Paola Bertolazzi

Upward drawings of triconnected digraphs

A polynomial-time algorithm for testing if a triconnected directed graph has an upward drkwing is presented. An upward drkwing is a planar drkwing such that all the edges flow in a common direction (e.g., from bottom to top). The problem arises in the fields of automatic graph drkwing and ordered sets, and has been open for several years. The proposed algorithm is based on a new combinatorial characterization that maps the problem into a max-flow problem on a sparse network; the time complexity isO(n+r2), wheren is the number of vertices andr is the number of sources and sinks of the directed graph. If the directed graph has an upward drkwing, the algorithm allows us to construct one easily.

Computing orthogonal drawings with the minimum number of bends

We describe a branch-and-bound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such an algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds of the number of bends. We experiment with such algorithm on a large test suite and compare the results with the state of the art. The experiments show the feasibility of the approach and also its limitations. Further, the experiments show how minimizing the number of bends has positive effects on other quality measures of the effectiveness of the drawing. We also present a new method for dealing with vertices of degree larger than four.

A framework for dynamic graph drawing

In this paper we give a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing. We present dynamic algorithms for drawing planar graphs that use a variety of drawing standards (such as polyline, straight-line, orthogonal, grid, upward, and visibility drawings), and address aesthetic criteria that are important for readability, such as the display of planarity, symmetry, and reachability. Also, we provide techniques that are especially tailored for important subclasses of planar graphs such as trees and series-parallel digraphs. Our dynamic drawing algorithms have the important property of performing “smooth updates” of the drawing. Of special geometric interest is the possibility of performing point-location and window queries on the implicit representation of the drawing.

HOW TO DRAW A SERIES-PARALLEL DIGRAPH

Upward and dominance drawings of acyclic digraphs find important applications in the display of hierarchical structures such as PERT diagrams, subroutine-call charts, and is-a relationships. The combinatorial model underlying such hierarchical structures is often a series-parallel digraph. In this paper the problem of constructing upward and dominance drawings of series-parallel digraphs is investigated. We show that the area requirement of upward and dominance drawings of series-parallel digraphs crucially depends on the choice of planar embedding. Also, we present efficient sequential and parallel algorithms for drawing series-parallel digraphs. Our results show that while series-parallel digraphs have a rather simple and well understood combinatorial structure, naive drawing strategies lead to drawings with exponential area, and clever algorithms are needed to achieve optimal area.

Computing Orthogonal Drawings with the Minimum Number of Bends

We describe a branch-and-bound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds of the number of bends. We experiment such algorithm on a large test suite and compare the results with the state-of-the-art. The experiments show how minimizing the number of bends strongly improves several quality measures of the effectiveness of the drawing. We also present a graphic tool with animation that embodies the algorithm and allows interacting with all the phases of the computation.

Parametric graph drawing

A diagram is a drawing on the plane that represents a graph like structure, where nodes are represented by symbols and edges are represented by curves connecting pairs of symbols. An automatic layout facility is a tool that receives as input a graph like structure and is able to produce a diagram that nicely represents such a structure. Many systems use diagrams in the interaction with the users; thus, automatic layout facilities and algorithms for graphs layout have been extensively studied in the last years. We present a new approach in designing an automatic layout facility. Our approach is based on a modular management of a large collection of algorithms and on a strategy that, given the requirements of an application, selects a suitable algorithm for such requirements. The proposed approach has been used for designing the automatic layout facility of Diagram Server, a network server that offers to its clients several facilities for managing diagrams. >

How to draw a series-parallel digraph

Upward and dominance drawings of acyclic digraphs find important applications in the display of hierarchical structures such as PERT diagrams, subroutine-call charts, and is-a relationships. The combinatorial model underlying such hierarchical structures is often a series-parallel digraph. In this paper the problem of constructing upward and dominance drawings of series-parallel digraphs is investigated. We show that the area requirement of upward and dominance drawings of series-parallel digraphs crucially depends on the choice of planar embedding. Also, we present parallel and sequential drawing algorithms that are optimal with respect to both the time complexity and to the area achieved. Our results show that while series-parallel digraphs have a rather simple and well understood combinatorial structure, naive drawing strategies lead to drawings with exponential area, and clever algorithms are needed to achieve optimal area.

A parallel algorithm for the visibility problem from a point

Abstract The problem of the visibility from a point is defined as follows: given a set S of n nonintersecting line segments and a point x, determine the region of the plane that is visible from x. A simpler instance of this problem is the visibility from a point inside a simple polygon. In this paper we present an optimal parallel algorithm for determining the visibility from a point. The algorithm is based on a divide-and-conquer strategy and has time complexity O (log n) on a PRAM with O (n) processors.

Optimal Upward Planarity Testing of Single-Source Digraphs

A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of single-source digraphs: we provide a new combinatorial characterization of upward planarity, and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source digraph with

A systematic approach to the design of modular systolic arrays

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