Algorithms in Combinatorial Geometry

This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

[1]  Heinrich W. E. Jung.  Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. , 1901 .

[2]  M. Dehn.  Die Eulersche Formel im Zusammenhang mit dem Inhalt in der Nicht-Euklidischen Geometrie , 1906 .

[3]  C. Carathéodory.  Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen , 1907 .

[4]  J. Radon.  Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[5]  D. M. Y. Sommerville,et al.  The Relations Connecting the Angle-Sums and Volume of a Polytope in Space of n Dimensions , 1927 .

[6]  W. Ackermann.  Zum Hilbertschen Aufbau der reellen Zahlen , 1928 .

[7]  H. S. M. Coxeter,et al.  Vorlesungen über die Theorie der Polyeder , 1935 .

[8]  H. S. White.  The Convex Cells Formed by Seven Planes. , 1939, Proceedings of the National Academy of Sciences of the United States of America.

[9]  P. Halmos.  Finite-Dimensional Vector Spaces , 1960 .

[10]  H. Coxeter.  A Problem of Collinear Points , 1948 .

[11]  Ludwig Schl fli.  Theorie der vielfachen Kontinuit??t , 1950 .

[12]  G. Dirac.  COLLINEARITY PROPERTIES OF SETS OF POINTS , 1951 .

[13]  T. Motzkin.  The lines and planes connecting the points of a finite set , 1951 .

[14]  B. L. Waerden,et al.  Das Problem der dreizehn Kugeln , 1952 .

[15]  A. Heppes.  Beweis einer Vermutung von A. Vázsonyi , 1956 .

[16]  G. Ringel.  Teilungen der Ebene durch Geraden oder topologische Geraden , 1956 .

[17]  P. Erdos.  Some unsolved problems. , 1957 .

[18]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[19]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[20]  M. Shimrat.  Algorithm 112: Position of point relative to polygon , 1962, CACM.

[21]  Richard Hacker,et al.  Certification of Algorithm 112: Position of point relative to polygon , 1962, Commun. ACM.

[22]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[23]  A. Rényi,et al.  über die konvexe Hülle von n zufällig gewählten Punkten , 1963 .

[24]  V. Klee,et al.  Combinatorial Geometry in the Plane , 1964 .

[25]  J. Milnor.  On the Betti numbers of real varieties , 1964 .

[26]  W. Kellaway,et al.  Linear Algebra and Matrix Theory , 1964 .

[27]  H. R. Morton,et al.  Introduction to Topology , 1964, The Mathematical Gazette.

[28]  C. A. Rogers,et al.  Packing and Covering , 1964 .

[29]  H. Davenport,et al.  A Combinatorial Problem Connected with Differential Equations , 1965 .

[30]  L. A. Li︠u︡sternik.  Convex figures and polyhedra , 1966 .

[31]  W. W. Moss.  Some New Analytic and Graphic Approaches to Numerical Taxonomy, with an Example from the Dermanyssidae (ACARI) , 1967 .

[32]  R. Canham.  A theorem on arrangements of lines in the plane , 1969 .

[33]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[34]  H. Raynaud.  Sur L'enveloppe convexe des nuages de points aleatoires dans Rn . I , 1970 .

[35]  P. Scott.  On the Sets of Directions Determined by n Points , 1970 .

[36]  C. L. Liu,et al.  Introduction to Combinatorial Mathematics. , 1971 .

[37]  D. Barnette.  The minimum number of vertices of a simple polytope , 1971 .

[38]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[39]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[40]  Manuel Blum,et al.  Time Bounds for Selection , 1973, J. Comput. Syst. Sci..

[41]  P. Erdös,et al.  Dissection Graphs of Planar Point Sets , 1973 .

[42]  I. G. MacDonald,et al.  CONVEX POLYTOPES AND THE UPPER BOUND CONJECTURE , 1973 .

[43]  Alfred V. Aho,et al.  The design and analysis of algorithms , 1974 .

[44]  E. Szemerédi.  On a problem of Davenport and Schinzel , 1974 .

[45]  N. Sloane,et al.  The orchard problem , 1974 .

[46]  T. Zaslavsky.  Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes , 1975 .

[47]  Ronald L. Rivest,et al.  Expected time bounds for selection , 1975, Commun. ACM.

[48]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[49]  Michael Ian Shamos,et al.  Geometric complexity , 1975, STOC.

[50]  Tom C. Brown.  Common Transversals , 1976, J. Comb. Theory, Ser. A.

[51]  Richard J. Lipton,et al.  Multidimensional Searching Problems , 1976, SIAM J. Comput..

[52]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[53]  I. Hodder,et al.  Spatial Analysis in Archaeology. , 1977 .

[54]  Franco P. Preparata,et al.  Location of a Point in a Planar Subdivision and Its Applications , 1977, SIAM J. Comput..

