Edge crossings in drawings of bipartite graphs

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed barycenter heuristic for this problem is analyzed. An alternative method, the median heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.     

The diameter of Hanoi graphs

Many questions regarding the Tower of Hanoi problem have been posed and answered during the years. Variants of the classical puzzle, such as allowing more than 3 pegs, and imposing limitations on the possible moves among the pegs, raised the analogous questions for those variants. One such question is: given a variant, and a certain number of disks, find a pair of disk arrangements such that the minimal number of moves required for changing from the first to the second is maximal over all pairs. One of the main results of the paper is identifying these for the Cyclic_h variants—the variants with h pegs arranged along a uni-directional circle—to be the pairs of perfect configurations where the destination peg is right before the source peg.     

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G=(V,E) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v∈V. The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.     

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

A theory of rectangular dual graphs

Arectangular graph is a plane graph where all regions are four-sided and all edges are oriented in either the vertical or the horizontal direction. In addition the graph enclosure must also be rectangular. Given a plane graph representing a desired component connectivity, itsrectangular dual can be used to build afloorplan. This indicates that a system implementation can allocate rectangular floorplan regions to components that can be pairwise connected through common borders.This paper proves that constructing a rectangular dual graph is equivalent to a matching problem in a bipartite graph derived from the given plane graph. A simple existence theorem in terms of the graph structure is obtained as a corollary.This work was supported by the National Science Council, Taiwan, Republic of China, under Contract NSC 78-0404-E006-14.     

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g=3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g∈{4,5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g?6, both problems are APX-hard already for graphs with maximum degree 3.     

Shortest paths in euclidean graphs

We analyze a simple method for finding shortest paths inEuclidean graphs (where vertices are points in a Euclidean space and edge weights are Euclidean distances between points). For many graph models, the average running time of the algorithm to find the shortest path between a specified pair of vertices in a graph withV vertices andE edges is shown to beO(V) as compared withO(E +V logV) required by the classical algorithm due to Dijkstra.Support for the first author was provided in part by NSF Grant MCS-83-08806. Support for the second author was provided in part by NSF Grants MCS-81-05324 and DCR-84-03613, an NSF Presidential Young Investigator Award, an IBM research contract, and an IBM Faculty Development Award. Support for this research was also provided in part by an ONR and DARPA under Contract N00014-83-K-0146 and ARPA Order No. 4786. Equipment support was provided by NSF Grant MCS-81-218106.     

Some results on graphs without long induced paths

Many NP-hard graph problems remain difficult on P_k-free graphs for certain values of k. Our goal is to distinguish subclasses of P_k-free graphs where several important graph problems can be solved in polynomial time. In particular, we show that the independent set problem is polynomial-time solvable in the class of (P_k,K_1,n)-free graphs for any positive integers k and n, thereby generalizing several known results.     

Evaluating esthetics for user-sketched layouts of clustered graphs with known clustering information

This paper aims to empirically analyze the esthetics for user-sketched layouts of clustered graphs with known clustering information. In our experiments, given not only the adjacency list of a clustered graph but also its predefined clustering information, each participant was asked to manually sketch clustered graphs “nicely” from scratch on a tablet system using a stylus. Different from previous works, the main concern in this paper is on which graph drawing esthetics people favor when sketching their own drawings of clustered graphs with known clustering information. Another concern of this paper is on the esthetics of clustered graph layouts employed by participants which include not only characteristics and structures of the final graph layouts but also the behavior of user's sketching process (including layout creation and adjustment). By observing all layouts and drawing processes, the drawing strategies which participants applied and the drawing esthetics are analyzed. Results show that most participants were unsurprisingly able to draw graphs with clear presence of bridge edges and clustering cohesiveness; more importantly, to distinguish clusters within the restricted-size tablet screen during the drawing process, some of the participants were still able to make each cluster with fewer edge crossings, more symmetries, and more alignment of grid in a smaller drawing area where the cluster spreads. Our results support that to alleviate user's complex drawing tasks, aside from the grid-based editing function suggested by the previous work, graph drawing systems should also provide the clustering information if the structure of the graph to be drawn is known.     

A layout algorithm for undirected compound graphs

We present an algorithm for the layout of undirected compound graphs, relaxing restrictions of previously known algorithms in regards to topology and geometry. The algorithm is based on the traditional force-directed layout scheme with extensions to handle multi-level nesting, edges between nodes of arbitrary nesting levels, varying node sizes, and other possible application-specific constraints. Experimental results show that the execution time and quality of the produced drawings with respect to commonly accepted layout criteria are quite satisfactory. The algorithm has also been successfully implemented as part of a pathway integration and analysis toolkit named PATIKA, for drawing complicated biological pathways with compartmental constraints and arbitrary nesting relations to represent molecular complexes and various types of pathway abstractions.     

