This paper defines the Arrwwid number of a recursive tiling (or space-filling curve) as the smallest number w such that any ball Q can be covered by w tiles (or curve sections) with total volume O(vol(Q)). Recursive tilings and space-filling curves with low Arrwwid numbers can be applied to optimise disk, memory or server access patterns when processing sets of points in d-dimensional space. This paper presents recursive tilings and space-filling curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube tilings and space-filling curves cannot have optimal Arrwwid number, and we see how to construct alternatives with better Arrwwid numbers.
Jagadish, Kumar and studied how well space-filling curves succeed in keeping the number of fragments needed to cover a query range low @cite_3 @cite_8 @cite_2 @cite_20 . The curve quality measures in their work are based on the number of fragments needed to cover a query range, averaged over a selection of query ranges that depends on the underlying tiling. As a result their results can only be used to analyse space-filling curves with the same underlying tiling; in particular they assume a tiling that subdivides squares into smaller squares, expanded down to a fixed level of recursion. This class of curves includes well-known curves such as the Hilbert curve @cite_15 and Z-order, also known as Morton order or Lebesgue order @cite_16 . However, it does not include, for example, Peano's curve @cite_11 , which is based on subdividing squares into squares and seems to be the curve of choice in certain applications @cite_9 @cite_1 . The work by Jagadish, Kumar and does not enable a comparison between, for example, Hilbert's curve and Peano's curve.
@cite_8 @cite_9 @cite_1 @cite_3 @cite_2 @cite_15 @cite_16 @cite_20 @cite_11
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This paper presents bounds on the quality of partitions induced by space-filling curves. We compare the surface that surrounds an arbitrary index range with the optimal partition in the grid, i.e. the square. It is shown that partitions induced by Lebesgue and Hilbert curves behave about 1.85 times worse with respect to the length of the surface. The Lebesgue indexing gives better results than the Hilbert indexing in worst case analysis. Furthermore, the surface of partitions based on the Lebesgue indexing are at most 5 2??3 times larger than the optimal in average case. A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set S@?R^d is a set (A"i,B"i) of pairs of subsets of S such that for each i, there are balls D"A"""i and D"B"""i containing A"i and B"i respectively such that d(D"A"""i,D"B"""i)>=s@?min(radius(D"A"""i),radius(D"B"""i)), and for any two points p,q@?S there is a unique index i such that p@?A"i and q@?B"i or vice versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set S@?R^d of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with O(nlogn (t-1)^d) edges that can be computed in O(nlogn (t-1)^d) time. If all balls have the same radius, the number of edges reduces to O(n (t-1)^d). Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(n^2log^2n) time using O(nlogn) space and answers a query in O(n^1^ ^2^+^@e) time, for any @e>0. By reducing the preprocessing time to O(n^1^+^@e) and using O(nlog^2n) space, the query can be answered in O(n^3^ ^4^+^@e) time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems. Abstract We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ⩾ K will have curvature ⩾ K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ⩾ K will have rough curvature ⩾ K . We apply our results to concrete examples of homogeneous planar graphs. In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes).