A browser for directed graphs

A general-purpose browser for directed graphs is described. The browser provides operations to examine and edit graphs and to generate a layout for a graph automatically that minimizes edge crossings. Two layout algorithms were implemented. A hierarchical graph layout algorithm was found to be best for directed graphs. The graph browser also has facilities that allow it to be integrated with other applications (e.g. a program browser). These facilities and our experiences building a program call-graph browser are described.     

A note on leader election in directed split-stars and directed alternating group graphs

A leader node is defined to be any node of the network unambiguously identified by some characteristics. In this paper, we first present a distributed algorithm for finding a leader node of a directed split-star. Moreover, an efficient self-stabilizing leader election algorithm that converges with linear rounds is proposed for directed split-stars. Actually, the distributed algorithm and the self-stabilizing algorithm are also applicable to the problem of directed alternating group graphs. As far as we know, no self-stabilizing leader election algorithm was known for the two graphs.     

A space-filling visualization technique for multivariate small-world graphs

We introduce an information visualization technique, known as GreenCurve, for large multivariate sparse graphs that exhibit small-world properties. Our fractal-based design approach uses spatial cues to approximate the node connections and thus eliminates the links between the nodes in the visualization. The paper describes a robust algorithm to order the neighboring nodes of a large sparse graph by solving the Fiedler vector of its graph Laplacian, and then fold the graph nodes into a space-filling fractal curve based on the Fiedler vector. The result is a highly compact visualization that gives a succinct overview of the graph with guaranteed visibility of every graph node. GreenCurve is designed with the power grid infrastructure in mind. It is intended for use in conjunction with other visualization techniques to support electric power grid operations. The research and development of GreenCurve was conducted in collaboration with domain experts who understand the challenges and possibilities intrinsic to the power grid infrastructure. The paper reports a case study on applying GreenCurve to a power grid problem and presents a usability study to evaluate the design claims that we set forth.     

Dominating sets in directed graphs

We consider the problem of incrementally computing a minimal dominating set of a directed graph after the insertion or deletion of a set of arcs. Earlier results have either focused on the study of the properties that minimum (not minimal) dominating sets preserved or lacked to investigate which update affects a minimal dominating set and in what ways. In this paper, we first show how to incrementally compute a minimal dominating set on arc insertions. We then reduce the case of computing a minimal dominating set on arc deletions to the case of insertions. Some properties on minimal dominating sets are provided to support the incremental strategy. Lastly, we give a new bound on the size of minimum dominating sets based on those results.     

Combinatorial algorithms for feedback problems in directed graphs

Given a weighted directed graph G=(V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A′⊆A such that the directed graph (V,A?A′) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.     

Gaussian conditional random fields extended for directed graphs

For many real-world applications, structured regression is commonly used for predicting output variables that have some internal structure. Gaussian conditional random fields (GCRF) are a widely used type of structured regression model that incorporates the outputs of unstructured predictors and the correlation between objects in order to achieve higher accuracy. However, applications of this model are limited to objects that are symmetrically correlated, while interaction between objects is asymmetric in many cases. In this work we propose a new model, called Directed Gaussian conditional random fields (DirGCRF), which extends GCRF to allow modeling asymmetric relationships (e.g. friendship, influence, love, solidarity, etc.). The DirGCRF models the response variable as a function of both the outputs of unstructured predictors and the asymmetric structure. The effectiveness of the proposed model is characterized on six types of synthetic datasets and four real-world applications where DirGCRF was consistently more accurate than the standard GCRF model and baseline unstructured models.     

Optimization of hierarchical directed graphs

A machine- and language-independent representation for programs suitable for use within a compiler is presented. This representation is the hierarchical directed graph. Powerful code optimization techniques are presented in the context of this representation. An experimental optimizer has been built to validate the ideas discussed. The power of the representation is evident in the compactness of the code implementing the optimizations.     

Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log² n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.     

On a game in directed graphs

Inspired by recent algorithms for electing a leader in a distributed system, we study the following game in a directed graph: each vertex selects one of its outgoing arcs (if any) and eliminates the other endpoint of this arc; the remaining vertices play on until no arcs remain. We call a directed graph lethal if the game must end with all vertices eliminated and mortal if it is possible that the game ends with all vertices eliminated. We show that lethal graphs are precisely collections of vertex-disjoint cycles, and that the problem of deciding whether or not a given directed graph is mortal is NP-complete (and hence it is likely that no “nice” characterization of mortal graphs exists).     

Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log² n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.     

