Lecture 2 (CGAL) Panos Giannopoulos, Dror Atariah AG TI WS 2012/13
|
|
|
- Silas Reed
- 5 years ago
- Views:
Transcription
1 Lecture 2 (CGAL) Panos Giannopoulos Dror Atariah AG TI WS 2012/13
2 New CGAL release 4.1 Do the exercises! Campus Management?
3 Outline Numerics Number types Kernels Convex Hull Exercise 2 FU Berlin Lecture 2 (CGAL) WS 2012/13 3
4 Numerics Number types Kernels([E. Fogel]) Cgal Numerical Issues t y p e d e f CGAL : : C a r t e s i a n <NT> K e r n e l ; NT s q r t 2 = s q r t (NT ( 2 ) ) ; Kernel : : Point 2 p (0 0) q( sqrt2 sqrt2 ) ; K e r n e l : : C i r c l e 2 C( p 4 ) ; a s s e r t (C. h a s o n b o u n d a r y ( q ) ) ; OK if NT supports exact sqrt. Assertion violation otherwise. Note: assert() will only run in debug mode Use: -DCMAKE_BUILD_TYPE=Debug with cmake (2nd run) FU Berlin Lecture 2 (CGAL) WS 2012/
5 Numerics Number types Kernels Example FU Berlin Lecture 2 (CGAL) WS 2012/13 5
6 Numerics Number types Kernels([E. Fogel]) Cgal Pre-defined Cartesian Kernels Support construction of points from d o u b l e Cartesian coordinates. Support exact geometric predicates. Handle geometric constructions differently: CGAL : : E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l Geometric constructions may be inexact due to round-off errors. It is however more efficient and sufficient for most Cgal algorithms. CGAL : : E x a c t p r e d i c a t e s e x a c t c o n s t r u c t i o n s k e r n e l CGAL : : E x a c t p r e d i c a t e s e x a c t c o n s t r u c t i o n s k e r n e l w i t h s q r t Its number type supports the exact square-root operation. Computational Geometry Algorithm Library FU Berlin Lecture 2 (CGAL) WS 2012/
7 Numerics Number types Kernels([E. Fogel]) Computing the Orientation imperative style #i n c l u d e <CGAL/ E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l. h> t y p e d e f CGAL : : E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l typedef Kernel : : Point 2 Kernel ; Point 2 ; i n t main ( ) { Point 2 p (0 0) q (10 3) r (12 1); r e t u r n (CGAL : : o r i e n t a t i o n ( q p r ) == CGAL : : LEFT TURN )? 0 : 1 ; } precative style #i n c l u d e <CGAL/ E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l. h> t y p e d e f CGAL : : E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l typedef Kernel : : Point 2 typedef Kernel : : Orientation 2 Kernel ; Point 2 ; Orientation 2 ; i n t main ( ) { Kernel kernel ; Orientation 2 orientation = kernel. orientation 2 object (); Point 2 p (0 0) q (10 3) r (12 1); r e t u r n ( o r i e n t a t i o n ( q p r ) == CGAL : : LEFT TURN )? 0 : 1 ; } FU Berlin Lecture 2 (CGAL) WS 2012/
8 Numerics Number types Kernels Orientation1 Orientation2 FU Berlin Lecture 2 (CGAL) WS 2012/13 8
9 Convex Hull Terms and Definitions Definition (convex hull) The convex hull of a set of points P Rd denoted as conv(p) is the smallest (inclusionwise) convex set containing P. When an elastic band stretched open to encompass the input points is released it assumes the shape of the convex hull. n h the number of input points. the number of points in the hull. Time complexities of convex hull computation: Optimal output sensitive: O(n log h). QuickHull (expected): O(n log n). FU Berlin Lecture 2 (CGAL) WS 2012/13 [Chan6] [BDH6] 52
10 Convex Hull Terms and Definitions Definition (convex hull) The convex hull of a set of points P Rd denoted as conv(p) is the smallest (inclusionwise) convex set containing P. When an elastic band stretched open to encompass the input points is released it assumes the shape of the convex hull. n h the number of input points. the number of points in the hull. Time complexities of convex hull computation: Optimal output sensitive: O(n log h). QuickHull (expected): O(n log n). FU Berlin Lecture 2 (CGAL) WS 2012/13 [Chan6] [BDH6] 53
11 Convex Hull Properties A subset S IR d is convex the line segment pq S for any two points p q S. The convex hull of a set S is the smallest convex set containing S. The convex hull of a set of points P is a convex polygon with vertices in P. Input: set of points P (or objects). Output: the convex hull S P of P. FU Berlin Lecture 2 (CGAL) WS 2012/13 54
