通过嵌套集合的灰度编码
Gray-coding through nested sets
摘要：
我们考虑下面的组合问题。设$$ S_0 \subset S_1 \subset S_2 \subset ...\subset S_m $$为嵌套集合，其中# $(S_i) = i$。移动包括以某种方式更改集合$S_i$、$1 \le i \le m-1$之一，以使嵌套条件仍然成立，并且# $(S_i)$仍然是$i$。我们的目标是找到一个序列的移动，用尽所有子集的$S_m$（除了初始集$S_i$）没有重复。我们称之为“通过嵌套集合的格雷编码”，因为它与弗兰克·格雷的理论类似，即通过整数耗尽，同时一次只改变一位。我们的主要结果是一个有效的算法，解决了这个问题。作为副产品，我们通过二进制$m$位整数产生新的循环格雷码族。
英文摘要：
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.
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