Gray-coding through nested sets

TL;DR

This paper introduces an efficient algorithm for Gray-coding through nested sets, enabling exhaustive, non-repetitive traversal of all subsets within a nested set structure, and produces new cyclic Gray codes for binary integers.

Contribution

The paper presents a novel algorithm for Gray-coding through nested sets and introduces new families of cyclic Gray codes for binary m-bit integers.

Findings

Algorithm efficiently exhausts all subsets without repetition.

New cyclic Gray codes for binary m-bit integers are constructed.

The approach generalizes Gray coding to nested set structures.

Abstract

We consider the following combinatorial question. Let $$ S_0 \subset S_1 \subset S_2 \subset ...\subset S_m $$ be nested sets, where #$(S_i) = i$. A move consists of altering one of the sets $S_i$, $1 \le i \le m-1$, in a manner so that the nested condition still holds and #$(S_i)$ is still $i$. Our goal is to find a sequence of moves that exhausts through all subsets of $S_m$ (other than the initial sets $S_i$) with no repeats. We call this "Gray-coding through nested sets" because of the analogy with Frank Gray's theory of exhausting through integers while altering only one bit at a time. Our main result is an efficient algorithm that solves this problem. As a byproduct, we produce new families of cyclic Gray codes through binary $m$-bit integers.