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Parametric and Kinetic Minimum Spanning Trees
"... We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * r ..."
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Cited by 29 (7 self)
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We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2=3 log4=3 n) per combinatorial change in the tree (or randomized O(n2=3 log n) per change). Our time bounds reduce to O(n1=2 log3=2 n) per change (O(n1=2 log n) randomized) for planar graphs or other minorclosed families of graphs, and O(n1=4 log3=2 n) per change (O(n1=4 log n) randomized) for planar graphs with weight changes but no insertions or deletions.
Rectilinear and Polygonal pPiercing and pCenter Problems
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axispa ..."
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Cited by 28 (1 self)
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We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex coriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) pcenter problem, in which we are given a set S of n points in the plane, and wish to find p axisparallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a lineartime solution for the rectilinear 3center problem (by showing that this problem can be formulated as an LPtype problem and by exhibiting a relation to Helly numbers). We give O(n log n...
Motion Planning for a Convex Polygon in a Polygonal Environment
 Geom
, 1997
"... We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which i ..."
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Cited by 13 (7 self)
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We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also nearquadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...
Mathematical Snapshots From the Computational Geometry Landscape
"... the lines between vertices, and the 2cells are the open convex polygons left after removing the lines from the plane. More generally, for a collection H = fh 1 ; h 2 ; : : : ; h n g of sets in R d , the arrangement of H is a decomposition of R d into connected cells, where each cell is a conn ..."
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the lines between vertices, and the 2cells are the open convex polygons left after removing the lines from the plane. More generally, for a collection H = fh 1 ; h 2 ; : : : ; h n g of sets in R d , the arrangement of H is a decomposition of R d into connected cells, where each cell is a connected component of the set of points lying in all of the sets h i with i 2 I and in no h j with j 62 I, for some index set I ` f1; 2; : : : ; ng. In computational geometry, the most general sets considered in the role of the h i 's are usually the socalled surface patches, which means (d \Gamma 1)dimensional closed semialgebr