Optimal Worst-case Time Complexity Research Articles

A Novel Negative Selection Algorithm with Optimal Worst-case Training Time Complexity for R-chunk Detectors

Objectives: To generate complete and non-redundant detector set with optimal worst-case time complexity. Methods: In this study, a novel exact matching and string-based Negative Selection Algorithm utilizing r-chunk detectors is proposed. Improved algorithms are tested on some data sets; the experiments’ results are compared with recently published ones. Moreover, algorithms’ complexities are also proved mathematically. Findings: For string-based Artificial Immune Systems, r-chunk detector is the most common detector type and their generation complexity is one of the important factors considered in the literature. We proposed optimal algorithms based on automata to present all detectors. Novelty/applications: The algorithm could generate the representation of complete and nonredundant detector set with optimal worst-case time complexity. To the best of our knowledge, the algorithm is the first one to possess such worst-case training time complexity. Keywords: Artificial Immune Systems, Negative Selection Algorithms, Positive Selection Algorithms, Detector Sets, Self, Non-self.

An optimal coarse-grained arc consistency algorithm

The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: fine-grained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables, and coarse-grained algorithms where the removal of a value will be propagated to the related variables. One big advantage of coarse-grained algorithms, like AC-3, over fine-grained algorithms, like AC-4, is the ease of integration when implementing an algorithm in a constraint solver. However, fine-grained algorithms usually have optimal worst case time complexity while coarse-grained algorithms do not. For example, AC-3 is an algorithm with non-optimal worst case complexity although it is simple, efficient in practice, and widely used. In this paper we propose a coarse-grained algorithm, AC2001/3.1, that is worst case optimal and preserves as much as possible the ease of its integration into a solver (no heavy data structure to be maintained during search). Experimental results show that AC2001/3.1 is competitive with the best fine-grained algorithms such as AC-6. The idea behind the new algorithm can immediately be applied to obtain a path consistency algorithm that has the best-known time and space complexity. The same idea is then extended to non-binary constraints.

Saving Support-Checks Does Not Always Save Time

Arc-consistency algorithms are the workhorse of backtrackers that maintain arc-consistency (MAC). This paper will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arc-consistency algorithm to have an optimal worst-case time-complexity. Sacrificing this optimality allows MAC solvers that (1) do not need additional data structures during search, (2) have an excellent average time-complexity, and (3) have a space-complexity that improves significantly on that of MAC solvers that have optimal arc-consistency components. Results will be presented from an experimental comparison between MAC-2001, MAC-3d and related algorithms. MAC-2001 has an arc-consistency component with an optimal worst-case time-complexity, whereas MAC-3d does not. MAC-2001 requires additional data structures during search, whereas MAC-3d does not. MAC-3d has a O(e+nd) of space-complexity, where n is the number of variables, d the maximum domain size, and e the number of constraints. We shall demonstrate that MAC-2001's space-complexity is O(edmin(n,d)). Our experimental results indicate that MAC-2001 was slower than MAC-3d for easy and hard random problems. For real-world problems things were not as clear.

Arc-consistency and arc-consistency again

There is no need to show the importance of arc-consistency in constraint networks. Mohr and Henderson [9] have proposed AC-4, an algorithm with optimal worst-case time complexity. But it has two drawbacks: its space complexity and average time complexity. In problems with many solutions, where constraints are large, these drawbacks become so important that users often replace AC-4 by AC-3 [8], a non-optimal algorithm. In this paper, we propose a new algorithm, AC-6, which keeps the optimal worst-case time complexity of AC-4 while working out the drawback of space complexity. Moreover, the average time complexity of AC-6 is optimal for constraint networks where nothing is known about the constraint semantics.

All-to-all broadcast by flooding in communications networks

The author establishes the worst-case time complexity of all-to-all broadcast, which involves the simultaneous broadcast of messages from each node in a network to each other node, as with the dissemination of network status information for adaptive routing in ARPANET, by flooding. It it shown to be almost two times the optimum. This suboptimality of flooding results from the selection method permitting some message(s) to be consistently blocked by others. It is possible to avoid such blocking and attain the optimal worst-case time complexity by flooding within a spanning subtree. Simulations examine the performance of all-to-all broadcast by flooding using different selection methods. The best overall performance came using random selection, with FIFO next best, followed by LIFO, and in-order last. This relative performance is consistent with the hypothesis that flooding will generally perform better if implemented with a selection method that tends to avoid having some message(s) consistently blocked by others. Simulations show that the optimal worst-case time complexity is rarely exceeded, and then by only a relatively small amount.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Plane-sweep solves the closest pair problem elegantly

Optimal algorithms for the closest pair problem are often complex because they are derived from algorithms that solve more general problems, such as all-nearest-neighbors or computing the Voronoi diagram. The only known direct algorithm with optimal worst-case time complexity (in the algebraic decision tree model of computation) uses divide-and-conquer and involves a complicated merge step. We show how the two-dimensional closest pair problem is solved elegantly in asymptotically optimal time using a plane-sweep algorithm which is easier to understand and implement. We explain why sweeping generalizes easily, but not efficiently, to multi-dimensional closest pair problems.