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A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
Abstract

Cited by 520 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. 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 softwar ..."
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Cited by 34 (4 self)
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Abstract. 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 singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
A NearLinear Algorithm for the Planar Segment Center Problem
, 1996
"... Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in tim ..."
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Cited by 16 (7 self)
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Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(n1+"), for any " ? 0, improving the previous solution of Agarwal et al. [3] by nearly a factor of O(n).
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
Abstract

Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Optimal Cooperative Search In Fractional Cascaded Data Structures
, 1995
"... Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total size n. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying ..."
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Cited by 8 (3 self)
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Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total size n. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel. The preprocessing takes O(log n) time with n/log n processors on an EREW PRAM. For a balanced binary tree cooperative search along roottoleaf paths can be done in O((logn)/logp) time using p processors on a CREW PRAM.
Subquadratic Algorithms for the Weighted Maximin Facility Location Problem (Extended Abstract)
, 1995
"... Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p). We consider the problem of positioning a point x inside a compact region R # R 2 such that min{ w(p) 1 d(x, p) ; p # S } is maximized. Based on the parametric search paradigm, we give the first subquadra ..."
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Cited by 7 (3 self)
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Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p). We consider the problem of positioning a point x inside a compact region R # R 2 such that min{ w(p) 1 d(x, p) ; p # S } is maximized. Based on the parametric search paradigm, we give the first subquadratic algorithms for this problem, with running time O(n log 4 n). Furthermore, we shall introduce the concept of `exact approximation' as the bit model counterpart to parametric search. Exploiting ideas from exact computation, we show that the considered problem can be solved in time O(L(L)n log n), where L denotes the maximal bitsize of input numbers, and (L) the complexity of multiplying two Lbit integers.
I/Oefficient strong connectivity and depthfirst search for directed planar graphs
 In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science
, 2003
"... We present the first I/Oefficient algorithms for the following fundamental problems on directed planar graphs: finding the strongly connected components, finding a simplepath 2 3separator, and computing a depthfirst spanning (DFS) tree. Our algorithms for the first two problems perform O(sort(N ..."
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Cited by 6 (6 self)
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We present the first I/Oefficient algorithms for the following fundamental problems on directed planar graphs: finding the strongly connected components, finding a simplepath 2 3separator, and computing a depthfirst spanning (DFS) tree. Our algorithms for the first two problems perform O(sort(N)) I/Os, where N = V + E and sort(N) = Θ((N/B)log M/B (N/B)) is the number of I/Os required to sort N elements. The DFSalgorithm performs O(sort(N)log(N/M)) I/Os, where M is the number of elements that fit into main memory. 1.
An externalmemory data structure for shortest path queries
 Path Queries, Diplomarbeit, FriedrichSchillerUniversitit Jena, Nov,1998
, 1998
"... ii An ExternalMemory Data Structure for Shortest Path Queries ..."
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Cited by 5 (1 self)
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ii An ExternalMemory Data Structure for Shortest Path Queries
Lower Bounds and Parallel Algorithms for Planar Orthogonal Grid Drawings (Extended Abstract)
 In Proc. IEEE Symposium on Parallel and Distributed Processing
, 1991
"... ) Roberto Tamassia Department of Computer Science Brown University Providence, R.I. 029121910 Ioannis G. Tollis Department of Computer Science The University of Texas at Dallas Richardson, Texas 750830688 Jeffrey S. Vitter Department of Computer Science Brown University Providence, R.I. 029 ..."
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Cited by 4 (3 self)
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) Roberto Tamassia Department of Computer Science Brown University Providence, R.I. 029121910 Ioannis G. Tollis Department of Computer Science The University of Texas at Dallas Richardson, Texas 750830688 Jeffrey S. Vitter Department of Computer Science Brown University Providence, R.I. 029121910 Proc. IEEE Symposium on Parallel and Distributed Processing, 1991 Abstract We consider the problem of constructing a planar orthogonal grid drawing (or more simply, layout) of an nvertex graph, with the goal of minimizing the number of bends along the edges. We exhibit graphs that require \Omega\Gamma n) bends in any layout, and show that there exist optimal drawings that require \Omega\Gamma n) bends and have all of them on a single edge of length\Omega\Gamma n 2 ). On the other side of the coin, we present a parallel algorithm that runs on a CREW PRAM in O(logn) time with n= log n processors and constructs layouts with O(n) maximum edge length and O(n 2 ) area. For bic...
Space efficient algorithms for directed seriesparallel graphs
, 2002
"... The subclass of directed seriesparallel graphs plays an important role in computer science. Whether a given graph is seriesparallel is a well studied problem in algorithmic graph theory, for which fast sequential and parallel algorithms have been developed in a sequence of papers. Also methods are ..."
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Cited by 4 (0 self)
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The subclass of directed seriesparallel graphs plays an important role in computer science. Whether a given graph is seriesparallel is a well studied problem in algorithmic graph theory, for which fast sequential and parallel algorithms have been developed in a sequence of papers. Also methods are known to solve the reachability and the decomposition problem for seriesparallel graphs time efficiently. However, no dedicated results have been obtained for the space complexity of these problems when restricted to seriesparallel graphs – the topic of this paper. Deterministic algorithms are presented for the recognition, reachability, decomposition and the path counting problem for seriesparallel graphs that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in Logspace the main contribution of this work are novel deterministic path finding routines that work correctly in seriesparallel graphs, and a characterization of seriesparallel graphs by forbidden subgraphs that can be tested spaceefficiently. The space bounds are best possible, i.e. the decision problem is shown to be £complete with respect to AC^0reductions. They have also implications for the parallel time complexity of these problems when restricted to seriesparallel graphs. Finally, we sketch how these results can be generalised to extension of the seriesparallel graph family: to graphs with multiple sources or multiple sinks and to the class of minimal vertex seriesparallel graphs.