12 Cgal Convex Hull #i n c l u d e <CGAL/ E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l. h> #i n c l u d e <CGAL/ c o n v e x h u l l 2. h> t y p e d e f CGAL : : E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l K e r n e l ; typedef Kernel : : Point 2 Point 2 ; i n t main ( ) { CGAL : : s e t a s c i i m o d e ( s t d : : c i n ) ; CGAL : : s e t a s c i i m o d e ( s t d : : c o u t ) ; s t d : : i s t r e a m i t e r a t o r <P o i n t 2> i n s t a r t ( s t d : : c i n ) ; s t d : : i s t r e a m i t e r a t o r <P o i n t 2> i n e n d ; s t d : : o s t r e a m i t e r a t o r <P o i n t 2> o u t ( s t d : : c o u t \n ) ; CGAL : : c o n v e x h u l l 2 ( i n s t a r t i n e n d o u t ) ; return 0; } p3 p5 p2 p4 p1 FU Berlin Lecture 2 (CGAL) WS 2012/13 55
13 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 56
14 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 57
15 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 58
16 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 5
17 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 60
18 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 61
19 Incremental Convex Hull The edge pq is visible from r orientation(p q r ) < 0 The edge pq is weakly visible from r orientation(p q r ) 0 Maintain the current convex hull S of a set of points seen so far 1. Initialize S to the counter-clockwise sequence {a b c} P 2. Remove a b and c from P 3. for all r P do 4. if there is an edge e visible from r then 5. Compute the sequence of edges {vi vi+1... vj 1 vj } weakly visible from r 6. Replace the sequence {vi+1... vj 1 } by r The sequence of edges weakly visible from r {vi vi+1... vj 1 vj } is a consecutive chain FU Berlin Lecture 2 (CGAL) WS 2012/13 62
20 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 = ( ) p5 p1 p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. p1 p5 p2 p3 [KMP+ 08] FU Berlin Lecture 2 (CGAL) WS 2012/13 63
21 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. p1 p6 p5 p2 p3 [KMP+ 08] FU Berlin Lecture 2 (CGAL) WS 2012/13 64
22 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 65
23 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 66
24 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 67
25 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 68
26 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 6
27 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 70
28 Wrong Incremental Convex Hull p1 = ( ) p2 = ( ) p3 = ( ) p4 = ( ) p5 p1 p5 = ( ) p6 = (6 6) p4 (p1 p2 p3 p4 ) form a convex quadrilateral. p5 is truly inside this quadrilateral. orientation (p4 p1 p5 ) < 0. orientation (p4 p5 p6 ) < 0. p1 p6 p5 p2 p3 orientation (p5 p1 p6 ) > 0. orientation (p1 p2 p6 ) < 0. FU Berlin Lecture 2 (CGAL) WS 2012/13 [KMP+ 08] 71
29 Graham s Scan Convex Hull Andrew s variant of Graham s scan algorithm Compute the convex hull S of a set of points P Sort the points in lexicographic order resulting in the sequence p1... pn Supper {p1 p2 } for i 3 to n while Supper > 2 &!rightturn(q r pi ) q and r are the last points of Supper Supper Supper \ {r } Supper Supper {pi } FU Berlin Lecture 2 (CGAL) WS 2012/13 72
30 Graham s Scan Convex Hull Andrew s variant of Graham s scan algorithm Compute the convex hull S of a set of points P Sort the points in lexicographic order resulting in the sequence p1... pn Supper {p1 p2 } for i 3 to n while Supper > 2 &!rightturn(q r pi ) q and r are the last points of Supper Supper Supper \ {r } Supper Supper {pi } Slower {pn pn 1 } for i n 2 downto 1 while Slower > 2 &!rightturn(q r pi ) q and r are the last points of Slower Slower Slower \ {r } Slower Slower {pi } Remove the last points from Slower and Supper S Supper Slower FU Berlin Lecture 2 (CGAL) WS 2012/13 73
31 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 74
32 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 75
33 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 76
