Theory and Practice of I/O efficient Algorithms for Multidimensional Batched Searching Problems

Extended AbstractWe describe a powerful framework for designing efficient batch algorithms for certain large-scale dynamic problems that must be solved using external memory. The class of problems we consider, which we call colorable external decomposable problems, include rectangle intersection, orthogonal line segment intersection, range searching, and point location. We are particularly interested in these problems in two and higher dimensions. They have numerous applications in geographic information systems (GIS), spatial databases, and VLSI and CAD design. We present simplified algorithms for problems previously solved by more complicated approaches (such as rectangle intersection), and we present efficient algorithms for problems not previously solved in an efficient way (such as point location and higher dimensional versions of range searching and rectangle intersection). We give experimental results concerning the running time for our approach applied to the red-blue rectangle intersection problem, which is a key component of the extremely important database operation spatial join. Our algorithm scales well with the problem size, and for large problems sizes it greatly outperforms the well-known sweepline approach

Utilization of the discrete differential evolution for optimization in multidimensional point clouds

The Differential Evolution (DE) is a widely used bioinspired optimization algorithm developed by Storn and Price. It is popular for its simplicity and robustness. This algorithm was primarily designed for real-valued problems and continuous functions, but several modified versions optimizing both integer and discrete-valued problems have been developed. The discrete-coded DE has been mostly used for combinatorial problems in a set of enumerative variants. However, the DE has a great potential in the spatial data analysis and pattern recognition. This paper formulates the problem as a search of a combination of distinct vertices which meet the specified conditions. It proposes a novel approach called the Multidimensional Discrete Differential Evolution (MDDE) applying the principle of the discrete-coded DE in discrete point clouds (PCs). The paper examines the local searching abilities of the MDDE and its convergence to the global optimum in the PCs. The multidimensional discrete vertices cannot be simply ordered to get a convenient course of the discrete data, which is crucial for good convergence of a population. A novel mutation operator utilizing linear ordering of spatial data based on the space filling curves is introduced. The algorithm is tested on several spatial datasets and optimization problems. The experiments show that the MDDE is an efficient and fast method for discrete optimizations in the multidimensional point clouds.Web of Scienceart. no. 632953

An Improved Technique for Multi-Dimensional Constrained Gradient Mining

Multi-dimensional Constrained Gradient Mining, which is an aspect of data mining, is based on mining constrained frequent gradient pattern pairs with significant difference in their measures in transactional database. Top-k Fp-growth with Gradient Pruning and Top-k Fp-growth with No Gradient Pruning were the two algorithms used for Multi-dimensional Constrained Gradient Mining in previous studies. However, these algorithms have their shortcomings. The first requires construction of Fp-tree before searching through the database and the second algorithm requires searching of database twice in finding frequent pattern pairs. These cause the problems of using large amount of time and memory space, which retrogressively make mining of database cumbersome.  Based on this anomaly, a new algorithm that combines Top-k Fp-growth with Gradient pruning and Top-k Fp-growth with No Gradient pruning is designed to eliminate these drawbacks. The new algorithm called Top-K Fp-growth with support Gradient pruning (SUPGRAP) employs the method of scanning the database once, by searching for the node and all the descendant of the node of every task at each level. The idea is to form projected Multidimensional Database and then find the Multidimensional patterns within the projected databases. The evaluation of the new algorithm shows significant improvement in terms of time and space required over the existing algorithms.  &nbsp

Stochastic Constriction Cockroach Swarm Optimization for Multidimensional Space Function Problems

The effect of stochastic constriction on cockroach swarm optimization (CSO) algorithm performance was examined in this paper. A stochastic constriction cockroach swarm optimization (SCCSO) algorithm is proposed. A stochastic constriction factor is introduced into CSO algorithm for swarm stability enhancement; control cockroach movement from one position to another while searching for solution to avoid explosion; enhanced local and global searching capabilities. SCCSO performance was tested through simulation studies and its performance on multidimensional functions is compared with that of original CSO, modified cockroach swarm optimization (MCSO), and one of the well-known global optimization techniques in the literature known as line search restart techniques (LSRS). Standard benchmarks that have been widely used for global optimization problems are considered for evaluating the proposed algorithm. The selected benchmarks were solved up to 3000 dimensions by the proposed algorithm

Minimax Theorems

Import 22/07/2015V této bakalářské práci se zabývám hledáním stacionárních bodů funkcí více proměnných pomocí vybraných vět o minimaxu. Na začátku vysvětlím podmínky existence stacionárního bodu a možnosti jeho nalezení. Poté důkladně popíši algoritmus založený na minimaxu, který se k nalezení těchto bodů používá. Uvedu několik řešených příkladů pro lepší pochopení a nakonec vysvětlím, jak se dá tento postup aplikovat při hledání slabých řešení okrajových úloh.This bachelor thesis deals with searching for stationary points of multidimensional functions, using the minimax theorems. Firstly, it explains conditions of existance of stationary points and finding them. Secondly, it thoroughly explains the algorithm based on minimax, which solves those problems. Several solved problems are shown for better understanding. And at last, it explains how previously introduced theorems and algorithm can find weak solutions of boundary value problems.470 - Katedra aplikované matematikyvelmi dobř

Solving constrained Procrustes problems: a conic optimization approach

Procrustes problems are matrix approximation problems searching for a~transformation of the given dataset to fit another dataset. They find applications in numerous areas, such as factor and multivariate analysis, computer vision, multidimensional scaling or finance. The known methods for solving Procrustes problems have been designed to handle specific sub-classes, where the set of feasible solutions has a special structure (e.g. a Stiefel manifold), and the objective function is defined using a specific matrix norm (typically the Frobenius norm). We show that a wide class of Procrustes problems can be formulated and solved as a (rank-constrained) semi-definite program. This includes balanced and unbalanced (weighted) Procrustes problems, possibly to a partially specified target, but also oblique, projection or two-sided Procrustes problems. The proposed approach can handle additional linear, quadratic, or semi-definite constraints and the objective function defined using the Frobenius norm but also standard operator norms. The results are demonstrated on a set of numerical experiments and also on real applications

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30