We investigate fault-tolerant shortest path problem in the hypercube between two nodes where some nodes are faulty (or blocked) and thus cannot be used in routing. Previously, several similar problems were studied where proposed algorithms are distributed and local-information-based, that is, each node in the network knows only its neighbor's status (faulty or not) and they also look for optimal or near-optimal paths. There have been studies that established some sufficient conditions for these paths to exist. Since these conditions are only sufficient, there could be shortest paths that will be missed by these conditions. We study the problem under the assumption that for two given nodes, a source node s, a target node t, only s requires to have a global information of the network in order to find a shortest path to t, should it exist. A shortest path is defined as the Hamming distance between s and t. This problem can be solved by trivial algorithms. The first is to try all possible paths. In an n-dimensional hypercube with 2ⁿ vertices, this method would cost at least n! time. Another method is to perform a standard shortest path finding algorithm, which would require at least 2ⁿ time. A routing algorithm has been previously developed which is efficient in certain situations. However, in the worst case, its running time could be exponential in the hypercube dimension. We propose an efficient algorithm with running time of O(n³m²), polynomial in n, the hypercube dimension, and m, number of blocking nodes. We gain our efficiency by reducing the routing problem to a permutation problem which can be solved using inclusion-exclusion principle. We finally use dynamic programming technique to optimally count the terms. With the proposed algorithm, not only can we find a shortest path, if such a path does exist, but we can also count all possible shortest paths.

中文翻译：

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具有阻塞/故障节点的超多维数据集上的有效最短路径路由

我们研究了两个节点之间的超多维数据集中的容错最短路径问题，其中某些节点存在故障（或阻塞），因此无法在路由中使用。以前，曾研究过几种类似的问题，这些问题是在分布式算法和基于本地信息的算法下进行的，也就是说，网络中的每个节点仅知道其邻居的状态（有无故障），并且它们还会寻找最佳或接近最佳的路径。已经有研究为这些路径的存在建立了一些充分的条件。由于这些条件仅够用，因此这些条件可能会错过最短的路径。我们在以下假设下研究问题：对于两个给定节点，源节点s，目标节点t，仅s要求拥有网络的全局信息，以便找到t的最短路径（如果存在）。最短路径定义为s和t之间的汉明距离。这个问题可以通过简单的算法解决。首先是尝试所有可能的路径。在具有2^个n顶点的n维超立方体中，此方法至少要花费n！时间。另一种方法是执行标准的最短路径查找算法，这至少需要2 ⁿ时间。先前已经开发了在某些情况下有效的路由算法。但是，在最坏的情况下，其运行时间在超立方体维度上可能是指数级的。我们提出了一种运行时间为O（n ³ m ²），n中为多项式，超立方体维数以及m为阻塞节点数的有效算法。通过将路由问题简化为排列问题，可以使用包含-排除原理解决该问题，从而提高了效率。最后，我们使用动态编程技术来优化对术语的计数。使用提出的算法，我们不仅可以找到最短路径（如果确实存在这样的路径），还可以计算所有可能的最短路径。