Shortest paths along a sequence of line segments and connected orthogonal convex hulls.

Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.Shortest paths along a sequence of line segments and connected orthogonal convex hulls.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 HANOI 2022 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, un- der 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, Viet- nam 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 Com- puting, 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 Institute- Level (Professor Nguyen Dong Yen, Professor Le Dung Muu, Associate Pro- fessor Truong Xuan Duc Ha, Associate Professor Nguyen Nang Tam, Asso- ciate 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 v of Notation 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 Poly- gons 36 2.1 Straightest Paths 36 2.2 An Initial Value Problem on a Sequence of Adjacent Polygons 2.3 Conclusions 45 38 Chapter Finding the Connected Orthogonal Convex Hull of iii a Finite Planar Point Set 3.1 46 Orthogonal Convex Sets and their Properties .46 iv 3.2 Construction of the Connected Orthogonal Convex Hull of a Finite Planar Point Set 57 3.3 Algorithm, Implementation and Complexity .60 3.4 3.3.1 New Algorithm Based on Graham’s Convex Hull Algo- rithm 60 3.3.2 Complexity 66 3.3.3 Implementation 68 Conclusions 70 neral Conclusions Ge 71 List of Author’s Related Papers 72 v Table of Notations (X, ρ) [t0, t1], t0, t1 ∈ R γ l(γ) (E, ǁ.ǁ) e , e , , ek a, b, c, p, q, [p, q], p, q ∈ E x a , ya 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 ǁ.ǁ 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) P (a, b)(e1, ,ek ) a path joining a and b with respect to the sequence e1, , ek SP (a, b)(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 con- vex 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(a, b) a staircase path joining a and h an orthogonal convex (x, y)-polygon γ1 ∗ γ2 σ : t = τ0 < τ1 < · · · < τn = t Oabc A(a, b) s(a, b) F(K) P COCH(P ) Pah Pah T (P ) vi o-ext(COCH)(P ) the set of extreme points of COCH(P ) vii ... 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...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... constructed Graham’s convex hull algorithm Some advantages of Graham’s convex hull algorithm can be found in Allison and Noga [9] In case of connected orthogonal convex hulls, if we have a reasonably