Towards Optimal Locality in Mesh-Indexings

The eeciency of many algorithms in parallel processing, computational geometry, image processing, and several other elds relies on \locality-preserving" indexing schemes for meshes. We concentrate on the case where the maximum distance between two mesh nodes indexed i and j shall be a slow-growing function of ji ? jj (using the Euclidean, the maximum, and the Manhattan metric). In this respect, space-lling, self-similar curves like the Hilbert curve are superior to simple indexing schemes like \row-major." We present new tight results on 2-D and 3-D Hilbert indexings which are easy to generalize to a quite large class of curves. We then present a new indexing scheme we call H-indexing, which has superior locality. For example, with respect to the Euclidean metric the H-indexing provides locality approximately 50% better than the usually used Hilbert indexing. This answers an open question of Gots-man and Lindenbaum. In addition, H-indexings have the useful property to form a Hamiltonian cycle and they are optimally locality-preserving among all cyclic indexings.