Data schemata based on directed graphs

Rooted, labeled, directed graphs (RLDs) are taken as the basis for describing data structures. A constructive formalism is set up to describe RLDs, generate them by means of grammars, and carry out operations such as accessing nodes, inserting and deleting data items, and recognizing graph patterns. It is shown how the formalism can be translated into languages for data definition and data manipulation, of the type associated with data-base systems. The availability of the formalism allows a systematic development of such languages, and provides a method of incorporating consistency preconditions to structural operations and proving the correctness of the data structures that arise in using such languages.     

Parallel algorithms for minimal spanning trees of directed graphs

The main results of this paper are efficient parallel algorithms, MSP and LOCATE, for computing minimal spanning trees and locating minimal paths in directed graphs, respectively. Algorithm MSP has time complexityO(log³ n) usingO(n ³/logn) processors, while LOCATE has time complexityO(logn) usingO(n ²) processors. Algorithm MSP is derived from sequential algorithms, when the unbounded parallelism model is used.     

Drawing directed graphs using quadratic programming

We describe a new method for visualization of directed graphs. The method combines constraint programming techniques with a high performance force-directed placement (FDP) algorithm. The resulting placements highlight hierarchy in directed graphs while retaining useful properties of FDP; such as emphasis of symmetries and preservation of proximity relations. Our algorithm automatically identifies those parts of the digraph that contain hierarchical information and draws them accordingly. Additionally, those parts that do not contain hierarchy are drawn at the same quality expected from a nonhierarchical, undirected layout algorithm. Our experiments show that this new approach is better able to convey the structure of large digraphs than the most widely used hierarchical graph-drawing method. An interesting application of our algorithm is directional multidimensional scaling (DMDS). DMDS deals with low-dimensional embedding of multivariate data where we want to emphasize the overall flow in the data (e.g., chronological progress) along one of the axes.     

Experiments on drawing 2-level hierarchical graphs

This paper studies different heuristics for drawing 2-level hierarchical graphs. Especially, we compare the barycenter and the median heuristics. We show that the barycenter heuristic clearly outperforms the median heuristic, although only the latter has a proved bound for the maximum error done when two vertices are ordered. Moreover, we improve a known heuristic, called the greedy switching, by introducing the barycenter heuristic as a preprocessing phase for it.     

DiGVis: A system for comprehension and creation of directed graphs for the visually challenged

One of the major barriers for the social inclusion of blind persons is the limited access to graphics-based learning resources which are highly vision oriented. This paper presents a cost-effective tool which facilitates comprehension and creation of virtual directed graphs, such as flowcharts, using alternate modalities of audio and touch. It provides a physically accessible virtual spatial workspace and multimodal interface to non-visually represent directed graphs in interactive manner. The concept of spatial query is used to aid exploration and mental visualization through audio and tactile feedback. A unique aspect of the tool, named DiGVis, offers compatible representations of directed graphs for the sighted and non-sighted persons. A study with 28 visually challenged subjects indicates that the tool facilitates comprehension of layout and directional connectivity of elements in a virtual diagram. Further, in a pilot study, blind persons could independently comprehend a virtual flowchart layout and its logical steps. They were also able to create the flowchart data without sighted assistance using DiGVis. A comparison with sighted subjects using DiGVis for similar task demonstrates the effectiveness of the technique for inclusive education.     

Multi-level association rules and directed graphs for spatial data analysis

We propose a methodology that upgrades the methods of the Lagrangian analysis of surface sea-water parcels. This methodology includes data mining with efficient visualization techniques, namely, spatial–temporal association rules and multi-level directed graphs with different levels of space and time granularity. In the resulting multi-level directed graphs we can intertwine knowledge from various disciplines related to oceanography (in our application) and perform the mining of such graphs. We evaluate the proposed methodology on Lagrangian tracking of virtual particles in the velocity field of the numerical model called the Mediterranean Ocean Forecasting Model (MFS). We describe an efficient algorithm based on label propagation clustering, which finds cycles and paths in multi-level directed graphs and reveals how the number and size of the cycles depend on the seasons. In addition, we offer three interesting results of the visualization and mining of such graphs, that is, the 12 months periodicity of the exchange of water masses among sea areas, the separation of Mediterranean Sea circulation in summer and winter situations, obtained with the hierarchical clustering of multi-level directed graphs, and finally, with visualization with multi-level directed graphs we confirm the reversal of sea circulation in the Ionian Sea over the last decades. The aforementioned results received a very favorable evaluation from oceanographic experts.