Efficient and Optimal Algorithms for Tree Summarization with Weighted Terminologies
Xuliang Zhu, Xin Sunny Huang, Byron Koon Kau Choi, Jianliang Xu, William K. Cheung, Yanchun Zhang, Jiming Liu · IEEE Transactions on Knowledge and Data Engineering · 2021
Data summarization that presents a small subset of a dataset to users has been widely applied in numerous applications and systems. Many datasets are coded with hierarchical terminologies, e.g., gene ontology, disease ontology, to name a few. In this paper, we study the weighted tree summarization. We motivate and formulate our${\mathsf {kWTS}}$-${\mathsf {problem}}$as selecting a diverse set of$k$nodes tosummarize a hierarchicaltree$T$withweighted terminologies. We first propose an efficient greedy tree summarization algorithm${\mathsf {GTS}}$. It solves the problem with$(1-1/e)$-approximation guarantee. Although${\mathsf {GTS}}$achieves quality-guaranteed answers approximately, but it is still not optimal. To tackle the problem optimally, we further develop a dynamic programming algorithm${\mathsf {OTS}}$to obtain optimal answers for${\mathsf {kWTS}}$-${\mathsf {problem}}$in$O(nhk^3)$time, where$n, h$are the node size and height in tree$T$. The algorithm complexity and correctness of${\mathsf {OTS}}$are theoretically analyzed. In addition, we propose a useful optimization technique of tree reduction to remove useless nodes with zero weights and shrink the tree into a smaller one, which ensures the efficiency acceleration of both${\mathsf {GTS}}$and${\mathsf {OTS}}$in real-world datasets. Moreover, we illustrate one useful application of graph visualization based on the answer of$k$-sized tree summarization and show it in a novel case study. Extensive experimental results on real-world datasets show the effectiveness and efficiency of our proposed approximate and optimal algorithms for tree summarization. Furthermore, we conduct a usability evaluation of attractive topic recommendation on ACM Computing Classification System dataset to validate the usefulness of our model and algorithms.