Many real-world data sets change continuously, but their enormous size prohibits frequent re-processing of the whole data. Thus, there is an urgent need for efficient, fully dynamic data structures that maintain properties of the data set while supporting fast insertions and deletions. This is especially important for problems in data mining and network analysis, where a data structure often needs to fulfill new additional constraints that are not supported by classic data structures: (1) It should only use sublinear space, even if this leads to some small error in the answers. (2) As data sets frequently contain private information which needs to be protected, it should reveal nothing about individual data points, which is often modeled through differential privacy. Our ambitious goal is to design such groundbreaking new fully dynamic data structures for central problems on graphs and point sets.<br/><br/>Specifically, we will focus on problems with large practical relevance such as subgraph detection, k-core decomposition, and balanced graph partitioning as well as various clustering variants in general metric spaces. For these problems no fully dynamic data structures with small asymptotic running time are known and they have not even been studied in the small-space or differentially-private regime. However, using recent advanced in algorithms research it is now the right time to develop novel techniques to solve these challenging questions.<br/><br/>Thus, the goal of this project is to design algorithms for highly-relevant problems as well as advancing the field of data structures in general by moving it from a narrow focus on asymptotic complexity to a broader set of modern requirements with the goal of bridging the gap that currently exists between theory and practice. As data structures are used by every computer program the impact of this work will be far-reaching. With over 30 years of experience in algorithms research the PI is in the unique position to do so.