Convex Drawings of 3-Connected Plane Graphs

We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size The parameter depends on the Schnyder wood used for the drawing. This parameter is in the range The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to For a random 3-connected plane graph, tests show that the expected size of the drawing is      

Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

A bipartite graph G=(U,W,E) with vertex set V=U∪W is convex if there exists an ordering of the vertices of W such that for each u∈U, the neighbors of u are consecutive in W. A compact representation of a convex bipartite graph for specifying such an ordering can be computed in O(|V|+|E|) time. The paired-domination problem on bipartite graphs has been shown to be NP-complete. The complexity of the paired-domination problem on convex bipartite graphs has remained unknown. In this paper, we present an O(|V|) time algorithm to solve the paired-domination problem on convex bipartite graphs given a compact representation. As a byproduct, we show that our algorithm can be directly applied to solve the total domination problem on convex bipartite graphs in the same time bound.     

A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.     

A Linear-Time Algorithm for 7-Coloring 1-Plane Graphs

A graph G is 1-planar if it can be embedded in the plane in such a way that each edge crosses at most one other edge. Borodin showed that 1-planar graphs are 6-colorable, but his proof does not lead to a linear-time algorithm. This paper presents a linear-time algorithm for 7-coloring 1-plane graphs (which are 1-planar graphs already embedded in the plane). The main difficulty in the design of our algorithm comes from the fact that the class of 1-planar graphs is not closed under the operation of edge contraction. This difficulty is overcome by a structural lemma that may be useful in other problems on 1-planar graphs. This paper also shows that it is NP-complete to decide whether a given 1-planar graph is 4-colorable. The complexity of the problem of deciding whether a given 1-planar graph is 5-colorable is still unknown.     

A Linear-Time Algorithm for Finding Locally Connected Spanning Trees on Circular-Arc Graphs

Suppose that T is a spanning tree of a graph G. T is called a locally connected spanning tree of G if for every vertex of T, the set of all its neighbors in T induces a connected subgraph of G. In this paper, given an intersection model of a circular-arc graph, an O(n)-time algorithm is proposed that can determine whether the circular-arc graph contains a locally connected spanning tree or not, and produce one if it exists.     

Linear-Time Recognition of Circular-Arc Graphs

A graph G is a circular-arc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a linear-time algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.     

On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R ² (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. In this paper, we show that the classical Schnyder drawing in R ² of plane triangulations is greedy with respect to a simple natural metric function H(u,v) over R ² that is equivalent to Euclidean metric D _E (u,v) (in the sense that $D_{E}(u,v) \leq H(u,v) \leq2\sqrt{2}D_{E}(u,v)$ ). The drawing uses two integer coordinates between 0 and 2n?5, which can be represented by logn bits. We also show that the classical Schnyder drawing in R ² of 3-connected plane graphs is weakly greedy with respect to the same metric function H(?,?). The drawing uses two integer coordinates between 0 and f (where f is the number of internal faces of G).     

A Simpler Linear-Time Recognition of Circular-Arc Graphs

We give a linear-time recognition algorithm for circular-arc graphs based on the algorithm of Eschen and Spinrad (Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 128–137, 1993) and Eschen (PhD thesis, 1997). Our algorithm both improves the time bound of Eschen and Spinrad, and fixes some flaws in it. Our algorithm is simpler than the earlier linear-time recognition algorithm of McConnell (Algorithmica 37(2):93–147, 2003), which is the only linear time recognition algorithm previously known.     

New Linear-Time Algorithms for Edge-Coloring Planar Graphs

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max {Δ,9} colors. Thus the coloring is optimal for graphs with maximum degree Δ≥9. Moreover for Δ=4,5,6 we give linear-time algorithms that use Δ+2 colors. These results improve over the algorithms of Chrobak and Yung (J. Algorithms 10:35–51, 1989) and of Chrobak and Nishizeki (J. Algorithms 11:102–116, 1990) which color planar graphs using max {Δ,19} colors in linear time or using max {Δ,9} colors in time. R. Cole supported in part by NSF grants CCR0105678 and CCF0515127 and IDM0414763. Ł. Kowalik supported in part by KBN grant 4T11C04425. Part of this work was done while Ł. Kowalik was staying at the Max Planck Institute in Saarbruecken, Germany.     

A Linear-Time Approximation Scheme for Minimum Weight Triangulation of Convex Polygons

A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + ε from the optimum, where ε is an arbitrarily small positive constant. This is the first subcubic algorithm that guarantees such an approximation factor, and it has interesting applications. Received November 21, 1996; revised March 9, 1997.     

