(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls

(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls(Luận án tiến sĩ) Shortest paths along a sequence of line segments and connected orthogonal convex hulls

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHONG THI THU HUYEN SHORTEST PATHS ALONG A SEQUENCE OF LINE SEGMENTS AND CONNECTED ORTHOGONAL CONVEX HULLS DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2022 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHONG THI THU HUYEN SHORTEST PATHS ALONG A SEQUENCE OF LINE SEGMENTS AND CONNECTED ORTHOGONAL CONVEX HULLS Speciality: Applied Mathematics Speciality code: 46 01 12 DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Associate Professor PHAN THANH AN HANOI - 2022 Confirmation This dissertation was written on the basis of my research works carried out at Institute of Mathematics, Vietnam Academy of Science and Technology, under the supervision of Associate Professor Phan Thanh An All the presented results have never been published by others January 10, 2022 The author Phong Thi Thu Huyen i Acknowledgment First and foremost, I would like to thank my academic advisor, Associate Professor Phan Thanh An, for his guidance and constant encouragement The wonderful research environment of the Institute of Mathematics, Vietnam Academy of Science and Technology, and the excellence of its staff have helped me to complete this work within the schedule I would like to express my special appreciation to Professor Hoang Xuan Phu, Professor Nguyen Dong Yen, Associate Professor Ta Duy Phuong, and other members of the weekly seminar at Department of Numerical Analysis and Scientific Computing, Institute of Mathematics, as well as all the members of Associate Professor Phan Thanh An’s research group for their valuable comments and suggestions on my research results In particular, I would like to express my sincere thanks to Associate Professor Nguyen Ngoc Hai and PhD student Nguyen Thi Le for their significant comments and suggestions concerning the research related to Chapters 1, and Chapter of this dissertation I would like to thank the members of the Thesis Evaluation Committee at the Department-Level and the Thesis Evaluation Committee at the InstituteLevel (Professor Nguyen Dong Yen, Professor Le Dung Muu, Associate Professor Truong Xuan Duc Ha, Associate Professor Nguyen Nang Tam, Associate Professor Nguyen Trung Thanh, Associate Professor Nguyen Thi Thu Thuy, Doctor Hoang Nam Dung, Doctor Nguyen Duc Manh, Doctor Le Xuan Thanh and Doctor Le Hai Yen), and the two anonymous referees, for their careful readings of this dissertation and valuable comments Finally, I would like to thank my family for their endless love and unconditional support ii Contents Table of Notation v List of Figures vi Introduction ix Chapter Preliminaries 0.1 Paths 0.2 Graham’s Convex Hull Algorithm Chapter Shortest Paths with respect to a Sequence of Line Segments in Euclidean Spaces 1.1 Shortest Paths with respect to a Sequence of Ordered Line Segments 1.2 Concatenation of Two Shortest Paths 21 1.3 Conclusions 35 Chapter Straightest Paths on a Sequence of Adjacent Polygons 36 2.1 Straightest Paths 36 2.2 An Initial Value Problem on a Sequence of Adjacent Polygons 38 2.3 Conclusions 45 Chapter Finding the Connected Orthogonal Convex Hull of a Finite Planar Point Set 46 3.1 Orthogonal Convex Sets and their Properties iii 46 3.2 3.3 Construction of the Connected Orthogonal Convex Hull of a Finite Planar Point Set 57 Algorithm, Implementation and Complexity 60 3.3.1 3.4 New Algorithm Based on Graham’s Convex Hull Algorithm 60 3.3.2 Complexity 66 3.3.3 Implementation 68 Conclusions 70 General Conclusions 71 List of Author’s Related Papers 72 iv Table of Notations (X, ρ) [t0 , t1 ], t0 , t1 ∈ R γ l(γ) (E, k.k) e1 , e2 , , ek a, b, c, p, q, [p, q], p, q ∈ E xa , ya P (a, b)(e1 , ,ek ) SP (a, b)(e1 , ,ek ) γ1 ∗ γ2 σ : t0 = τ0 < τ1 < · · · < τn = t1 4abc `(a, b) s(a, b) F(K) P COCH(P ) Pah Pah T (P ) a metric space X with metric ρ an interval in R a path the length of a path γ an Euclidean space E with norm k.k a sequence of line segments in E some points in spaces a line segment between two points p and q two coordinates of a point a = (xa , ya ) a path joining a and b with respect to the sequence e1 , , ek a shortest path joining a and b with respect to the sequence e1 , , ek the concatenation of γ1 and γ2 a set of partitions of [t0 , t1 ] a triangle with three vertexes a, b, and c an orthogonal line through a and b in the sense of orthogonal convexity an orthogonal line segment through a and b in the sense of orthogonal convexity the family of all connected orthogonal convex hulls of the set K a planar finite point set the connected orthogonal convex hull of P the set of points in P in the quadrant of `(a, b) a staircase path joining a and h an orthogonal convex (x, y)-polygon v o-ext(COCH)(P ) the set of extreme points of COCH(P ) vi List of Figures 0.1 Illustration of a simple polygon 0.2 Illustration of convex and nonconvex sets 0.3 Illustration of a polytope 0.4 Convex hull problem 0.5 Illustration of a stack 0.6 An illustration for Graham’s convex hull algorithm 1.1 Illustration of a path 10 1.2 Illustration of Theorem 1.1 17 1.3 Illustration of Theorem 1.2 24 1.4 Illustration of Corollaries 1.7 and 1.8 32 1.5 Illustration of Theorem 1.3 32 2.1 Illustration for a straightest path 38 2.2 A counterexample for the existence of straightest paths 42 3.1 Orthogonal convex sets 47 3.2 Connected orthogonal convex hulls 48 3.3 Orthogonal lines 49 3.4 Semi-isolated points 50 3.5 Two forms of an orthogonal convex set 52 3.6 Example of the intersection of connected orthogonal convex sets 53 3.7 Illustration of Remark 3.1 54 3.8 Corners of an orthogonal line 55 3.9 Maximal elements 58 3.10 An extreme point 59 vii 3.11 The orthogonal line determined by two points in the sense of orthogonal convexity 61 3.12 Left and four cases of orthogonal lines 62 3.13 An example of Procedure Semi-Isolated-Point 63 3.14 Illustration of the connected orthogonal convex hull algorithm 67 3.15 Illustration of time complexity 68 viii ... dissertation studies shortest paths and straightest paths along a sequence of line segments in Euclidean spaces and connected orthogonal convex hulls of a finite planar point set They are meaningful problems... vertexes a, b, and c an orthogonal line through a and b in the sense of orthogonal convexity an orthogonal line segment through a and b in the sense of orthogonal convexity the family of all connected. ..VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHONG THI THU HUYEN SHORTEST PATHS ALONG A SEQUENCE OF LINE SEGMENTS AND CONNECTED ORTHOGONAL CONVEX HULLS Speciality: Applied Mathematics