6.851: Advanced Data Structures (Spring'12)

This lecture covers our second of two lower bounds. This one is work by Mihai Pătraşcu (former 6.851 TA) and myself. We'll show that maintaining a graph subject to edge insertion, edge deletion, and connectivity queries (are v & w connected by a path?) requires Ω(lg n) time per operation, even if the graph is just a bunch of paths. This in particular proves optimality of the logarithmic dynamic tree data structures, and shows that the O(lg² n) data structure we saw for general graphs is pretty good. The lower bound introduces a new but very simple technique, which at the time was the first “truly logarithmic” lower bound for a natural problem, yet the whole proof is relatively clean.