一种跟踪隐式曲面交线的算法*

传统的跟踪方法在求下一个跟踪点时一般是采用迭代法,而迭代法会出现初始值的选取和迭代收敛的问题。为此提出一种跟踪隐式曲面交线的算法。该方法最主要的优点是：在跟踪隐式曲面的交线时,在前一个跟踪交点已经求得的情况下,利用正方形与两个隐式曲面的交点,即可快速有效地求出下一个跟踪点,而不用涉及迭代收敛的判断。     

Efficient parallel recognition of some circular arc graphs, II

Based on Tucker's work, we present an accurate proof of the characterization of proper circular arc graphs and obtain the first efficient parallel algorithm which not only recognizes proper circular arc graphs but also constructs proper circular arc representations. The algorithm runs inO(log² n) time withO(n ³) processors on a Common CRCW PRAM. The sequential algorithm can be implemented to run inO(n ²) time and is optimal if the input graph is given as an adjacency matrix, so to speak. Portions of this paper appear in preliminary form in theProceedings of the 1989Workshop on Algorithms and Data Structures [2], and theProceedings of the 1994International Symposium on Algorithms and Computation [5].     

Fully dynamic biconnectivity in graphs

We present an algorithm for maintaining the biconnected components of a graph during a sequence of edge insertions and deletions. It requires linear storage and preprocessing time. The amortized running time for insertions and for deletions isO(m ^2/3 ), wherem is the number of edges in the graph. Any query of the form ‘Are the verticesu andv biconnected?’ can be answered in timeO(1). This is the first sublinear algorithm for this problem. We can also output all articulation points separating any two vertices efficiently. If the input is a plane graph, the amortized running time for insertions and deletions drops toO(√n logn) and the query time isO(log² n), wheren is the number of vertices in the graph. The best previously known solution takes timeO(n ^2/3 ) per update or query.     

Drawing graphs by eigenvectors: theory and practice

The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Two important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper, we explore spectral visualization techniques and study their properties from different points of view. We also suggest a novel algorithm for calculating spectral layouts resulting in an extremely fast computation by optimizing the layout within a small vector space.     

Situation awareness assessment in critical driving situations at intersections by task and human error analysis

The rapid development of sensor and tracking technology enables deployment of new Advanced Driver Assistance Systems (ADAS) that support the driver not just on highways but in urban areas as well. Intersections particularly present critical traffic scenarios where almost 35% of accidents occur, partially due to the present lack of in‐depth research about human errors and their determinants. The first step in ergonomic design of ADAS is to identify the specific situations in which drivers require support. To contribute to identification of such spots, situation awareness of 20 drivers in four critical intersection scenarios was explored. The study was conducted in the fixed‐base driving simulator. The applied approach consisted of assessing drivers' expectations and mental workload and of comparing theoretically correct cognitive behavior to experimentally collected data. Intersection scenarios were divided into five segments, and for each segment a task analysis was made. The study has shown that the driving simulator environment can be successfully deployed to provoke and explore various driver errors. The results have revealed that, in scenarios in which information is objectively missing, the majority of errors happened because the drivers had inaccurate mental models of particular scenarios. To the contrary, in the complex scenario the major cause of accidents was information overload. Furthermore, the task analysis disclosed applicable areas of intersection assistance. © 2010 Wiley Periodicals, Inc.     

A long trip in the charming world of graphs for Pattern Recognition

This paper is a historical overview of graph-based methodologies in Pattern Recognition in the last 40 years; history is interpreted with the aim of recognizing the rationale inspiring the papers published in these years, so as to roughly classify them. Despite the extent of scientific production in this field, it is possible to identify three historical periods, each having its own connotation common to most of the corresponding papers, which are called here as the pure, the impure and extreme periods.     

Chain packing in graphs

This paper discusses the complexity of packingk-chains (simple paths of lengthk) into an undirected graph; the chains packed must be either vertex-disjoint or edge-disjoint. Linear-time algorithms are given for both problems when the graph is a tree, and for the edge-disjoint packing problem when the graph is general andk = 2. The vertex-disjoint packing problem for general graphs is shown to be NP-complete even when the graph has maximum degree three andk = 2. Similarly the edge-disjoint packing problem is NP-complete even when the graph has maximum degree four andk = 3.This is a revised version of the technical report [15].