Orderly Spanning Trees with Applications

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

Elements of Design for Containers and Solutions in the LinBox Library

We describe in this paper new design techniques used in the \cpp exact linear algebra library \linbox, intended to make the library safer and easier to use, while keeping it generic and efficient. First, we review the new simplified structure for containers, based on our \emph{founding scope allocation} model. We explain design choices and their impact on coding: unification of our matrix classes, clearer model for matrices and submatrices, \etc Then we present a variation of the \emph{strategy} design pattern that is comprised of a controller--plugin system: the controller (solution) chooses among plug-ins (algorithms) that always call back the controllers for subtasks. We give examples using the solution \mul. Finally we present a benchmark architecture that serves two purposes: Providing the user with easier ways to produce graphs; Creating a framework for automatically tuning the library and supporting regression testing.Comment: 8 pages, 4th International Congress on Mathematical Software, Seoul : Korea, Republic Of (2014

Graphulo Implementation of Server-Side Sparse Matrix Multiply in the Accumulo Database

The Apache Accumulo database excels at distributed storage and indexing and is ideally suited for storing graph data. Many big data analytics compute on graph data and persist their results back to the database. These graph calculations are often best performed inside the database server. The GraphBLAS standard provides a compact and efficient basis for a wide range of graph applications through a small number of sparse matrix operations. In this article, we implement GraphBLAS sparse matrix multiplication server-side by leveraging Accumulo's native, high-performance iterators. We compare the mathematics and performance of inner and outer product implementations, and show how an outer product implementation achieves optimal performance near Accumulo's peak write rate. We offer our work as a core component to the Graphulo library that will deliver matrix math primitives for graph analytics within Accumulo.Comment: To be presented at IEEE HPEC 2015: http://www.ieee-hpec.org

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis