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
${\mathsf {kWTS}}$
-
${\mathsf {problem}}$
as selecting a diverse set of
$k$
nodes to
s
ummarize a hierarchical
t
ree
$T$
with
w
eighted 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.