Summary The fast Fourier transform (FFT) based on modular arithmetic can compute convolution without round‐off errors, which is desirable in many applications such as computational algebra and combinatory pattern matching. One of the critical challenges of the FFT is to enhance the performance. An effective approach is to optimize the high‐cost operations. Modular reduction is one of the most frequently used high‐cost operations that is a bottleneck of the FFT using modular arithmetic. In this article, we present three modular reduction methods and apply them to the implementation of the FFT. We use the strategy of delaying the modular reduction in the first method. We apply the Montgomery reduction to the FFT in the second method. The two methods both first transform the input, and then replace the modular reductions with lightweight replacements, and apply the reverse transform in the output stage to compute the right results. In the third method, we design an efficient modular reduction for the specific form of modular used in FFT. Experiments show that the incorporation of the new modular reductions speedups the FFT based on modular arithmetic.

Abstract In this article, we present an improved hybrid algorithm based on ant colony optimization and the polycephalum algorithm. First, we use improved the probability selection mechanism in the ant colony algorithm in order to improve the efficiency of next point searching. Second, in each iteration we update the pheromone concentration of the optimal route by using the polycephalum algorithm. We regard the starting point of the optimal route as the water injection point and the end point as the water outlet point. The hybrid algorithm is compared on multiple TSPLIB problems and flexible job shop scheduling problems. And experiments show that the improved algorithm has good application results and resultful in accuracy and optimal solutions.

Summary Given an m‐length query string q, approximate dictionary lookup searches for strings in a string dictionary D at a distance of 1 to q under some distances. In biological retrieval systems, the distances in such queries are defined by evolution operations on genomes. We consider the approximate dictionary lookup with one genome evolution operation including reversal and transposition, which searches for strings in D that can be generated from q by one reversal or one transposition. When the length of the reversed substring is confined to a constant α>1, we propose an O(m)‐time approach which uses bits space, where the dictionary D has d strings with totally |D| symbols. If the lengths of the reversals are in a range [α,β], the time for query is , and the space is words for any constant ε, in which occ is the number of matched strings. For problems allowing one transposition, when the length of the transposition is fixed to α, the time for a dictionary lookup is , while using words. In the case that the two swapped substrings are of the same length, the time for answering the query is O(m), while the space is bits.

In this paper, we introduce two improvements on ant colony optimization (ACO) algorithm: route optimization and individual variation. The first is an optimized implementation of ACO, by which the running time of ants routing is largely reduced. The results of the simulated experiments show that the improved algorithm not only reduces the number of routing in the ACO but also surpasses existing algorithms in performance in solving large-scale TSP problems. In the second improvement, we introduce individual variation to ACO, by which the ants have different routing strategies. Simulation results show that the speed of convergence of ACO algorithm could be enhanced greatly.