[55]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[56]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[57]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[58]  Der-Tsai Lee.  Proximity and reachability in the plane. , 1978 .

[59]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[60]  B. Fox,et al.  Construction of Voronoi Polyhedra , 1978 .

[61]  Heiko Harborth.  Konvexe Fünfecke in ebenen Punktmengen. , 1978 .

[62]  George B. Purdy.  Triangles in arrangements of lines , 1979, Discret. Math..

[63]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[64]  A. W. Schurle.  Topics in Topology , 1979 .

[65]  D. Harel.  A Linear Time Algorithm for the Lowest Common Ancestors Problem (Extended Abstract) , 1980, FOCS 1980.

[66]  T. Lewis.  Two counterexamples concerning transversals for convex subsets of the plane , 1980 .

[67]  D. Matula,et al.  Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane , 2010 .

[68]  Jon Louis Bentley,et al.  Decomposable Searching Problems I: Static-to-Dynamic Transformation , 1980, J. Algorithms.

[69]  Godfried T. Toussaint,et al.  The relative neighbourhood graph of a finite planar set , 1980, Pattern Recognit..

[70]  David P. Dobkin,et al.  The Complexity of Linear Programming , 1980, Theor. Comput. Sci..

[71]  Jan van Leeuwen,et al.  Dynamization of Decomposable Searching Problems , 1980, Inf. Process. Lett..

[72]  Richard Pollack,et al.  On the Combinatorial Classification of Nondegenerate Configurations in the Plane , 1980, J. Comb. Theory, Ser. A.

[73]  David P. Dobkin,et al.  Efficient uses of the past , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[74]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[75]  Jan van Leeuwen,et al.  Maintenance of Configurations in the Plane , 1981, J. Comput. Syst. Sci..

[76]  Joseph O'Rourke,et al.  An on-line algorithm for fitting straight lines between data ranges , 1981, CACM.

[77]  Franco P. Preparata,et al.  A New Approach to Planar Point Location , 1981, SIAM J. Comput..

[78]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[79]  R. Seidel.  A Convex Hull Algorithm Optimal for Point Sets in Even Dimensions , 1981 .

[80]  Mark H. Overmars,et al.  Dynamization of Order Decomposable Set Problems , 1981, J. Algorithms.

[81]  Gerald L. Alexanderson,et al.  Arrangements of planes in space , 1981, Discret. Math..

[82]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[83]  W. Nef.  Zur Einführung der Eulerschen Charakteristik , 1981 .

[84]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[85]  Dan E. Willard,et al.  Polygon Retrieval , 1982, SIAM journal on computing (Print).

[86]  G. E. Martin.  Transformation Geometry: An Introduction to Symmetry , 1982 .

[87]  A. Brøndsted.  An Introduction to Convex Polytopes , 1982 .

[88]  C. H. Edwards,et al.  Calculus and Analytic Geometry , 1982 .

[89]  J. Michael Steele,et al.  Lower Bounds for Algebraic Decision Trees , 1982, J. Algorithms.

[90]  Peter Ungar,et al.  2N Noncollinear Points Determine at Least 2N Directions , 1982, J. Comb. Theory, Ser. A.

[91]  Michael Ben-Or,et al.  Lower bounds for algebraic computation trees , 1983, STOC.

[92]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[93]  F. Frances Yao,et al.  A 3-space partition and its applications , 1983, STOC.

[94]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[95]  David G. Kirkpatrick,et al.  Fast Detection of Polyhedral Intersection , 1983, Theor. Comput. Sci..

[96]  J. Horton.  Sets with No Empty Convex 7-Gons , 1983, Canadian Mathematical Bulletin.

[97]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[98]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[99]  Kenneth J. Supowit,et al.  The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees , 1983, JACM.

[100]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[101]  Franz Aurenhammer,et al.  An optimal algorithm for constructing the weighted voronoi diagram in the plane , 1984, Pattern Recognit..

[102]  H. Edelsbrunner,et al.  Efficient algorithms for agglomerative hierarchical clustering methods , 1984 .

[103]  Micha Sharir,et al.  Nonlinearity of Davenport-Schinzel Sequences and of a Generalized Path Compression Scheme , 1984, FOCS.

[104]  Derick Wood,et al.  Paradigms and programming with PASCAL , 1984 .

[105]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[106]  Z. Füredi,et al.  Arrangements of lines with a large number of triangles , 1984 .

[107]  D. Avis.  The Number of Furthest Neighbour Pairs of a Finite Planar Set , 1984 .

[108]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[109]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[110]  Martin E. Dyer,et al.  Linear Time Algorithms for Two- and Three-Variable Linear Programs , 1984, SIAM J. Comput..

[111]  Bernard Chazelle,et al.  On the convex layers of a planar set , 1985, IEEE Trans. Inf. Theory.

[112]  Noga Alon,et al.  A Simple Proof of the Upper Bound Theorem , 1985, Eur. J. Comb..

