Hierarchical directed acyclic graph (DAG) is an essential model for representing terminologies and their hierarchical relationships, such as Disease Ontology and ImageNet categories. Due to the vast number of terminologies and complex structures in a large DAG, it becomes challenging for humans to effectively analyze and explore the hierarchical relationships they encode. Therefore, summarizing hierarchical DAGs is essential for enhancing the interpretability and visualization of the underlying hierarchy. Beyond visual data exploration, hierarchical DAG summarization also supports a range of applications, such as biomedical ontology analytics, snippet generation for information search, and summarized recommendation. In this article, we address a new problem of finding k representative vertices to summarize a hierarchical DAG. To capture diverse summarization and identify important vertices, we design a summary score function that reflects vertices diversity coverage and structure correlation. The studied problem is theoretically proven to be NP-hard. To tackle it efficiently, we propose a greedy algorithm with an approximation guarantee that iteratively adds vertices with significant summary contributions to the answers. To further enhance the answer quality, we introduce a subtree extraction-based method that is proven to achieve higher-quality answers. Additionally, we develop a scalable algorithm, \(\mathsf {k}\) - \(\mathsf {PCGS}\) , which employs candidate pruning and DAG compression for large-scale hierarchical DAGs. For the query-dependent problem, we propose an index-based method and several optimization techniques to improve efficiency. Extensive experiments on large real-world datasets demonstrate the effectiveness and efficiency of our proposed algorithms.

Interactive graph search (IGS) over DAGs aims to find a hidden target by asking interactive questions as few as possible. IGS is useful for many applications, e.g., facilitating supervised learning tasks by harnessing labeled data, image categorization, and product classification. However, most of the existing IGS methods only work for either single target search on DAGs or multiple targets search on simple trees. To overcome the gap, it motivates us to study a challenging and yet not solved problem of multiple targets search over DAGs. We analyze the new problem in-depth and propose a key concept of uncertain candidates. Based on it, we design an effective gain function to determine the best vertex to be asked questions and shrink the search space of potential targets greatly. Leveraging our uncertain candidates and gain function, we develop a unified k-EIS framework to search both single target and multiple targets. We analyze all algorithm complexities and theoretically show that our solution can significantly improve existing DFS-tree-based methods by asking O ( n ) questions to O (log 2 n ) questions in worst cases. To further improve IGS for multiple targets, we propose an advanced solution by dividing the whole DAG into k disjoint subgraphs with single targets and then tackling each subgraph one by one independently. Extensive experiments on real-world datasets validate that our proposed k-EIS framework can save lots of questions to search exact targets against four state-of-the-art IGS competitors.

The COVID-19 pandemic has caused the society lockdowns and a large number of deaths in many countries. Potential transmission cluster discovery is to find all suspected users with infections, which is greatly needed to fast discover virus transmission chains so as to prevent an outbreak of COVID-19 as early as possible. In this article, we study the problem of potential transmission cluster discovery based on the spatio-temporal logs. Given a query of patient user q and a timestamp of confirmed infection t q , the problem is to find all potential infected users who have close social contacts to user q before time t q . We motivate and formulate the potential transmission cluster model, equipped with a detailed analysis of transmission cluster property and particular model usability. To identify potential clusters, one straightforward method is to compute all close contacts on-the-fly, which is simple but inefficient caused by scanning spatio-temporal logs many times. To accelerate the efficiency, we propose two indexing algorithms by constructing a multigraph index and an advanced BCG-index. Leveraging two well-designed techniques of spatio-temporal compression and graph partition on bipartite contact graphs, our BCG-index approach achieves a good balance of index construction and online query processing to fast discover potential transmission cluster. We theoretically analyze and compare the algorithm complexity of three proposed approaches. Extensive experiments on real-world check-in datasets and COVID-19 confirmed cases in the United States validate the effectiveness and efficiency of our potential transmission cluster model and algorithms.

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.

Data summarization has wide applications in real world, e.g. attributes filter, image set labeling and personalized recommendation. In this work, we study a new problem HSD to summarize a dataset using k concepts in a hierarchical taxonomy. Different from the existed works of whole hierarchy summarization, we focus on the accurate coverage of the given query set Q. The objective is to cover more items in Q and less items not in Q. To tackle it, we first propose a dynamic programming based algorithm on the tree hierarchy, which is a simple instance of HSD problem. Furthermore, we propose a heuristic method to assign the vertex to one of its in-neighbors for HDAGs and apply the tree algorithm on it. The experimental results confirm the quality of our methods on both tree and HDAG datasets.

Interactive graph search leverages human intelligence to categorize target labels in a hierarchy, which is useful for image classification, product categorization, and database search. However, many existing interactive graph search studies aim at identifying a single target optimally, and suffer from the limitations of asking too many questions and not being able to handle multiple targets. To address these two limitations, in this paper, we study a new problem of budget constrained interactive graph search for multiple targets called kBM-IGS problem. Specifically, given a set of multiple targets T in a hierarchy and two parameters k and b , the goal is to identify a k -sized set of selections S , such that the closeness between selections S and targets T is as small as possible, by asking at most a budget of b questions. We theoretically analyze the updating rules and design a penalty function to capture the closeness between selections and targets. To tackle the kBM-IGS problem, we develop a novel framework to ask questions using the best vertex with the largest expected gain, which provides a balanced trade-off between target probability and benefit gain. Based on the kBM-IGS framework, we first propose an efficient algorithm STBIS to handle the SingleTarget problem, which is a special case of kBM-IGS. Then, we propose a dynamic programming based method kBM-DP to tackle the MultipleTargets problem. To further improve efficiency, we propose two heuristic but efficient algorithms, kBM-Topk and kBM-DP+. Experiments on large real-world datasets with ground-truths verify both the effectiveness and efficiency of our algorithms.

Directed acyclic graph (DAG) is an essentially important model to represent terminologies and their hierarchical relationships, such as Disease Ontology. Due to massive terminologies and complex structures in a large DAG, it is challenging to summarize the whole hierarchical DAG.

Hierarchical directed acyclic graph (HDAG) is an essential graph model to represent terminology relationships in a hierarchy, such as Disease Ontology, Gene Ontology, and Wikipedia. However, due to massive terminologies and complex structures in a HDAG, an end user might feel difficult to explore and summarize the whole graph, which is practically useful but less studied in the literature. In this demo, we develop an interactive system of HDAG-Explorer to help users summarize HDAG with highly important and diverse vertices. Our HDAG-Explorer system exhibits several useful features including summarized visualization, interactive exploration, and structural statistics report. All these features facilitate in-depth understanding of the HDAG data. We showcase the usability of the HDAG-Explorer through two real-world applications of summarized topic recommendation and visual data exploration.