We study the problem of finding flows in undirected graphs so as to minimize the weighted $p$-norm of the flow for any $p > 1$. When $p=2$, the problem is that of finding an electrical flow, and its dual is equivalent to solving a Laplacian linear system. The case $p = \infty$ corresponds to finding a min-congestion flow, which is equivalent to max-flows. A typical algorithmic construction for such problems considers vertex potentials corresponding to the flow conservation constraints, and has two simple types of update steps: cycle toggling, which modifies the flow along a cycle, and cut toggling, which modifies all potentials on one side of a cut. Both types of steps are typically performed relative to a spanning tree $T$; then the cycle is a fundamental cycle of $T$, and the cut is a fundamental cut of $T$. In this paper, we show that these simple steps can be used to give a novel efficient implementation for the $p = 2$ case and to find near-optimal $p$-norm flows in a low number of iterations for all values of $p > 1$. Compared to known faster algorithms for these problems, our algorithms are simpler, more combinatorial, and also expose several underlying connections between these algorithms and dynamic graph data structures that have not been formalized previously.
翻译:我们研究了在未定向图表中找到流动流量的问题,以便最大限度地减少任何美元=1美元的加权流动量。当美元=2美元时,问题在于寻找电流,其二是解决拉帕拉西亚线性系统。案例$p==\infty$相当于找到与最大流量相等的速流流。对于这些问题的典型算法结构考虑了与流量保护限制相对应的顶点潜力,并有两种简单的更新步骤:循环调整,改变循环中的流量,并削减,从而改变在截断的一边的所有潜力。两种步骤通常都与横跨树的美元线性系统相对进行;然后是基本循环,相当于最大流量。在本文中,我们用这些简单的步骤可以用来为美元=2美元案件提供新的高效执行,并且找到接近于周期的美元-opimal 美元- coloria 结构的近-opimalalalal 。在之前的图表中,这些最简单的算法性结构是我们所知道的“美元-corrupal ” 和“1美元” 之间,这些更简单的算法性结构中,这些是比更简单的数据更快的。