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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 627 (44 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.
On the Complexity of Optimal Microaggregation for Statistical Disclosure Control
 Statistical Journal of the United Nations Economic Comission for Europe
, 2001
"... Statistical disclosure control (SDC), also termed inference control two decades ago, is an integral part of data security dealing with the protection of statistical databases. The basic problem in SDC is to release data in a way that does not lead to disclosure of individual information (high securi ..."
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Cited by 31 (8 self)
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Statistical disclosure control (SDC), also termed inference control two decades ago, is an integral part of data security dealing with the protection of statistical databases. The basic problem in SDC is to release data in a way that does not lead to disclosure of individual information (high security) but preserves the informational content as much as possible (low information loss). SDC is dual with data mining in that progress of data mining techniques forces official statistics to a continual improvement of SDC techniques: the more powerful the inferences that can be made on a released data set, the more protection is needed so that no inference jeopardizes the privacy of individual respondents' numerical data. This paper deals with the computational complexity of optimal microaggregation, where optimal means yielding minimal information loss for a fixed security level. More specifically, we show that the problem of optimal microaggregation cannot be exactly solved in polynomial time. This result is relevant because it provides theoretical justification for the lack of exact optimal algorithms and for the current use of heuristic approaches.
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 22 (10 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 22 (4 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
The Minimum Broadcast Time Problem for Several Processor Networks
, 1994
"... Broadcasting is the information dissemination process in a communication network. A subset of processors V 0 V called originators knows an unique message which has to be transferred by calls between adjacent processors. Each call requires one time unit and each processor can participate in at most ..."
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Cited by 6 (0 self)
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Broadcasting is the information dissemination process in a communication network. A subset of processors V 0 V called originators knows an unique message which has to be transferred by calls between adjacent processors. Each call requires one time unit and each processor can participate in at most one call per time unit. The problem is to nd a schedule such that the time needed to inform all processors is less than or equal to a deadline k 2 IN. We present NPcompleteness results for this problem restricted to several communication networks (bipartite planar graphs, grid graphs, complete grid graphs, split graphs and chordal graphs) with constant deadline k = 2 or one originator V 0 = {v}.
One Strike Against the MinMax Degree Triangulation Problem
 Manuscript, Fachbereich IV, Mathematik und Informatik, Universitat Trier, Postfach 3825, W5500
, 2000
"... In this paper we analyze the computational complexity of the minmax degree triangulation problem. The problem arises in the generation of twodimensional meshes for plane objects. We show that the problem to triangulate a plane geometric graph with degree at most seven is NPcomplete. 1 Introductio ..."
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Cited by 6 (0 self)
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In this paper we analyze the computational complexity of the minmax degree triangulation problem. The problem arises in the generation of twodimensional meshes for plane objects. We show that the problem to triangulate a plane geometric graph with degree at most seven is NPcomplete. 1 Introduction First, we give some denitions. Let V be a set of n points in IR 2 . An edge is a closed line segment connecting two points of V . Let E be a set of edges. Then G = (V; E) is a geometric graph if for every edge ab 2 E, ab \ V = fa; bg. A geometric graph is called plane if for every two edges ab 6= cd in E, either ab \ cd = ; or ab \ cd is an endpoint of both edges. The connected components of IR 2 minus all points in V and on edges of E are the faces of G. If the edges in E are pairwise disjoint, then G is a matching and we have only one unbounded face. If V is xed and E is maximal such that no two edges cross, then G is a geometric triangulation of the convex hull of V . Then, the...
On stable cutsets in clawfree graphs and planar graphs
, 2008
"... A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let K4 and K1,3 (claw) denote the complete (bipartite) graph on 4 and 1 + 3 vertices. It is NPcomplete to decide whether a line graph (hence a clawfree graph) with maximum degree five or a K4free graph admi ..."
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Cited by 3 (1 self)
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A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let K4 and K1,3 (claw) denote the complete (bipartite) graph on 4 and 1 + 3 vertices. It is NPcomplete to decide whether a line graph (hence a clawfree graph) with maximum degree five or a K4free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a clawfree graph with maximum degree at most four or whether a (claw, K4)free graph admits a stable cutset. As a byproduct we obtain that the stable cutset problem is polynomially solvable for clawfree planar graphs, and also for planar line graphs. Thus, the computational complexity of the stable cutset problem is completely determined for clawfree graphs with respect to degree constraint, and for clawfree planar graphs. Moreover, we prove that the stable cutset problem remains NPcomplete for K4free planar graphs with maximum degree five.
SOME GEOMETRIC CLUSTERING PROBLEMS
 NORDIC JOURNAL OF COMPUTING 1(1994), 246–263
, 1994
"... This paper investigates the computational complexity of several clustering problems with special objective functions for point sets in the Euclidean plane. Our strongest negative result is that clustering a set of 3k points in the plane into k triangles with minimum total circumference is NPhard. ..."
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Cited by 3 (0 self)
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This paper investigates the computational complexity of several clustering problems with special objective functions for point sets in the Euclidean plane. Our strongest negative result is that clustering a set of 3k points in the plane into k triangles with minimum total circumference is NPhard. On the other hand, we identify several special cases that are solvable in polynomial time due to the special structure of their optimal solutions: The clustering of points on a convex hull into triangles; the clustering into equal–sized subsets of points on a line or on a circle with special objective functions; the clustering with minimal clusterdistances. Furthermore, we investigate clustering of planar point sets into convex quadrilaterals.
STATISTICAL COMMISSION and COMMISSION OF THE
, 1999
"... this paper we compare different definitions of disclosure risk and different models for the estimation of such risk from the point of view of the level of safety achieved ..."
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this paper we compare different definitions of disclosure risk and different models for the estimation of such risk from the point of view of the level of safety achieved