The efficiency of many data structures and algorithms relies on "locality-preserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slow-growing function of ji jj. We present a new 2-D indexing scheme we call H-indexing , which has superior (possibly optimal) locality in comparison with the well-known Hilbert indexings. H-indexings form a Hamiltonian cycle and we prove that they are optimally locality-preserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2-D and 3-D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.

Indexing schemes based on space filling curves like the Hilbert curve are a powerful tool for building efficient parallel algorithms on mesh-connected computers. The main reason is that they are locality-preserving, i.e., the Manhattan-distance between processors grows only slowly with increasing index differences. We present a simple and easy-to-verify proof that the Manhattan-distance of any indices i and j is bounded by 3 p ji \Gamma jj \Gamma 2 for the 2D-Hilbert curve. The technique used for the proof is then generalized for a large class of self-similar curves. We use this result to show a (quite tight) bound of 4:73458 3 p ji \Gamma jj \Gamma 3 for a 3D-Hilbert curve. 1 Introduction It has become increasingly clear that mesh-connected processor arrays, grids for short, are among the most realistic models of parallel computation [1, 4, 14, 18]. The indexing of the processors is an important aspect in the design of mesh algorithms. Several indexing schemes are well-known. Mos...