34 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 77
35 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 78
36 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 7
37 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 80
38 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 81
39 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 82
40 Graham s Scan Convex Hull Theorem (Convex Hull) The convex hull of a set of n points in the plane can be computed in O(n log n) time. FU Berlin Lecture 2 (CGAL) WS 2012/13 83
41 Cgal Convex-Hull Implementations Given n points and h extreme points CGAL : : c h a k l t o u s s a i n t ( ) CGAL : : c h b y k a t ( ) CGAL : : c h e d d y ( ) CGAL : : c h g r a h a m a n d r e w ( ) CGAL : : c h j a r v i s ( ) CGAL : : ch melkman ( ) FU Berlin Lecture 2 (CGAL) WS 2012/13 O(n log n) O(nh) O(nh) O(nlogn) O(nh) O(n) (simple polygon) 84
42 Upper Convex Hull #i n c l u d e <i o s t r e a m > #i n c l u d e <CGAL/ E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l. h> t y p e d e f CGAL : : E x a c t p r e d i c a t e s i n e x a c t c o n s t r u c t i o n s k e r n e l Kernel ; i n t main ( ) { Kernel kernel ; s t d : : v e c t o r <K e r n e l : : P o i n t 2> i n ; s t d : : v e c t o r <K e r n e l : : P o i n t 2> o u t ; u p p e r h u l l ( i n. b e g i n ( ) i n. end ( ) s t d : : b a c k i n s e r t e r ( o u t ) k e r n e l ) ; return 0; } Using Random Access Iterator. [ u p p e r c o n v e x h u l l 1. cpp] Provides both increment and decrement. Constant-time methods for moving forward and backward in arbitrary-sized steps ( [ ] operator). Maintaining the current point separately. [ u p p e r c o n v e x h u l l 2. cpp] Maintaining an iterator that points to the current point separately. [ u p p e r c o n v e x h u l l 3. cpp] More generic can be used for computing the lower hull. [ u p p e r c o n v e x h u l l 4. cpp] FU Berlin Lecture 2 (CGAL) WS 2012/13 85
43 Convex-Hull Bibliography C. Bradford Barber David P. Dobkin and Hannu T. Huhdanpaa The Quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 22(4): Timothy M. Chan Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry 16: Lutz Kettner Kurt Mehlhorn Sylvain Pion Stefan Schirra and Chee K. Yap. Classroom Examples of Robustness Problems in Geometric Computations. Computational Geometry: Theory and Applications 40(1): FU Berlin Lecture 2 (CGAL) WS 2012/13 86
44 Convex Hull Numerics Kernels FU Berlin Lecture 2 (CGAL) WS 2012/13 10
45 Exercise 2 1. FU Berlin Lecture 2 (CGAL) WS 2012/13 11
Randomization in Algorithms and Data Structures
Randomization in Algorithms and Data Structures 3 lectures (Tuesdays 14:15-16:00) May 1: Gerth Stølting Brodal May 8: Kasper Green Larsen May 15: Peyman Afshani For each lecture one handin exercise deadline
Interpolation via Barycentric Coordinates
Interpolation via Barycentric Coordinates Pierre Alliez Inria Some material from D. Anisimov Outline Barycenter Convexity Barycentric coordinates For Simplices For point sets Inverse distance (Shepard)
Convex Hull of Planar H-Polyhedra
Convex Hull of Planar H-Polyhedra Axel Simon and Andy King Computing Laboratory University of Kent, CT2 7NF, UK {a.simon,a.m.king}@ukc.ac.uk May 6, 2008 Abstract Suppose A i, c i are planar (convex) H-polyhedra,
Numerical optimization
Numerical optimization Lecture 4 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 2 Longest Slowest Shortest Minimal Maximal
Oblivious and Adaptive Strategies for the Majority and Plurality Problems
Oblivious and Adaptive Strategies for the Majority and Plurality Problems Fan Chung 1, Ron Graham 1, Jia Mao 1, and Andrew Yao 2 1 Department of Computer Science and Engineering, University of California,
New techniques for trail bounds and application to differential trails in Keccak
New techniques for trail bounds and application to differential trails in Keccak Silvia Mella 1,2 Joan Daemen 1,3 Gilles Van Assche 1 1 STMicroelectronics 2 University of Milan 3 Radboud University Fast
arxiv: v2 [math.co] 27 Aug 2016
