Optimal Cache-Oblivious Mesh Layout

This paper shows how to generate a cache-oblivious memory layout of a well-shaped finite-element mesh G. This cache-oblivious mesh layout enables asymptotically optimal mesh updates, in which each vertex communicates with all of its neighbors. Mesh updates is the building block of iterative linear system solver. For block size B and cache size M, the mesh update cost is O(1+ |G|/B), assuming the tall cache assumption M = Ω(Bd), where d is the dimensionality of the mesh’s geometric domain. The layout algorithm runs cache-obliviously in O(|G| log2 |G|) operations and O(1+ |G| log2 |G|/B) memory transfers with high probability. The approach combines ideas from VLSI theory, graph separators, and I/O-efficient computing, and presents simplified and improved methods for building fully-balanced decomposition trees from the VLSI literature and k-way partitioning from the graph-separator literature. We then introduce the relaxedbalanced decomposition tree and show that with M = Ω(Bd), mesh update cost is O(1+ |G|/B). We give a layout algorithm for relaxed-balanced decomposition trees, which runs cache-obliviously in O(|G| log |G| log log |G|) operations and O(1+ |G| log |G| log log |G|) memory transfers with high probability.