Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

A note on the complexity of an algorithm for tchebycheff approximation

Abstract: Some remarks are given concerning the complexity of an exchange algorithm for Tchebycheff Approximation. We consider an “exchange” algorithm that constructs the best polynomial of uniform approximation to a continuous function defined on a closed interval or a finite point set of real numbers. The first, and still popular, class of methods for this problem have been called “exchange algorithms”. We shall consider the simplest method of this class, a blood relative of the dual simplex method of linear programming, and a special case of the cutting plane method. The the idea of the method was initiated by Remes, [1] and [2]. See also Cheney [3], for further developments. Klee and Minty [4], (1972) showed by example that the number of steps in a Simplex method can be exponential in the dimension of the problem. Since then considerable effort has been expended trying to explain the efficiency experienced in practice. Recently, probabilistic models have been assumed that yield expected values for the number of steps with low order monomial behaviour. See for example, Borgwardt [5], and Smale [6]. Alternatively, one might ask can one somehow classify the good problems from the bad ones. We believe that this may be possible for the exchange algorithm.

Strict process machine complexity

Abstract: We introduce a notion of description for infinite sequences and their sets, and a corresponding notion of complexity. We show that for strict process machines, complexity of a sequence or of a subset of Cantor space is equal to its effective Hausdorff dimension.

The design of the algorithm of creating Sudoku puzzle

Abstract: Sudoku puzzle is a well-known and logical-based game. To generate some puzzles of varying difficulty with "unique solution" is not so easy. We make a standard of difficulty based on the player's position, that is, difficulty of solving methods. Then we develop an algorithm to generate puzzles satisfied the requirement. For the complexity of our algorithm, we divide it into two parts. One is the complexity of the algorithm to generate the complete grid. We discover the randomness of generating complete grid increases when the complexity increases, that is, the randomness higher and the complexity greater. We have developed an algorithm which guarantees the most important premise "unique solution" and ensures the complexity is low enough.

On complexity of fast convolution algorithms

Abstract: Complexity analysis is highly recommended to be based on suitable sequential and parallel machines using arbitrary resources. Traditional complexity predicates are changed completely with respect to fast hardware multipliers, complex arithmetic processing units and the forthcoming VLSI technology. Both signal flow graph derivative and program based complexity analysis are proposed. It is shown that sequential complexity will decrease whereas parallel complexity will increase. The analysis of the last transform step in a transform convolution system carries out an increase of the time complexity at all. Consequently the reduction of the last transform step is resulting in an improved performance of fast transform filter algorithms, especially when the transform size is of moderate small size. This is applicable to the prime factor DFT computation via fast convolution, the WFTA transforms and arbitrary FFT transformations in general.

Improving data-parallel construction of δ N -nets with maximum dispersion

Abstract: Linear nearest-neighbor search in high-dimensional data exposes high computational complexity In order to minimize search complexity we employ optimal δ N -nets of rank N, which consist of a sub set of N vectors out of an initial code book E, yet approximate all En vectors of E by the least error (denoted by the number theoretical concept of dispersion) of all possible selections of N vectors In this paper we demonstrate a substantially improved algorithm, which results in a reduction of computational complexity from previously O (En N Λ 2- N Λ 3) to O(En Λ 2) We demonstrate computational feasibility both on CPU-and GPGPU-based systems for any desired accuracy in the approximation of codebook E by some δ N -net for real-world problem sizes in the range of En = [10 Λ 710 Λ 8] In addition to the high speedup that can be achieved by the new algorithm, the GPGPU implementation typically shows speedups of one order of magnitude over a CPU implementation We also give a practical example of image segmentation within computed tomography scans in order to show one potential application of this new approach