Matching Points with Rectangles and Squares Sergey Bereg, Nikolaus - PowerPoint PPT Presentation

Introduction Rectangles Squares Matching Points with Rectangles and Squares Sergey Bereg, Nikolaus Mutsanas & Alexander Wolff SOFSEM’06 university-logo Bereg, Mutsanas & Wolff 1 17 Matching Points with Rectangles and Squares

Introduction Rectangles Squares Outline Introduction Matching in graphs and in the plane Already known... Open Problems Rectangles General position 1/2-Approximation Squares Is there a strong realization? Application to map-labeling NP-Completeness university-logo Bereg, Mutsanas & Wolff 2 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Matching in graphs Maximum Matching O ( √ nm ) [Micali & Vazirani] Euclidean Minimum-Weight Perfect Matching O ( n 2 . 5 log 4 n ) [Vaidya] O (( n /ε 3 ) log 6 n ) [Varadarajan & Agarwal] Matching with segments, rectangles, squares, disks... university-logo Bereg, Mutsanas & Wolff 3 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Matching in the plane Definition Matching is perfect: covers all points. Matching is strong: no overlap. university-logo Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Matching in the plane not perfect Definition Matching is perfect: covers all points. Matching is strong: no overlap. university-logo Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Matching in the plane strong and perfect Definition Matching is perfect: covers all points. Matching is strong: no overlap. university-logo Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Already known... Let P be a set of 2 n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P. university-logo Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Already known... Let P be a set of 2 n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P. university-logo Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Already known... Let P be a set of 2 n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P. university-logo Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Already known... Let P be a set of 2 n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P. university-logo Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Open Problems Questions – How many points can be matched strongly? – Does a given matching have a strong realization? matching size ex. strong realization? segments 100% O ( n log n ) rectangles ? O ( n log n ) n 2 ? n 2 squares ? disks ? ? university-logo Bereg, Mutsanas & Wolff 6 17 Matching Points with Rectangles and Squares

Introduction Matching in graphs and in the plane Rectangles Already known... Squares Open Problems Open Problems Questions – How many points can be matched strongly? – Does a given matching have a strong realization? matching size ex. strong realization? segments 100% O ( n log n ) rectangles 50% O ( n log n ) O ( n 2 log n ) n 2 ? n 2 squares disks ? ? university-logo Bereg, Mutsanas & Wolff 6 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares General position university-logo Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares General position university-logo Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares General position university-logo Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares General position university-logo Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares No general position university-logo Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares No general position university-logo Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares No general position university-logo Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares No general position university-logo Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares No general position π 2 university-logo Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together H 1 V 1 H 2 V 2 university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together H 1 V 1 H 2 V 2 university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together H 1 V 1 H 2 V 2 university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation Divide into subsets → match subsets → put together university-logo Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation - worst case (almost) Worst Case university-logo Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation - worst case (almost) Worst Case H 1 V 1 H 2 V 2 H 3 V 3 H 4 V 4 university-logo Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

Introduction General position Rectangles 1/2-Approximation Squares 1/2-Approximation - worst case (almost) Worst Case H 1 V 1 H 2 V 2 H 3 V 3 H 4 V 4 university-logo Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares