Faster replacement paths
Faster replacement paths
复制标题
更快的更换路径
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
V. V. Williams
中科院分区：
文献类型：
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作者：
V. V. Williams
有向图的替换路径问题是对于给定的节点S和t以及它们之间的最短路径上的每条边，找到S和t之间的最短路径，从而避免了e。对于n个顶点上的未加权有向图，最著名的算法运行时是Roditty和Zwick的2.5。Weimann和Yuster证明了对于在{−，...，M}中具有整数权的图，可以使用快速矩阵乘法，并且可以在(MN<sup>2.584</sup>)时间内解决这个问题，如果矩阵乘法的指数ω为2，则运行时间为<O<sup>O</sup>(<sup>2</sup>)。
我们对这两种算法都进行了改进。我们的新算法还依赖于快速矩阵乘法，运行时间为ω+0(1)。我们的结果表明，至少对于较小的整数权，有向图中的路径替换问题可能比相关的有向图中的所有对最短路径问题容易，因为后者的当前最佳运行时是Ω(n<sup>2.5</sup>)时间，即使ω=2。
The replacement paths problem for directed graphs is to find for given nodes <i>s</i> and <i>t</i> and every edge <i>e</i> on the shortest path between them, the shortest path between <i>s</i> and <i>t</i> which avoids <i>e</i>. For unweighted directed graphs on <i>n</i> vertices, the best known algorithm runtime was <i>Õ</i>(<i>n</i><sup>2.5</sup>) by Roditty and Zwick. For graphs with integer weights in {− <i>M</i>,..., <i>M</i>}, Weimann and Yuster showed that one can use fast matrix multiplication and solve the problem in <i>O</i>(<i>Mn</i><sup>2.584</sup>) time, a runtime which would be <i>O</i>(<i>Mn</i><sup>2.33</sup>) if the exponent ω of matrix multiplication is 2.
We improve on both of these algorithms. Our new algorithm also relies on fast matrix multiplication and runs in <i>Mn</i><sup>ω+<i>o</i>(1)</sup> time. Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current best runtime for the latter is Ω(<i>n</i><sup>2.5</sup>) time even if ω = 2.