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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 ..."
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Cited by 516 (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.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 139 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Dynamic Graph Algorithms
, 1999
"... Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamicall ..."
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Cited by 55 (1 self)
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Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. This chapter is intended as an overview of this field. In a typical dynamic graph problem one would like to answer queries on graphs that are undergoing a sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, rather than having to recompute it from scratch each time. Given their powerful versatility, it is not surprising that dynamic algorithms and dynamic data structures are often more difficult to design and analyze than their static c
Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation
 JOURNAL OF THE ACM
, 1999
"... This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new d ..."
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Cited by 52 (0 self)
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This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique which combines a novel graph decomposition with randomization. They are LasVegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of \Omega\Gamma m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O(p log 3 n) 1 for connectivity and bipartiteness. The worstcase time for one query is O(log n= log log n). For the kedge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p log 3 n) and expected time for q queries is O(qk log 3 n). Given a grap...
Planar Separators and Parallel Polygon Triangulation
"... We show how to construct an O ( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O ( p n)separator of that subgraph. We also show how to construct an O(n)way decomposition tree in ..."
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Cited by 52 (8 self)
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We show how to construct an O ( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O ( p n)separator of that subgraph. We also show how to construct an O(n)way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+)separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n = log n) processors on a CRCW PRAM.
Randomized Dynamic Graph ALgorithms with Polylogarithmic Time per Operation
 PROC. 33RD ANNUAL SYMP. ON FOUNDATIONS OF COMPUTER SCIENCE
, 1995
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Authenticated Data Structures for Graph and Geometric Searching
 IN CTRSA
, 2001
"... Following in the spirit of data structure and algorithm correctness checking, authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being maintained by a remote host. We present techniques for authenticatin ..."
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Cited by 46 (18 self)
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Following in the spirit of data structure and algorithm correctness checking, authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being maintained by a remote host. We present techniques for authenticating data structures that represent graphs and collection of geometric objects. We use a model where a data structure maintained by a trusted source is mirrored at distributed directories, with the directories answering queries made by users. When a user queries a directory, it receives a cryptographic proof in addition to the answer, where the proof contains statements signed by the source. The user verifies the proof trusting only the statements signed by the source. We show how to efficiently authenticate data structures for fundamental problems on networks, such as path and connectivity queries, and on geometric objects, such as intersection and containment queries.
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 43 (9 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Dynamic Generators of Topologically Embedded Graphs
, 2003
"... We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g ..."
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Cited by 35 (1 self)
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We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log 4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for lowgenus graphs, and to find in linear time a treedecomposition of lowgenus lowdiameter graphs.
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
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Cited by 34 (4 self)
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Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17