PATRICIA桥梁
PATRICIA bridges
摘要：
当在二叉树的叶子中存储不同的无限二进制字时，基数排序树出现，使得对于任何两个字，它们的公共前缀与对应的两个叶子的公共前缀一致。如果删除基数排序树中的出度$1$顶点并“闭合间隙”，则得到的PATRICIA树保持将无限单词排序为字典顺序所需的所有信息。我们调查的PATRICIA链-树值马尔可夫链时，出现连续建设PATRICIA树的收集无限二进制字$Z_1,\ldots, Z_n$，$n=1,2,\ldots$，源字$Z_1, Z_2,\ldots$是独立的，并有一个共同的扩散分布在$\{0,1\}^\infty$。事实证明，PATRICIA链共享一个共同的后向转移概率集合，并且这些概率与雷米为连续生成具有越来越多叶子的均匀随机二叉树而引入的链的概率相同。这意味着任何帕特里西亚链的无限桥（即，通过将帕特里西亚链限定在其遥远的未来而得到的链）与雷米链的无限桥重合。在Evans、Grübel和Wakolbinger 2017中，对雷米链的无限桥进行了具体的描述，我们在这里回顾了这一描述，同时添加了一些细节和说明。
英文摘要：
A radix sort tree arises when storing distinct infinite binary words in the leaves of a binary tree such that for any two words their common prefixes coincide with the common prefixes of the corresponding two leaves. If one deletes the out-degree $1$ vertices in the radix sort tree and "closes up the gaps", then the resulting PATRICIA tree maintains all the information that is necessary for sorting the infinite words into lexicographic order. We investigate the PATRICIA chains -- the tree-valued Markov chains that arise when successively building the PATRICIA trees for the collection of infinite binary words $Z_1,\ldots, Z_n$, $n=1,2,\ldots$, where the source words $Z_1, Z_2,\ldots$ are independent and have a common diffuse distribution on $\{0,1\}^\infty$. It turns out that the PATRICIA chains share a common collection of backward transition probabilities and that these are the same as those of a chain introduced by Rémy for successively generating uniform random binary trees with larger and larger numbers of leaves. This means that the infinite bridges of any PATRICIA chain (that is, the chains obtained by conditioning a PATRICIA chain on its remote future) coincide with the infinite bridges of the Rémy chain. The infinite bridges of the Rémy chain are characterized concretely in Evans, Grübel, and Wakolbinger 2017 and we recall that characterization here while adding some details and clarifications.
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