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Computing Hereditary Convex Structures

Bernard Chazelle, Wolfgang Mulzer
2011 Discrete & Computational Geometry  
More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them.  ...  Color red and blue the n vertices of a convex polytope P in R 3 . Can we compute the convex hull of each color class in o(n log n) time? What if we have more than two colors?  ...  To study the complexity of hereditary computing is part of a broader attempt to understand what makes what hard.  ... 
doi:10.1007/s00454-011-9346-8 fatcat:udq7zkbadzdwjkeb7smf7kpnbq

Computing hereditary convex structures

Bernard Chazelle, Wolfgang Mulzer
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09  
More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them.  ...  Color red and blue the n vertices of a convex polytope P in R 3 . Can we compute the convex hull of each color class in o(n log n) time? What if we have more than two colors?  ...  Hereditary algorithms are nothing new. Given a subset of a simple polygon, Chan [8] showed how to compute its convex hull in linear time 2 and how to triangulate it in O(n log * n) time.  ... 
doi:10.1145/1542362.1542374 dblp:conf/compgeom/ChazelleM09 fatcat:sre33etsv5e5dczxobeyxg3uiq

LexBFS-orderings of distance-hereditary graphs with application to the diametral pair problem

Feodor F Dragan, Falk Nicolai
2000 Discrete Applied Mathematics  
As an application we present an algorithm which computes the diameter and a diametral pair of vertices of a distance-hereditary graph in linear time. ?  ...  Moreover, we characterize those distance-hereditary graphs by forbidden subgraphs for which every LexBFS-ordering of the graph is a common perfect elimination ordering of all powers.  ...  In [15] a tree structure for distance-hereditary graphs was given, i.e. distance-hereditary graphs were characterized as the graphs for which the family of all maximal connected cographs is a dual hypertree  ... 
doi:10.1016/s0166-218x(99)00157-2 fatcat:q2zg3t4ghffqxin7hqqusiebie

Guest Editor's Foreword

John Hershberger
2011 Discrete & Computational Geometry  
The papers were revised and carefully reviewed according to the high standards of Discrete & Computational Geometry.  ...  and deep explorations of the boundaries of Asymptopia, results for the plane and for high-dimensional spaces, and topics ranging from classical convex hulls to rigidity theory to topological persistence  ...  They show that these hereditary convex hull problems can often be solved faster than computing the convex hull of the subset from scratch.  ... 
doi:10.1007/s00454-011-9339-7 fatcat:enbn2xpx2bbgpbtlvlsfsum2p4

Approximating Hereditary Discrepancy via Small Width Ellipsoids [chapter]

Aleksandar Nikolov, Kunal Talwar
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
We show that up to this approximation factor, the hereditary discrepancy of a matrix A is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest ℓ∞ norm  ...  Matoušek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing this bound.  ...  However, the structure and robustness of hereditary discrepancy explain its more tractable nature.  ... 
doi:10.1137/1.9781611973730.24 dblp:conf/soda/NikolovT15 fatcat:d6yyaykggffrnfvwy647yldmai

Approximating Hereditary Discrepancy via Small Width Ellipsoids [article]

Aleksandar Nikolov, Kunal Talwar
2014 arXiv   pre-print
We show that up to this approximation factor, the hereditary discrepancy of a matrix A is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest ℓ_∞ norm  ...  Matousek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing this bound.  ...  However, the structure and robustness of hereditary discrepancy explain its more tractable nature.  ... 
arXiv:1311.6204v2 fatcat:sgoa4xsgyrdc5pvyvnj2jts57a

LexBFS-orderings and powers of graphs [chapter]

Feodor F. Dragan, Falk Nicolai, Andreas Brandstädt
1997 Lecture Notes in Computer Science  
As an application we get a linear time approximation of the diameter for weak bipolarizable graphs, a subclass of HHD-free graphs containing all chordal graphs, and an algorithm which computes the diameter  ...  and a diametral pair of vertices of a distance-hereditary graph in linear time.  ...  We use the BFS-tree rooted at z which was already computed in step (9) . Let b : V -+ IN be the numbering of the vertices of G produced by BFS where b(z) = 1.  ... 
doi:10.1007/3-540-62559-3_15 fatcat:gowuukh7ofhnfkqv2c5ctyomue

Page 2908 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
(E-ZRGZM-SM; Zaragoza) The order of points on the second convex hull of a simple polygon. (English summary) Discrete Comput. Geom. 14 (1995), no. 2, 185-205.  ...  An analysis is given of s the topological and measure theoretic structure of the set F, of farthest points from a given point x on a convex surface.  ... 

