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...tions of this paper and suggest future work. 2 Historical Survey and Related Work G. Peano, in 1890, discovered the existence of a continuous curve which passes through every point of a closed square =-=[21]-=-. According to Jordan's precise notion (in 1887) of continuous curves, Peano's curve is a continuous mapping of the closed unit interval I = [0; 1] into the closed unit square S = [0; 1] 2 . Curves of...

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...ttention of mathematicians long before computer came into existence. Socalled space-filling curves try to solve exactly the above optimization problem and it is well-known that the PeanoHilbert curve =-=[50]-=-, [26] is among the space-filling curves which provide the best optimization of the above formula. In our first experiments, we therefore used the PeanoHilbert curve (cf. Fig. 7a) to present the data....

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... do not provide useful results. More useful are techniques which provide a better clustering of closely related data items such as space-filling curves (e.g., the well-known curves by Peano & Hilbert =-=[46]-=- [47] and Morton [48]). For data mining even more important are techniques that provide nice clustering properties as well as an arrangement which is semantically meaningful. An example for a techniqu...

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...+F--F+F+F+F--F+F--F+ . . . +F--F+F--F+F+F+F--F+F) Figure 3-6: Generation of the Koch island 68sCurves with this property are called Peano’s space-filling curves or simply spacefilling curves. 5 Peano =-=[49]-=- constructed the first such curve, which can be generated by an Lsystem with an axiom L and a set of rules L → LFRFL-F-RFLFR+F+LFRFL and R → RFLFR+F+LFRFL-F-RFLFR as shown in Figure 3-7 (a). The limit...

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...to appropriately decomposing its one-dimensional representation. The partitioning scheme first defines an extendable, ordered index space using extendible hashing techniques [1]. Space-filling curves =-=[2, 3]-=- are then used to define a mapping from the multi-dimensional grid hierarchy to the linear index-space. The self-similar nature of these mappings is exploited to maintain locality across levels of the...

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... ; t 0 R ); U(t j ; kC ; t C ; t 0 C )) 2.2 The Morton layout, LMO Our second nonlinear layout function has been variously described as being based either on quadtrees [17] or on space-filling curves =-=[25, 43, 48]-=-. This layout is known in parallel computing as the Morton ordering and has been used for load balancing purposes [5, 27, 28, 44, 49, 59]. It has also been applied for bandwidth reduction in informati...

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...s a small array tile. The above considerations have led authors to investigate a class of array layout functions variously described as being based either on quadtrees [12] or on space-filling curves =-=[18, 32, 34]-=-. Instances of this family are familiar in parallel computing under the names Morton ordering and Hilbert ordering. Despite the dilation effect described above, canonical layout functions have one maj...

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... recursive control structures as a means of delivering high and robust performance for parallel dense linear algebra. The use of quad- or oct-trees (or, in a dual interpretation, space-filling curves =-=[26, 43]-=-) is known in parallel computing [2, 28, 29, 44, 47, 52] for improving both load balance and locality. They have also been applied for bandwidth reduction in information theory [4], for graphics appli...

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... entire adaptive grid hierarchy. The design of these data-structures uses a linear representation of the hierarchical, multi-dimensional grid structure which is generated using space-filling mappings =-=[1, 2]-=-. Operations on the grid hierarchy such as grid creation, grid refinement or coarsening, grid partitioning and dynamic re-partitioning, are efficiently defined on this one-dimensional representation. ...