![arxiv: v2 [math.co] 27 Aug 2016](/thumbs/86/94454872.jpg "arxiv: v2 [math.co] 27 Aug 2016")
On the Erdős-Szekeres convex polygon problem Andrew Suk arxiv:1604.08657v2 [math.co] 27 Aug 26 Abstract Let ES(n) be the smallest integer such that any set of ES(n) points in the plane in general position
Knuth-Morris-Pratt Algorithm
Knuth-Morris-Pratt Algorithm Jayadev Misra June 5, 2017 The Knuth-Morris-Pratt string matching algorithm (KMP) locates all occurrences of a pattern string in a text string in linear time (in the combined
14.1 Finding frequent elements in stream
Chapter 14 Streaming Data Model 14.1 Finding frequent elements in stream A very useful statistics for many applications is to keep track of elements that occur more frequently. It can come in many flavours
Decidability of consistency of function and derivative information for a triangle and a convex quadrilateral
Decidability of consistency of function and derivative information for a triangle and a convex quadrilateral Abbas Edalat Department of Computing Imperial College London Abstract Given a triangle in the
Covering the Convex Quadrilaterals of Point Sets
Covering the Convex Quadrilaterals of Point Sets Toshinori Sakai, Jorge Urrutia 2 Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuyaku, Tokyo 5-8677, Japan 2 Instituto
Algorithms, Probability, and Computing Final Exam HS14
Institute of Theoretical Computer Science Thomas Holenstein, Emo Welzl, Peter Widmayer Algorithms, Probability, and Computing Final Exam HS14 Candidate First name:.........................................................
PRINCIPLE OF MATHEMATICAL INDUCTION
Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION 4.1 Overview Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of
Lecture 28 The Main Sources of Error
Lecture 28 The Main Sources of Error Truncation Error Truncation error is defined as the error caused directly by an approximation method For instance, all numerical integration methods are approximations
The Traveling Salesman Problem with Few Inner Points
The Traveling Salesman Problem with Few Inner Points Vladimir G. Deĭneko 1,, Michael Hoffmann 2, Yoshio Okamoto 2,, and Gerhard J. Woeginger 3, 1 Warwick Business School, The University of Warwick, Conventry
0 1 d 010. h 0111 i g 01100
CMSC 5:Fall 07 Dave Mount Solutions to Homework : Greedy Algorithms Solution : The code tree is shown in Fig.. a 0000 c 0000 b 00 f 000 d 00 g 000 h 0 i 00 e Prob / / /8 /6 / Depth 5 Figure : Solution
Fall 2017 November 10, Written Homework 5
CS1800 Discrete Structures Profs. Aslam, Gold, & Pavlu Fall 2017 November 10, 2017 Assigned: Mon Nov 13 2017 Due: Wed Nov 29 2017 Instructions: Written Homework 5 The assignment has to be uploaded to blackboard
4 a b 1 1 c 1 d 3 e 2 f g 6 h i j k 7 l m n o 3 p q 5 r 2 s 4 t 3 3 u v 2
Round Solutions Year 25 Academic Year 201 201 1//25. In the hexagonal grid shown, fill in each space with a number. After the grid is completely filled in, the number in each space must be equal to the
11 Heavy Hitters Streaming Majority
11 Heavy Hitters A core mining problem is to find items that occur more than one would expect. These may be called outliers, anomalies, or other terms. Statistical models can be layered on top of or underneath
Algorithms, Design and Analysis. Order of growth. Table 2.1. Big-oh. Asymptotic growth rate. Types of formulas for basic operation count
Types of formulas for basic operation count Exact formula e.g., C(n) = n(n-1)/2 Algorithms, Design and Analysis Big-Oh analysis, Brute Force, Divide and conquer intro Formula indicating order of growth
The Multi-Agent Rendezvous Problem - The Asynchronous Case
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas WeB03.3 The Multi-Agent Rendezvous Problem - The Asynchronous Case J. Lin and A.S. Morse Yale University
Minimum-Dilation Tour (and Path) is NP-hard
Minimum-Dilation Tour (and Path) is NP-hard Panos Giannopoulos Christian Knauer Dániel Marx Abstract We prove that computing a minimum-dilation (Euclidean) Hamilton circuit or path on a given set of points
