A parallel two-list algorithm for the knapsack problem

An n-element knapsack problem has 2(n) possible solutions to search over, so a task which can be accomplished in 2(n) trials if an exhaustive search is used. Due to the exponential time in solving the knapsack problem, the problem is considered to be very hard. In the past decade, much effort has been done in order to find techniques which could lead to practical algorithms with reasonable running time. In 1994, Chang et al. proposed a brilliant parallel algorithm, which needs O(2(n/8)) processors to solve the knapsack problem in O(2(n/2)) time; that is, the cost of Chang et al.'s parallel algorithm is O(2(5n/8)). In this paper, we propose a parallel algorithm to improve Chang et al.'s parallel algorithm by reducing the time complexity to be O(2(3n/8)) under the same O(2(n/8)) processors available. Thus, the proposed parallel algorithm has a cost of O(2(n/2)). It is an improvement over previous literature. We believe that the proposed parallel algorithm is pragmatically feasible at the moment when multiprocessor systems become more and more popular.