We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Power consumption is one of the most critical problems in data centers. One effective way to reduce power consumption is to consolidate the hosting workloads and shut down physical machines which become idle after consolidation. Server consolidation is a NP-hard problem. In this paper, a new algorithms Dynamic Round-Robin (DRR), is proposed for energy-aware virtual machine scheduling and consolidation.We compare this strategy with the GREEDY, ROUNDROBIN and POWERSAVE scheduling strategies implemented in the Eucalyptus Cloud system. Our experiment results show that the Dynamic Round-Robin algorithm reduce a significant amount of power consumption compared with the three strategies in Eucalyptus.

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Let G be an n-node planar graph. In a visibility representation of G, each node of G is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated G yields a visibility representation of G no wider than 22n−40 15 . Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether 3n−6 2 is a worst-case lower bound on the required width. Also, if G has no degree-three (respectively, degree-five) internal node, then our visibility representation for G is no wider than 4n−9 3 (respectively, 4n−7 3 ). Moreover, if G is four-connected, then our visibility representation for G is no wider than n − 1, matching the best known result of Kant and He. As a by-product, we give a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Saëc, and Mosbah.

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