A Linear-Time Approximation Scheme for Maximum Weight Triangulation of Convex Polygons

In this paper we present a linear-time approximation scheme for determining the maximum weight triangulation of a convex polygon. Our algorithm is simple and can be implemented easily.     

一种线性原地二路归并算法

和其它排序算法相比,二路归并最适合于两个有序子表的排序。但经典原地二路归并算法的时间性能是乘积型的,尚有改进空间。文章介绍了改进经典原地二路归并算法所需的基本技术,提出了一种线性原地二路归并算法。归并长度分别为m和n的两个有序子表,谈算法最多需要2．5m 1．5n 4．5√m n次比较和8m 7n-3√m n次移动。     

A Total Order Heuristic-Based Convex Hull Algorithm for Points in the Plane

Computing the convex hull of a set of points is a fundamental operation in many research fields, including geometric computing, computer graphics, computer vision, robotics, and so forth. This problem is particularly challenging when the number of points goes beyond some millions. In this article, we describe a very fast algorithm that copes with millions of points in a short period of time without using any kind of parallel computing. This has been made possible because the algorithm reduces to a sorting problem of the input point set, what dramatically minimizes the geometric computations (e.g., angles, distances, and so forth) that are typical in other algorithms. When compared with popular convex hull algorithms (namely, Graham’s scan, Andrew’s monotone chain, Jarvis’ gift wrapping, Chan’s, and Quickhull), our algorithm is capable of generating the convex hull of a point set in the plane much faster than those five algorithms without penalties in memory space.     

A Linear-Time Algorithm for the Perfect Phylogeny Haplotype Problem

The inference of evolutionary trees from binary species-character matrices is a fundamental computational problem in classical phylogenetic studies. Several problems arising in this field lead to different variants of the inference problem; some of these concern input data with missing values or incomplete matrices. A model of inference from incomplete data that has recently gained a remarkable interest is the Perfect Phylogeny Haplotype problem (PPH) introduced in [1] and successfully applied to infer haplotypes from genotype data. A stated open issue in this research field is the linear-time solution of this inference problem. In this paper we solve this question and give an O(nm)-time algorithm to complete matrices of n rows and m columns to represent PPH solutions: we show that solving the problem requires recognizing special posets of width 2.     

基于区域正交化分割的平面点集凸包算法EI北大核心CSCD

为解决实际工程应用中具有超大规模的平面点集的凸包计算问题,提出了一种基于点集所在区域正交化分割的新算法.利用点集几何结构的部分极点对平面点集进行正交化分割,以获取不相干的点集子集簇,再对所有点集子集分别计算其凸包极点,最后合并极点得到凸包点集.在不同层级的正交化分割过程中,根据已知极点的信息,逐层舍去对于凸包极点生成没有贡献的无效点,进而提高算法运行效率.在与目前常用凸包算法的对比实验中,该算法处理超大规模的平面点集时稳定性高且速度更快.     

Linear-Time Recognition of Helly Circular-Arc Models and Graphs

A circular-arc model ℳ is a circle C together with a collection A\mathcal{A} of arcs of C. If A\mathcal{A} satisfies the Helly Property then ℳ is a Helly circular-arc model. A (Helly) circular-arc graph is the intersection graph of a (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear-time recognition algorithms have been described both for the general class and for some of its subclasses. However, for Helly circular-arc graphs, the best recognition algorithm is that by Gavril, whose complexity is O(n ³). In this article, we describe different characterizations for Helly circular-arc graphs, including a characterization by forbidden induced subgraphs for the class. The characterizations lead to a linear-time recognition algorithm for recognizing graphs of this class. The algorithm also produces certificates for a negative answer, by exhibiting a forbidden subgraph of it, within this same bound.     

A Linear-Time Algorithm for the Feasibility of Pebble Motion on Trees

We consider the following pebble motion problem. We are given a tree T with n vertices and two arrangements and of k<n distinct pebbles numbered 1, . . ., k on distinct vertices of the tree. Pebbles can move along edges of T provided that at any given time at most one pebble is traveling along an edge and each vertex of T contains at most one pebble. We are asked the following question: Is arrangement reachable from ? We present an algorithm that, on input two arrangements of k pebbles on a tree with n vertices, decides in time O(n) whether the two arrangements are reachable from one another. We also give an algorithm that, on input two reachable configurations, returns a sequence of moves that transforms one configuration into the other. The pebble motion problem on trees has various applications including memory management in distributed systems, robot motion planning, and deflection routing. Received August 10, 1996; revised October 1, 1997, and February 17, 1998.