The Triangle Algorithm: A Geometric Approach to Systems of Linear Equations
: A Geometric Approach to Systems of Linear Equations Thomas 1 Bahman Kalantari 2 1 Baylor University 2 Rutgers University July 19, 2013 Rough Outline Quick Refresher of Basics Convex Hull Problem Solving
CS 372: Computational Geometry Lecture 14 Geometric Approximation Algorithms
CS 372: Computational Geometry Lecture 14 Geometric Approximation Algorithms Antoine Vigneron King Abdullah University of Science and Technology December 5, 2012 Antoine Vigneron (KAUST) CS 372 Lecture
ILP Formulations for the Lazy Bureaucrat Problem
the the PSL, Université Paris-Dauphine, 75775 Paris Cedex 16, France, CNRS, LAMSADE UMR 7243 Department of Statistics and Operations Research, University of Vienna, Vienna, Austria EURO 2015, 12-15 July,
Numerical optimization. Numerical optimization. Longest Shortest where Maximal Minimal. Fastest. Largest. Optimization problems
1 Numerical optimization Alexander & Michael Bronstein, 2006-2009 Michael Bronstein, 2010 tosca.cs.technion.ac.il/book Numerical optimization 048921 Advanced topics in vision Processing and Analysis of
Theory of Computation
Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 This course aims to cover... the development of computability theory using an extremely simple
POINTS, LINES, DISTANCES
POINTS, LINES, DISTANCES NIKOS APOSTOLAKIS Examples/Exercises: (1) Find the equation of the line that passes through (4, 5), (4, ) () Find the equation of the line that passes through the points (1, ),
Convex Hull-Based Metric Refinements for Topological Spatial Relations
ABSTRACT Convex Hull-Based Metric Refinements for Topological Spatial Relations Fuyu (Frank) Xu School of Computing and Information Science University of Maine Orono, ME 04469-5711, USA fuyu.xu@maine.edu
1 Approximate Quantiles and Summaries
CS 598CSC: Algorithms for Big Data Lecture date: Sept 25, 2014 Instructor: Chandra Chekuri Scribe: Chandra Chekuri Suppose we have a stream a 1, a 2,..., a n of objects from an ordered universe. For simplicity
AFFINE COMBINATIONS, BARYCENTRIC COORDINATES, AND CONVEX COMBINATIONS
On-Line Geometric Modeling Notes AFFINE COMBINATIONS, BARYCENTRIC COORDINATES, AND CONVEX COMBINATIONS Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University
CSE 421 Greedy Algorithms / Interval Scheduling
CSE 421 Greedy Algorithms / Interval Scheduling Yin Tat Lee 1 Interval Scheduling Job j starts at s(j) and finishes at f(j). Two jobs compatible if they don t overlap. Goal: find maximum subset of mutually
The Ellipsoid Algorithm
The Ellipsoid Algorithm John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA 9 February 2018 Mitchell The Ellipsoid Algorithm 1 / 28 Introduction Outline 1 Introduction 2 Assumptions
The τ-skyline for Uncertain Data
CCCG 2014, Halifax, Nova Scotia, August 11 13, 2014 The τ-skyline for Uncertain Data Haitao Wang Wuzhou Zhang Abstract In this paper, we introduce the notion of τ-skyline as an alternative representation
Final Exam, Machine Learning, Spring 2009
Name: Andrew ID: Final Exam, 10701 Machine Learning, Spring 2009 - The exam is open-book, open-notes, no electronics other than calculators. - The maximum possible score on this exam is 100. You have 3
Poisson line processes. C. Lantuéjoul MinesParisTech
Poisson line processes C. Lantuéjoul MinesParisTech christian.lantuejoul@mines-paristech.fr Bertrand paradox A problem of geometrical probability A line is thrown at random on a circle. What is the probability
libsupermesh version 1.0.1
libsupermesh version 1.0.1 James R. Maddison School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, UK Iakovos Panourgias EPCC, University of Edinburgh, UK, Patrick
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Catie Baker Spring 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Catie Baker Spring 2015 Today Registration should be done. Homework 1 due 11:59pm next Wednesday, April 8 th. Review math
This chapter reviews some basic geometric facts that we will need during the course.
