Generation Matrix: An Embeddable Matrix Representation for Hierarchical Trees (2201.11297v2)

Abstract: Starting from the local structures to study hierarchical trees is a common research method. However, the cumbersome analysis and description make the naive method challenging to adapt to the increasingly complex hierarchical tree problems. To improve the efficiency of hierarchical tree research, we propose an embeddable matrix representation for hierarchical trees, called Generation Matrix. It can transform the abstract hierarchical tree into a concrete matrix representation and then take the hierarchical tree as a whole to study, which dramatically reduces the complexity of research. Mathematical analysis shows that Generation Matrix can simulate various recursive algorithms without accessing local structures and provides a variety of interpretable matrix operations to support the research of hierarchical trees. Applying Generation Matrix to differential privacy hierarchical tree release, we propose a Generation Matrix-based optimally consistent release algorithm (GMC). It provides an exceptionally concise process description so that we can describe its core steps as a simple matrix expression rather than multiple complicated recursive processes like existing algorithms. Our experiments show that GMC takes only a few seconds to complete a release for large-scale datasets with more than 10 million nodes. The calculation efficiency is increased by up to 100 times compared with the state-of-the-art schemes.