使用统计编码实现前所未有的树简洁性
Using statistical encoding to achieve tree succinctness never seen before
摘要：
我们提出了一个新的简洁的表示标记树，它表示一个树T使用|T| H_k（T）的位数（加上一些较小的顺序项），其中|T| H_k（T）表示k阶（树标签）熵，如Ferragina等人2005年所定义的。我们的表示采用了一种新的，简单的方法划分的树，既保留了树的形状和节点度。以前，使用|T| H_k（T）位的唯一表示是基于XBWT，这是一种将树标签线性化为单个字符串的变换，并结合了压缩增强。所提出的表示比基于XBWT的表示简单得多，基于XBWT的表示使用隐藏在“更小阶项”概念中的附加线性空间（以0.01n为界），作为使用零阶熵编码器的伪像;我们的表示使用次线性附加空间（对于k的合理值和标签字母表σ的大小）。所提出的表示可以自然地扩展到一个简洁的树数据结构，它使用|T| H_k（T）加上额外的O（|T| k log_σ/ log_σ |T| + |T| log_σ |T| / log_σ |T|）位，并支持所有常见的导航查询在常数时间。以将查询时间增加到O（log log |T| / log |T|）为代价，我们可以进一步将空间冗余减少到O（|T| log log |T| / log_σ |T|）位，假设k <= log_σ |T|。这是对基于XBWT的表示的重大改进：即使基于XBWT的表示使用|T| H_k（T）位，支持导航查询的结构所需的空间也要大得多：（.）
英文摘要：
We propose a new succinct representation of labeled trees which represents a tree T using |T|H_k(T) number of bits (plus some smaller order terms), where |T|H_k(T) denotes the k-th order (tree label) entropy, as defined by Ferragina at al. 2005. Our representation employs a new, simple method of partitioning the tree, which preserves both tree shape and node degrees. Previously, the only representation that used |T|H_k(T) bits was based on XBWT, a transformation that linearizes tree labels into a single string, combined with compression boosting. The proposed representation is much simpler than the one based on XBWT, which used additional linear space (bounded by 0.01n) hidden in the "smaller order terms" notion, as an artifact of using zeroth order entropy coder; our representation uses sublinear additional space (for reasonable values of k and size of the label alphabet σ). The proposed representation can be naturally extended to a succinct data structure for trees, which uses |T|H_k(T) plus additional O(|T|k log_σ/ log_σ |T| + |T| log log_σ |T|/ log_σ |T|) bits and supports all the usual navigational queries in constant time. At the cost of increasing the query time to O(log log |T|/ log |T|) we can further reduce the space redundancy to O(|T| log log |T|/ log_σ |T|) bits, assuming k <= log_σ |T|. This is a major improvement over representation based on XBWT: even though XBWT-based representation uses |T|H_k(T) bits, the space needed for structure supporting navigational queries is much larger: (...)
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