Worst-Case Optimal Tree Layout in a Memory Hierarchy

Consider laying out a fixed-topology tree of N nodes into external memory with block size B so as to minimize the worst-case number of block memory transfers required to traverse a path from the root to a node of depth D. We prove that the optimal number of memory transfers is Θ   D lg(1+B) when D = O(lgN) lgN lg ( 1+ lg N D ) when D = Ω(lgN) and D = O(B lgN) D B when D = Ω(B lgN)  . This bound can be achieved even when B is unknown to the (cache-oblivious) layout algorithm.