In plenty of data analysis tasks, a basic and time-consuming process is to produce a large number of solutions and feed them into downstream processing. Various enumeration algorithms have been developed for this purpose. An enumeration algorithm produces all solutions of a problem instance without repetition. To be a statistically meaningful representation of the solution space, solutions are required to be enumerated in uniformly random order. This paper studies a set of self-reducible NP-problems in three hierarchies, where the problems are polynomially countable ($Sr_{NP}^{FP}$), admit FPTAS ($Sr_{NP}^{FPTAS}$), and admit FPRAS ($Sr_{NP}^{FPRAS}$), respectively. The trivial algorithm based on a (almost) uniform generator is in fact inefficient. We provide a new insight that the (almost) uniform generator is not the end of the story. More efficient algorithmic frameworks are proposed to enumerate solutions in uniformly random order for problems in these three hierarchies. (1) For problems in $Sr_{NP}^{FP}$, we show a random-order enumeration algorithm with polynomial delay (PDREnum); (2) For problems in $Sr_{NP}^{FPTAS}$, we show a Las Vegas random-order enumeration algorithm with expected polynomial delay (PDLVREnum); (3) For problems in $Sr_{NP}^{FPRAS}$, we devise a fully polynomial delay Atlantic City random-order enumeration algorithm with expected delay polynomial in the input size and the given error probability $\delta$ (FPACREnum), which has a probability of at least $1-\delta$ becoming a Las Vegas random-order enumeration algorithm. Finally, to further improve the efficiency of the random-order enumeration algorithms, based on the master/slave paradigm, we present a parallelization with 1.5-optimal enumeration delay and running time, along with the theoretical analysis.
翻译:在大量数据分析任务中,一个基本和耗时的过程是生成大量解决方案并将其输入下游处理。为此,已经开发了各种查点算法。一个查点算法生成了问题实例的所有解决方案,而不再重复。为了在统计上有意义地展示解决方案空间,需要以统一的随机顺序来罗列解决方案。本文研究三个等级的一组自降NP问题,其中的问题可以多盘数地计算 (Sr<unk> NP<unk> FP}$)、接受FPTAS (Sr<unk> NP<unk> FTAS}$ 美元)、并分别接受FPRAS (Sr<unk> NPZ<unk> FTRAS}$ 。基于一个(近于)统一生成器的微小的算法实际上效率。我们提供了一个新的洞察,(最接近的)统一生成器不是故事的结尾。建议更高效的算法框架可以以统一的随机顺序来计算这三种等级的问题。 (1) 与美元(Plational-Plickr)的问题,我们展示了一个Orental-ral_FAR_deal 的奥序算算法问题, 和一个O-ral-rendal-ral-ral-ral-ral-ral-rational-ral-ral-ral_ral dal dal dal dal dalxxxxxxx。</s>