Harmonious Hilbert curves and other extradimensional space-filling curves

This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d' < d, the d-dimensional curve is compatible with the d'-dimensional curve with respect to the order in which the curves visit the points of any d'-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation---unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper.Comment: 40 pages, 10 figures, pseudocode include

Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints

Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H\"{o}lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique.Comment: 29 pages, 5 figure

Computation of Lebesgue’s Space-Filling Curve

The means of realizing or approximating the Lebesgue space-filling curve (SFC) with binary arithmetic on a uniformly spaced binary grid are not obvious, one problem being its formulation in terms of ternary representations; that impediment can be overcome via use of a binary-oriented Cantor set.&nbsp; A second impediment, namely the Devil’s Staircase feature, also created by the role of the Cantor set, can be overcome via the definition of a “working inverse”, thereby providing means of achieving compatibility with such a grid. The results indicate an alternative way to proceed, in realizing an approximation to Lebesgue’s SFC, which circumvents any complication raised by Cantor sets and is compatible with binary and integer arithmetic. Well-known constructions such as the z-curve or Morton order, sometimes considered in association with Lebesgue’s SFC, are treated as irrelevant

Efficient Algorithms for Online Task Placement on Runtime Partially Reconfigurable FPGA

Recent generations of FPGAs allow run-time partial reconfiguration. One of the challenging problems in such a multitasking systems is online placement of task. Many online task placement algorithms designed for such partially reconfigurable systems have been proposed to provide efficient and fast task placement. In this paper two different approaches are being used to place the incoming tasks. The first method is uses a run-length based representation that defines the vacant slots on the FPGA. This compact representation allows the algorithm to locate a vacant area suitable to accommodate the incoming task quickly. In the proposed FPGA model, the CLBs are numbered according to Peano Space filling curve model. The second approach is based on harmonic packing. Simulation experiments indicate that proposed techniques result in low ratio of task rejection compared to existing techniques

Scanning and Sequential Decision Making for Multi-Dimensional Data - Part I: the Noiseless Case

We investigate the problem of scanning and prediction ("scandiction", for short) of multidimensional data arrays. This problem arises in several aspects of image and video processing, such as predictive coding, for example, where an image is compressed by coding the error sequence resulting from scandicting it. Thus, it is natural to ask what is the optimal method to scan and predict a given image, what is the resulting minimum prediction loss, and whether there exist specific scandiction schemes which are universal in some sense. Specifically, we investigate the following problems: First, modeling the data array as a random field, we wish to examine whether there exists a scandiction scheme which is independent of the field's distribution, yet asymptotically achieves the same performance as if this distribution was known. This question is answered in the affirmative for the set of all spatially stationary random fields and under mild conditions on the loss function. We then discuss the scenario where a non-optimal scanning order is used, yet accompanied by an optimal predictor, and derive bounds on the excess loss compared to optimal scanning and prediction. This paper is the first part of a two-part paper on sequential decision making for multi-dimensional data. It deals with clean, noiseless data arrays. The second part deals with noisy data arrays, namely, with the case where the decision maker observes only a noisy version of the data, yet it is judged with respect to the original, clean data.Comment: 46 pages, 2 figures. Revised version: title changed, section 1 revised, section 3.1 added, a few minor/technical corrections mad

The Peano software---parallel, automaton-based, dynamically adaptive grid traversals

We discuss the design decisions, design alternatives, and rationale behind the third generation of Peano, a framework for dynamically adaptive Cartesian meshes derived from spacetrees. Peano ties the mesh traversal to the mesh storage and supports only one element-wise traversal order resulting from space-filling curves. The user is not free to choose a traversal order herself. The traversal can exploit regular grid subregions and shared memory as well as distributed memory systems with almost no modifications to a serial application code. We formalize the software design by means of two interacting automata—one automaton for the multiscale grid traversal and one for the application-specific algorithmic steps. This yields a callback-based programming paradigm. We further sketch the supported application types and the two data storage schemes realized before we detail high-performance computing aspects and lessons learned. Special emphasis is put on observations regarding the used programming idioms and algorithmic concepts. This transforms our report from a “one way to implement things” code description into a generic discussion and summary of some alternatives, rationale, and design decisions to be made for any tree-based adaptive mesh refinement software