  The approximate nearest neighbor search (ANNS) is a fundamental and essential
component in data mining and information retrieval, with graph-based
methodologies demonstrating superior performance compared to alternative
approaches. Extensive research efforts have been dedicated to improving search
efficiency by developing various graph-based indices, such as HNSW
(Hierarchical Navigable Small World). However, the performance of HNSW and most
graph-based indices become unacceptable when faced with a large number of
real-time deletions, insertions, and updates. Furthermore, during update
operations, HNSW can result in some data points becoming unreachable, a
situation we refer to as the `unreachable points phenomenon'. This phenomenon
could significantly affect the search accuracy of the graph in certain
situations.
  To address these issues, we present efficient measures to overcome the
shortcomings of HNSW, specifically addressing poor performance over long
periods of delete and update operations and resolving the issues caused by the
unreachable points phenomenon. Our proposed MN-RU algorithm effectively
improves update efficiency and suppresses the growth rate of unreachable
points, ensuring better overall performance and maintaining the integrity of
the graph. Our results demonstrate that our methods outperform existing
approaches. Furthermore, since our methods are based on HNSW, they can be
easily integrated with existing indices widely used in the industrial field,
making them practical for future real-world applications. Code is available at
\url{https://github.com/xwt1/MN-RU.git}
