6.851: Advanced Data Structures (Spring'21)

This lecture continues our theme of dynamic graphs. Beyond surveying results for several such problems, we'll focus on dynamic connectivity, where you can insert and/or delete edges, and the query is to determine whether two vertices are in the same connected component (i.e., have a path between them). We'll cover a few different results for this problem:

1. Link-cut trees (from last lecture) already solve trees in O(lg n).
2. Euler-tour trees are a simpler way to solve trees in O(lg n), which allow us to aggregate over subtrees instead of paths.
3. If we just insert or just delete edges, we can solve trees in O(1), using our friend leaf trimming plus some simple bit-vector tricks.
4. For general graphs, we'll see how to support updates in O(lg² n) and queries in O(lg n / lg lg n).

This will be our culmination of data structures for dynamic graphs; the next (final) lecture is about matching lower bounds.