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arXiv 1707.04867cs.DS
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单源最短路径的近最优容权子图

Near Optimal Sized Weight Tolerant Subgraph for Single Source Shortest Path

Diptarka Chakraborty, Debarati Das

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摘要：

在本文中，我们解决的问题，计算一个加权有向图的稀疏子图，使指定的源顶点到所有其他顶点的确切距离保持有界的权重增量。寻找一个小尺寸的子图，保持任何对顶点之间的距离是一个很好的研究问题。由于在现实世界中，任何网络都容易出现故障，因此研究上述问题的容错版本是很自然的。不幸的是，事实证明，可能并不总是存在这样的稀疏子图，即使在单边故障[Demetrescu \emph{et al.} '08]。然而，在实际应用中，网络中的链路（边缘）并不总是完全故障。相反，可能发生的是，一些链路变得更加拥塞，这可以通过增加相应边缘上的权重来轻松捕获。因此，在上述权重增量模型下，尝试构造稀疏距离保持子图是有意义的。据我们所知，这个问题迄今为止还没有被研究过。本文证明了：给定任意一个有$n$顶点和一个源顶点的加权有向图，我们可以构造一个至多包含$e \cdot (k-1)!2^kn$多条边的子图，使得只要总的权重增量由$k$限定，并且我们只允许具有整数值，每个边上的权重（可以是负的）以及边的权重也只能增加某个正整数。接下来，我们显示了一个下界的$c\cdot 2^kn$，对于一些常数$c \ge 5/4$，对大小的子图。我们还认为，限制整数值的权重和整数值的权重增量实际上是必不可少的，如果我们删除这两个限制中的任何一个，我们可能需要存储$Ω(n^2)$边缘，以保持距离。

英文摘要：

In this paper we address the problem of computing a sparse subgraph of a weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a small sized subgraph that preserves distances between any pair of vertices is a well studied problem. Since in the real world any network is prone to failures, it is natural to study the fault tolerant version of the above problem. Unfortunately, it turns out that there may not always exist such a sparse subgraph even under single edge failure [Demetrescu \emph{et al.} '08]. However in real applications it is not always the case that a link (edge) in a network becomes completely faulty. Instead, it can happen that some links become more congested which can easily be captured by increasing weight on the corresponding edges. Thus it makes sense to try to construct a sparse distance preserving subgraph under the above weight increment model. To the best of our knowledge this problem has not been studied so far. In this paper we show that given any weighted directed graph with $n$ vertices and a source vertex, one can construct a subgraph that contains at most $e \cdot (k-1)!2^kn$ many edges such that it preserves distances between the source and all other vertices as long as the total weight increment is bounded by $k$ and we are allowed to have only integer valued (can be negative) weight on each edge and also weight of an edge can only be increased by some positive integer. Next we show a lower bound of $c\cdot 2^kn$, for some constant $c \ge 5/4$, on the size of the subgraph. We also argue that restriction of integer valued weight and integer valued weight increment are actually essential by showing that if we remove any one of these two restrictions we may need to store $Ω(n^2)$ edges to preserve distances.

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