A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l uw . (Namely, the order of the orthogonal projections of the vertices of P uw on l uw is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205 ) × O(n 1.205 ). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n × 12n. The smaller drawing size is achieved by a new simple Path Draw algorithm, and a procedure that carefully assigns primitive vectors to the paths of the input tree T .We also show that there exists a tree T 0 such that any monotone drawing of T 0 must use a grid of size Ω(n) × Ω(n). So the size of our monotone drawing of trees is asymptotically optimal.

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A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l. (Namely, the order of the orthogonal projections of the vertices of P uw on l is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205

In this paper, we propose a novel collaborative filtering approach for predicting the unobserved links in a network (or graph) with both topological and node features. Our approach improves the well-known compressed sensing based matrix completion method by introducing a new multiple-independent-Bernoulli-distribution model as the data sampling mask. It makes better link predictions since the model is more general and better matches the data distributions in many real-world networks, such as social networks like Facebook. As a result, a satisfying stability of the prediction can be guaranteed. To obtain an accurate multiple-independent-Bernoulli-distribution model of the topological feature space, our approach adjusts the sampling of the adjacency matrix of the network (or graph) using the clustering information in the node feature space. This yields a better performance than those methods which simply combine the two types of features. Experimental results on several benchmark datasets suggest that our approach outperforms the best existing link prediction methods.