An efficient algorithm of compressing decimal notations for tree structures

A decimal notation satisfies many simple mathematical properties and it is a useful tool in the analysis of trees. A practical method is presented that compresses the decimal codes while maintaining the fast determination of relations (e.g. ancestor, descendant, brother, etc.). A special node, called a kernel node, including many common subcodes of the other codes, is defined and a compact data structure is presented using the kernel nodes. For the case where n(m) is the number of the total (kernel) nodes, it is proved that encoding a decimal code is a constant time, that the worst-case time complexity of compressing the decimal codes is O(n+m²), and that the size of the data structure is proportional to m. From the experimental results for some hierarchical semantic primitives for natural language processing, it is shown that the ratio m/ n is extremely small, ranging from 0.047 to 0.13