In the area of graph drawing, the One-Sided Crossing Minimization Problem (OSCM) is defined on a bipartite graph with both vertex sets aligned parallel to each other and all edges being drawn as straight lines. The task is to find a permutation of one of the node sets such that the total number of all edge-edge intersections, called crossings, is minimized. Usually, the degree of the nodes of one set is limited by some constant k, with the problem then abbreviated to OSCM-k. In this work, we study an online variant of this problem, in which one of the node sets is already given. The other node set and the incident edges are revealed iteratively and each node has to be inserted into placeholders, which we call slots. The goal is again to minimize the number of crossings in the final graph. Minimizing crossings in an online way is related to the more empirical field of dynamic graph drawing. Note the slotted OSCM problem makes instances harder to solve for an online algorithm but in the offline case it is equivalent to the version without slots. We show that the online slotted OSCM-k is not competitive for any k greater or equal 2 and subsequently limit the graph class to that of 2-regular graphs, for which we show a lower bound of 4/3 and an upper bound of 5 on the competitive ratio.
翻译:在图形绘制领域,单向交叉最小化问题(OSCM)定义在双面图上,双面图上标定了双向顶端对齐,所有边缘都作为直线绘制。任务是找到一个节点组合的变形,这样可以最小化所有边缘交叉点的总数,称为交叉点。通常,一个组合的节点的程度受到某种恒定 k 的限制,问题随后缩略到OSCM-k 。在这项工作中,我们研究了这一问题的在线变式,其中已经给出了一个节点组合。另一个节点和事件边缘被迭接地显示,每个节点必须插入到占位的占位器中,我们称之为插座。目标是再次将最后图表中所有边缘交叉点的总数减少到最小化。以在线方式将过境点最小化与更具经验性的动态图形绘制领域有关。注意到已配置的OSCM问题使得在线算法更难解决,但在离线的案例中,一个节点已经给出了一个节点。其他节点和事件边缘的边缘点已被显示为不具有竞争力的版本。我们所显示的KMMMF2级或正位图的上,我们所显示的正位的正位为相等。