Towards optimal locality in mesh-indexings

Abstract

The e ciency of many data structures and algorithms relies on locality-preserving" indexing schemes for meshes. We concentrate on the case where the maximal distance between two mesh nodes indexed i and j shall be a slow-growing function of ji,jj. W e present a new 2-D indexing scheme we call H-indexing, which has superior supposedly 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 l o w er bounds for indexings without any restriction. Further investigations include lower bound results for 3-D mesh indexings. Finally, exempli ed b y i n v estigations concerning 2-D and 3-D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.

New Applications of the Incompressibility Method (Extended Abstract)

“…We present the first "average-case" bound here. Our incompressibility argument is in fact simpler than the worst-case analysis in [18]. Theorem 2.…”

“…Niedermeier, Reinhardt and Sanders recently studied the following problem [18]: In an n x n mesh, consider a computable curve fitting scheme that maps the numbers from { 1, ... , n 2 } into the mesh, with each number occupying a unique point in the mesh. The goal is to minimize the Euclidean distance between numbers relative to their absolute difference.…”

“…Many algorithms in parallel computing, computational geometry, and image processing depend on such "locality-preserving" curve fitting schemes for meshes. [18] shows that for any curve fitting scheme, there exists a pair i and j such that d(i, j) ~ j3.5ji -jj -1, where d is Euclidean distance. However, for both theoretical and practical reasons, it would be more interesting to establish distance bounds that hold not only for one pair of i, j's but for "many" pairs.…”

“…The question of such an "average-case analysis" was open. In fact, many experiments have been performed in the past by researchers in order to determine the average distance between two numbers in a curve fitting scheme [18]. We present the first "average-case" bound here.…”

Enhanced Spatial Stability with Hilbert and Moore Treemaps

“…To resolve this issue, some newly developed algorithms create a layout with non-rectangular items. For example, Voronoi treemaps [15] use arbitrary polygons rather than rectangles for visualising software metrics, and the Jigsaw layout [14] uses the space-filling H-curve [16] to create a treemap with non-rectangular puzzle-piece shaped items (in our paper we use the space-filling Hilbert and Moore curves [10]). Wattenberg [14] also provides a mathematical analysis of the use of space-filling curves for supporting space-filling visualisations.…”