Induced layered clusters, hereditary mappings, and convex geometries

B. Mirkin, I. Muchnik
2002 Applied Mathematics Letters  
This framework appears to be related to order-theoretic concepts of the hereditary mappings and convex geometries, which enables us to give characterizations of those in terms of the monotone linkage functions  ...  A method for structural clustering proposed by the authors is extended to the case when there are externally defined restrictions on the relations between sets and their elements.  ...  The work was done in the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University. The authors thank DIMACS' Director Dr. F. Roberts for his support.  ... 
doi:10.1016/s0893-9659(01)00133-1 fatcat:x6cjypebhfb7piatqbpfhumzmq

Page 1445 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews  
(English summary) Graph-theoretic concepts in computer science ( Aachen, 1995), 381-395, Lecture Notes in Comput. Sci., 1017, Springer, Berlin, 1995.  ...  An arbitrary finite point set M is divided into convexity levels. The first convexity level M, contains all the points lying on the boundary of the convex hull of M.  ... 

Page 8413 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Nicolai implies that being distance-hereditary is equivalent to having a certain tree structure defined in terms of the maximal complement-reducible (cograph) subgraphs, paralleling the characterization  ...  “In a companion paper [in Graph-theoretic concepts in computer science (Ascona, 1999), 135-147, Springer, Berlin, 1999] we showed that every distance-hereditary graph has clique-width at most 3, and a  ... 

A Multi-Objective Unsupervised Feature Selection using Genetic Algorithm

2017 International Journal of Science and Research (IJSR)  
The group feature selection method for the group structure may be formulated. It performs the task for filtering purpose for group structure technique.  ...  The underlying structure has been neglected by the previous feature selection method and it determines the feature separately.  ...  Gather structure is an accumulation of elements. This serves to builds exactness and reductions computational time.  ... 
doi:10.21275/art20164612 fatcat:ptyqn6pljvf2znr2g5bahikwti

Vertex-Weighted Matching in Two-Directional Orthogonal Ray Graphs [chapter]

C. Gregory Plaxton
2013 Lecture Notes in Computer Science  
Given such a compact representation of G, and a (possibly negative) weight for each vertex, we show how to compute a maximum weight matching of G in O(n log 2 n) time.  ...  As an application of our more general result, we obtain an O(n log 2 n)-time algorithm for computing the VCG outcome of a sealed-bid unit-demand auction in which each item has two associated numerical  ...  Then an MWM of G can be computed in O(n log 2 n) time. Proof. Since any two O(1)-word integers can be compared in constant time, we can construct a representation in reps(ξ * , G) in O(n log n) time.  ... 
doi:10.1007/978-3-642-45030-3_49 fatcat:ekrg335yazhmnj3ybw6rtuv5su

Rebuilding convex sets in graphs

José Cáceres, Alberto Márquez, Ortrud R. Oellermann, María Luz Puertas
2005 Discrete Mathematics  
We show that the contour vertices in distance hereditary graphs form a geodetic set.  ...  In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices.  ...  Among the wide variety of structures that have been studied under abstract convexity are metric spaces, ordered sets or lattices and graphs, the last being the focus of this paper.  ... 
doi:10.1016/j.disc.2005.03.020 fatcat:6ortapfxm5a33lmya63dngavni

Page 3066 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
Comput. Oper. Res. 26 (1999), no. 10-11, 1113-1124.  ...  2000d:90121 of node-deletion problems for hereditary properties.  ... 
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