All kind of networks, e.g., Internet of Things, social networks, wireless sensor networks, transportation networks, 4g/5G, etc., are around us to benefit and help our daily life. The multistate flow network (MFN) is always used to model network structures and applications. The level d reliability, Rd, of the MFN is the success probability of sending at least d units of integer flow from the source node to the sink node. The reliability Rd is a popular index for designing, managing, controlling, and evaluating MFNs. The traditional indirect algorithms must have all d-MPs (special connected vectors) or d-MCs (special disconnected vectors) first, then use Inclusion-Exclusion Technique (IET) or Sum-of-disjoint Product (SDP) in terms of found d-MPs or d-MCs to calculate Rd. The above four procedures are all NP-Hard and #P-Hard and cannot calculate Rd for all d at the same time A novel algorithm based on the binary-addition-tree algorithm (BAT) is proposed to calculate the Rd directly for all d at the same time without using any of the above four procedures. The time complexity and demonstration of the proposed algorithm are analyzed, and examples are provided. An experiment is also conducted to compare the proposed algorithm and existing algorithms based on d-MPs, d-MCs, IET, and/or SDP to validate the proposed algorithm.

中文翻译：

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用于计算多状态流网络的同时各级可靠性的新型直接算法

各种网络，例如物联网、社交网络、无线传感器网络、交通网络、4g/5G 等，都在我们身边造福和帮助我们的日常生活。多态流网络 (MFN) 始终用于对网络结构和应用程序进行建模。MFN 的 d 级可靠性 Rd 是从源节点向汇聚节点发送至少 d 个单位的整数流的成功概率。可靠性 Rd 是用于设计、管理、控制和评估 MFN 的流行指标。传统的间接算法必须先有所有的d-MPs（特殊连通向量）或d-MCs（特殊不连通向量），然后使用Inclusion-Exclusion Technique (IET) 或Sum-of-disjoint Product (SDP) 找到d -MPs 或 d-MCs 来计算 Rd。以上四个过程都是NP-Hard和#P-Hard，不能同时计算所有d的Rd同时不使用上述四种程序中的任何一种。分析了所提出算法的时间复杂度和论证，并给出了实例。还进行了一个实验来比较所提出的算法和现有的基于 d-MPs、d-MCs、IET 和/或 SDP 的算法，以验证所提出的算法。并提供了示例。还进行了一个实验来比较所提出的算法和现有的基于 d-MPs、d-MCs、IET 和/或 SDP 的算法，以验证所提出的算法。并提供了示例。还进行了一个实验来比较所提出的算法和现有的基于 d-MPs、d-MCs、IET 和/或 SDP 的算法，以验证所提出的算法。