To handle dynamic updates to the input tree, such as edge insertions or deletions, the parallel algorithms can be extended by incorporating techniques for maintaining the integrity of the dendrogram structure.
For edge insertions, the algorithm can be modified to dynamically update the dendrogram by adding the new edge to the appropriate clusters and recalculating the SLD as needed. This process may involve identifying the affected clusters, updating the characteristic spines in the heaps, and performing the necessary merges to reflect the changes in the tree structure.
Similarly, for edge deletions, the algorithm can adjust the dendrogram by removing the deleted edge from the clusters and updating the SLD accordingly. This may require reevaluating the characteristic spines, reorganizing the heap structures, and potentially undoing previous merges that involved the deleted edge.
By implementing efficient data structures and algorithms to handle these dynamic updates, the parallel algorithms can maintain the accuracy and consistency of the dendrogram in response to changes in the input tree.