[113]  GARRET SWART,et al.  Finding the Convex Hull Facet by Facet , 1985, J. Algorithms.

[114]  Herbert Edelsbrunner,et al.  Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane , 1985, Inf. Process. Lett..

[115]  Raimund Seidel,et al.  Finding the optimal shadows of a convex polytope , 1985, SCG '85.

[116]  Richard Cole,et al.  Partitioning Point Sets in 4 Dimensions , 2018, ICALP.

[117]  J. Reif,et al.  Shortest Paths in Euclidean Space with Polyhedral Obstacles. , 1985 .

[118]  Bernard Chazelle,et al.  The power of geometric duality , 1985, BIT Comput. Sci. Sect..

[119]  G. W. Peck.  On 'k-sets' in the plane , 1985, Discret. Math..

[120]  Mark H. Overmars,et al.  Batched Dynamic Solutions to Decomposable Searching Problems , 1985, J. Algorithms.

[121]  Hiroshi Imai,et al.  Voronoi Diagram in the Laguerre Geometry and its Applications , 1985, SIAM J. Comput..

[122]  Ted Lewis,et al.  Geometric permutations for convex sets , 1985, Discret. Math..

[123]  Herbert Edelsbrunner.  Finding Transversals for Sets of Simple Geometric Figures , 1985, Theor. Comput. Sci..

[124]  Bernard Chazelle,et al.  Optimal Solutions for a Class of Point Retrieval Problems , 1985, J. Symb. Comput..

[125]  Herbert Edelsbrunner,et al.  On the Number of Line Separations of a Finite Set in the Plane , 1985, J. Comb. Theory, Ser. A.

[126]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[127]  David Avis,et al.  Triangulating simplicial point sets in space , 1986, SCG '86.

[128]  Kenneth L. Clarkson,et al.  Linear Programming in O(n * (3_d)_2) Time , 1986, Information Processing Letters.

[129]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[130]  Pravin M. Vaidya.  An optimal algorithm for the all-nearest-neighbors problem , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[131]  Wilfred J. Hansen,et al.  Data structures in Pascal , 1986 .

[132]  D. T. Lee,et al.  Generalized delaunay triangulation for planar graphs , 1986, Discret. Comput. Geom..

[133]  F. Frances Yao,et al.  Point Retrieval for Polygons , 1986, J. Algorithms.

[134]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1986, Discret. Comput. Geom..

[135]  Noga Alon,et al.  The number of small semispaces of a finite set of points in the plane , 1986, J. Comb. Theory, Ser. A.

[136]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[137]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[138]  E. Bolker,et al.  Generalized Dirichlet tessellations , 1986 .

[139]  David Haussler,et al.  The complexity of cells in three-dimensional arrangements , 1986, Discret. Math..

[140]  Hungwen Li,et al.  Fast Hough transform: A hierarchical approach , 1986, Comput. Vis. Graph. Image Process..

[141]  Herbert Edelsbrunner,et al.  The number of extreme pairs of finite point-sets in Euclidean spaces , 1986, J. Comb. Theory, Ser. A.

[142]  Herbert Edelsbrunner,et al.  Computing a Ham-Sandwich Cut in Two Dimensions , 1986, J. Symb. Comput..

[143]  Alok Aggarwal,et al.  Geometric Applications of a Matrix Searching Algorithm , 1986, Symposium on Computational Geometry.

[144]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[145]  Mikhail J. Atallah,et al.  Efficient Algorithms for Common Transversals , 1987, Inf. Process. Lett..

[146]  Arto Salomaa,et al.  Formal languages , 1973, Computer science classics.

[147]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[148]  David P. Dobkin,et al.  Space Searching for Intersecting Objects , 1987, J. Algorithms.

[149]  I. Bárány,et al.  Empty Simplices in Euclidean Space , 1987, Canadian Mathematical Bulletin.

[150]  David P. Dobkin,et al.  Searching for empty convex polygons , 1988, SCG '88.

[151]  Micha Sharir,et al.  Separating two simple polygons by a sequence of translations , 2015, Discret. Comput. Geom..

[152]  Paul Erdös,et al.  Repeated distances in space , 1988, Graphs Comb..

[153]  Herbert Edelsbrunner,et al.  Tetrahedrizing Point Sets in Three Dimensions , 1988, ISSAC.

[154]  Micha Sharir,et al.  Planar realizations of nonlinear davenport-schinzel sequences by segments , 1988, Discret. Comput. Geom..

[155]  Micha Sharir,et al.  The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis , 2011, Discret. Comput. Geom..

[156]  H. Edelsbrunner,et al.  On the number of furthest neighbour pairs in a point set , 1989 .

[157]  Robert E. Tarjan,et al.  Planar Point Location Using Persistent Search Trees a , 1989 .

[158]  B.,et al.  Halfspace Range Search : An Algorithmic Application of k-Sets , 2022 .