A Simple Greedy Algorithm for Dynamic Graph Orientation
A Simple Greedy Algorithm for Dynamic Graph Orientation
复制标题
动态图定向的简单贪心算法
作者：
Edvin Berglin;G. Brodal
低输出度的图向是有效存储稀疏图的几种方法之一。如果图形允许插入和删除边，则可能需要翻转一些边的方向，以防止放大最大出度。我们使用树性作为稀疏度度量。利用一种非常简单的贪心算法，我们得到了最坏情况下抛掷次数和出度之间的参数化权衡界，而这种权衡界以前只存在于抛掷平摊次数之间。我们匹配了之前的最佳最坏情况算法（在Ologndocumentclass[12pt]{minimal} uspackage {amsmath} uspackage {wasysym} uspackage {amsfonts} uspackage {amssymb} uspackage {amssymb} uspackage {upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left(logn
ight) $$end{document} flips中）几乎所有的树性值，并击败了它以获得常树性或超对数树性。我们还匹配了之前的最佳平摊结果，至少具有对数性，并给出最坏情况下的第一个结果O1documentclass[12pt]{minimal} uspackage {amsmath} uspackage {wasysym} uspackage {amsfonts} uspackage {amssymb} uspackage {amssymb} uspackage {amssfs} uspackage {upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left(1
ight) $$end{document}和Ologndocumentclass[12pt]{minimal} uspackage {amsmath} uspackage {wasysym} uspackage {amssymb} uspackage {amsssy} uspackage {mathrsfs} uspackage {upgreek}setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left(sqrt{log n}
ight) $$end{document}将几乎匹配的出界翻转到它们各自的平摊解。
Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in Ologndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left( log n
ight) $$end{document} flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left( 1
ight) $$end{document} and Ologndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}left( sqrt{log n}
ight) $$end{document} flips nearly matching out-degree bounds to their respective amortized solutions.