fmt.Println(p)
	}
}
// Output:
// [1 2 0]
// [2 0 1]
// [0 2 1]
// [2 1 0]
// [0 1 2]
// [1 0 2]

Permutation of slice:

func main() {
	words := []rune("ABC")
	g, err := perm.Iter(ss)
	if err != nil {
		log.Fatal(err)
	}

	for ; !g.Done(); g.Next() {
		fmt.Printf("%v\t%s\n", g.Index(), string(words))
	}
}
// Output:
// [1 2 0]	ABC
// [2 0 1]	BCA
// [0 2 1]	CBA
// [2 1 0]	BAC
// [0 1 2]	CAB
// [1 0 2]	ACB

Performance

This algorithm is 40~50% faster than naive recursive algorithm.

	time/op
Primitive-8	33.3µs ± 2%
Generator-8	39.1µs ± 2%
PermRecursive-8	66.9µs ± 1%

func recPerm(p []int, n int, ignore []bool) {
	if n == 0 {
		return
	}
	for i := 0; i < len(p); i++ {
		if !ignore[i] {
			p[n-1] = i
			ignore[i] = true
			recPerm(p, n-1, ignore)
			ignore[i] = false
		}
	}
}

Background

See paper¹ for details.

I also explained details on my blog post(Japanese only). ²

Cayley graph on permutation group generated by sigma and tau has a size two disjoint cycle cover and Hamilton path.

sigma(rotation): (1, 2, ..., n) -> (2, 3, ..., n, 1)
tau(swap):       (1, 2, ..., n) -> (2, 1, ..., n)

Below is the Hamilton path created by split and join the two cycles.

By observing above Hamilton path, you can discover local rule for generating that.

See primitive.go or this text³² for details.

License

MIT

Copyright 2019 deltam

Footnotes

1. Hamiltonicity of the Cayley Digraph on the Symmetric Group Generated by σ = (1 2 ... n) and τ = (1 2) ↩ ↩²

2. 2つの操作のみで全順列を列挙する：対称群のグラフ上のハミルトン路にもとづく順列生成の紹介と実装 : サルノオボエガキ ↩ ↩²

3. The Williams Construction: Superpermutations — Greg Egan ↩