2,537 research outputs found

    Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort

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    Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of O(N(logN)M1)O(N (\log N)^{M - 1}), with the Best Order Sort algorithm, which has the runtime of O(N2M)O(N^2 M) but demonstrates the best practical performance out of quadratic algorithms. Empirical evaluation shows that the hybrid's running time is typically not worse than of both original algorithms, while for large numbers of points it outperforms them by at least 20%. For smaller numbers of objectives, the speedup can be as large as four times.Comment: A two-page abstract of this paper will appear in the proceedings companion of the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017

    ND-Tree-based update: a Fast Algorithm for the Dynamic Non-Dominance Problem

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    In this paper we propose a new method called ND-Tree-based update (or shortly ND-Tree) for the dynamic non-dominance problem, i.e. the problem of online update of a Pareto archive composed of mutually non-dominated points. It uses a new ND-Tree data structure in which each node represents a subset of points contained in a hyperrectangle defined by its local approximate ideal and nadir points. By building subsets containing points located close in the objective space and using basic properties of the local ideal and nadir points we can efficiently avoid searching many branches in the tree. ND-Tree may be used in multiobjective evolutionary algorithms and other multiobjective metaheuristics to update an archive of potentially non-dominated points. We prove that the proposed algorithm has sub-linear time complexity under mild assumptions. We experimentally compare ND-Tree to the simple list, Quad-tree, and M-Front methods using artificial and realistic benchmarks with up to 10 objectives and show that with this new method substantial reduction of the number of point comparisons and computational time can be obtained. Furthermore, we apply the method to the non-dominated sorting problem showing that it is highly competitive to some recently proposed algorithms dedicated to this problem.Comment: 15 pages, 21 figures, 3 table

    Finding Associations and Computing Similarity via Biased Pair Sampling

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    This version is ***superseded*** by a full version that can be found at http://www.itu.dk/people/pagh/papers/mining-jour.pdf, which contains stronger theoretical results and fixes a mistake in the reporting of experiments. Abstract: Sampling-based methods have previously been proposed for the problem of finding interesting associations in data, even for low-support items. While these methods do not guarantee precise results, they can be vastly more efficient than approaches that rely on exact counting. However, for many similarity measures no such methods have been known. In this paper we show how a wide variety of measures can be supported by a simple biased sampling method. The method also extends to find high-confidence association rules. We demonstrate theoretically that our method is superior to exact methods when the threshold for "interesting similarity/confidence" is above the average pairwise similarity/confidence, and the average support is not too low. Our method is particularly good when transactions contain many items. We confirm in experiments on standard association mining benchmarks that this gives a significant speedup on real data sets (sometimes much larger than the theoretical guarantees). Reductions in computation time of over an order of magnitude, and significant savings in space, are observed.Comment: This is an extended version of a paper that appeared at the IEEE International Conference on Data Mining, 2009. The conference version is (c) 2009 IEE

    Threesomes, Degenerates, and Love Triangles

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    The 3SUM problem is to decide, given a set of nn real numbers, whether any three sum to zero. It is widely conjectured that a trivial O(n2)O(n^2)-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ω(n2)\Omega(n^2) lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is O(n3/2logn)O(n^{3/2}\sqrt{\log n}) and give two subquadratic 3SUM algorithms, a deterministic one running in O(n2/(logn/loglogn)2/3)O(n^2 / (\log n/\log\log n)^{2/3}) time and a randomized one running in O(n2(loglogn)2/logn)O(n^2 (\log\log n)^2 / \log n) time with high probability. Our results lead directly to improved bounds for kk-variate linear degeneracy testing for all odd k3k\ge 3. The problem is to decide, given a linear function f(x1,,xk)=α0+1ikαixif(x_1,\ldots,x_k) = \alpha_0 + \sum_{1\le i\le k} \alpha_i x_i and a set ARA \subset \mathbb{R}, whether 0f(Ak)0\in f(A^k). We show the decision tree complexity of this problem is O(nk/2logn)O(n^{k/2}\sqrt{\log n}). Finally, we give a subcubic algorithm for a generalization of the (min,+)(\min,+)-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-O(n5/2logn)O(n^{5/2}\sqrt{\log n}) decision tree for this problem, as well as an algorithm running in time O(n3(loglogn)2/logn)O(n^3 (\log\log n)^2/\log n)

    Development of an Algorithm for Multicriteria Optimization of Deep Learning Neural Networks

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    Nowadays, machine learning methods are actively used to process big data. A promising direction is neural networks, in which structure optimization occurs on the principles of self-configuration. Genetic algorithms are applied to solve this nontrivial problem. Most multicriteria evolutionary algorithms use a procedure known as non-dominant sorting to rank decisions. However, the efficiency of procedures for adding points and updating rank values in non-dominated sorting (incremental non-dominated sorting) remains low. In this regard, this research improves the performance of these algorithms, including the condition of an asynchronous calculation of the fitness of individuals. The relevance of the research is determined by the fact that although many scholars and specialists have studied the self-tuning of neural networks, they have not yet proposed a comprehensive solution to this problem. In particular, algorithms for efficient non-dominated sorting under conditions of incremental and asynchronous updates when using evolutionary methods of multicriteria optimization have not been fully developed to date. To achieve this goal, a hybrid co-evolutionary algorithm was developed that significantly outperforms all algorithms included in it, including error-back propagation and genetic algorithms that operate separately. The novelty of the obtained results lies in the fact that the developed algorithms have minimal asymptotic complexity. The practical value of the developed algorithms is associated with the fact that they make it possible to solve applied problems of increased complexity in a practically acceptable time. Doi: 10.28991/HIJ-2023-04-01-011 Full Text: PD

    High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

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    We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes
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