Efficient and Optimal Algorithms for Tree Summarization With Weighted Terminologies

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 <inline-formula><tex-math notation="LaTeX">${\mathsf {kWTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">kWTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq1-3120722.gif"/></alternatives></inline-formula>-<inline-formula><tex-math notation="LaTeX">${\mathsf {problem}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">problem</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq2-3120722.gif"/></alternatives></inline-formula> as selecting a diverse set of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq3-3120722.gif"/></alternatives></inline-formula> nodes to <underline>s</underline>ummarize a hierarchical <underline>t</underline>ree <inline-formula><tex-math notation="LaTeX">$T$</tex-math><alternatives><mml:math><mml:mi>T</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq4-3120722.gif"/></alternatives></inline-formula> with <underline>w</underline>eighted terminologies. We first propose an efficient greedy tree summarization algorithm <inline-formula><tex-math notation="LaTeX">${\mathsf {GTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">GTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq5-3120722.gif"/></alternatives></inline-formula>. It solves the problem with <inline-formula><tex-math notation="LaTeX">$(1-1/e)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>e</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zhu-ieq6-3120722.gif"/></alternatives></inline-formula>-approximation guarantee. Although <inline-formula><tex-math notation="LaTeX">${\mathsf {GTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">GTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq7-3120722.gif"/></alternatives></inline-formula> achieves quality-guaranteed answers approximately, but it is still not optimal. To tackle the problem optimally, we further develop a dynamic programming algorithm <inline-formula><tex-math notation="LaTeX">${\mathsf {OTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">OTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq8-3120722.gif"/></alternatives></inline-formula> to obtain optimal answers for <inline-formula><tex-math notation="LaTeX">${\mathsf {kWTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">kWTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq9-3120722.gif"/></alternatives></inline-formula>-<inline-formula><tex-math notation="LaTeX">${\mathsf {problem}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">problem</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq10-3120722.gif"/></alternatives></inline-formula> in <inline-formula><tex-math notation="LaTeX">$O(nhk^3)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mi>h</mml:mi><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zhu-ieq11-3120722.gif"/></alternatives></inline-formula> time, where <inline-formula><tex-math notation="LaTeX">$n, h$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="zhu-ieq12-3120722.gif"/></alternatives></inline-formula> are the node size and height in tree <inline-formula><tex-math notation="LaTeX">$T$</tex-math><alternatives><mml:math><mml:mi>T</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq13-3120722.gif"/></alternatives></inline-formula>. The algorithm complexity and correctness of <inline-formula><tex-math notation="LaTeX">${\mathsf {OTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">OTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq14-3120722.gif"/></alternatives></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">${\mathsf {GTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">GTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq15-3120722.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">${\mathsf {OTS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">OTS</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq16-3120722.gif"/></alternatives></inline-formula> in real-world datasets. Moreover, we illustrate one useful application of graph visualization based on the answer of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="zhu-ieq17-3120722.gif"/></alternatives></inline-formula>-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.