Chapter 1 Some Basic Geometry This chapter reviews some basic geometric facts that we will need during the course. 1.1 Affine Geometry We will assume that you are familiar with the basic notions of linear
1 Sequences and Summation
1 Sequences and Summation A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. For example, a m, a m+1,...,
Linear Algebra. Mark Dean. Lecture Notes for Fall 2014 PhD Class - Brown University
Linear Algebra Mark Dean Lecture Notes for Fall 2014 PhD Class - Brown University 1 Lecture 1 1 1.1 Definition of Linear Spaces In this section we define the concept of a linear (or vector) space. The
Design and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 4: Divide and Conquer (I) Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Divide and Conquer ( DQ ) First paradigm or framework DQ(S)
Bernstein polynomials of degree N are defined by
SEC. 5.5 BÉZIER CURVES 309 5.5 Bézier Curves Pierre Bézier at Renault and Paul de Casteljau at Citroën independently developed the Bézier curve for CAD/CAM operations, in the 1970s. These parametrically
Construction of `Wachspress type' rational basis functions over rectangles
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, No. 1, February 2000, pp. 69±77. # Printed in India Construction of `Wachspress type' rational basis functions over rectangles P L POWAR and S S RANA Department
Math 530 Lecture Notes. Xi Chen
Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary
Convergence of Fixed-Point Iterations
Convergence of Fixed-Point Iterations Instructor: Wotao Yin (UCLA Math) July 2016 1 / 30 Why study fixed-point iterations? Abstract many existing algorithms in optimization, numerical linear algebra, and
34 th United States of America Mathematical Olympiad
34 th United States of America Mathematical Olympiad 1. Determine all composite positive integers n for which it is possible to arrange all divisors of n that are greater than 1 in a circle so that no
Algorithms and Complexity Theory. Chapter 8: Introduction to Complexity. Computer Science - Durban - September 2005
Algorithms and Complexity Theory Chapter 8: Introduction to Complexity Jules-R Tapamo Computer Science - Durban - September 2005 Contents 1 Introduction 2 1.1 Dynamic programming...................................
Linear Classification: Linear Programming
Linear Classification: Linear Programming Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong 1 / 21 Y Tao Linear Classification: Linear Programming Recall the definition
On Empty Convex Polygons in a Planar Point Set
On Empty Convex Polygons in a Planar Point Set Rom Pinchasi Radoš Radoičić Micha Sharir March 21, 2005 Abstract Let P be a set of n points in general position in the plane. Let X k (P) denote the number
Mathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: Exercise Answers
Mathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: 978-1-118-71219-1 Updated /17/15 Exercise Answers Chapter 1 1. Four right-handed systems: ( i, j, k), ( i, j,
CS360 Homework 12 Solution
CS360 Homework 12 Solution Constraint Satisfaction 1) Consider the following constraint satisfaction problem with variables x, y and z, each with domain {1, 2, 3}, and constraints C 1 and C 2, defined
Math Requirements for applicants by Innopolis University
Math Requirements for applicants by Innopolis University Contents 1: Algebra... 2 1.1 Numbers, roots and exponents... 2 1.2 Basics of trigonometry... 2 1.3 Logarithms... 2 1.4 Transformations of expressions...
The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative,
Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1] Objectives In this section you will learn
Gordon solutions. n=1 1. p n
Gordon solutions 1. A positive integer is called a palindrome if its base-10 expansion is unchanged when it is reversed. For example, 121 and 7447 are palindromes. Show that if we denote by p n the nth
On the existence of a convex point subset containing one triangle in the plane
Discrete Mathematics 305 (2005) 201 218 www.elsevier.com/locate/disc On the existence of a convex point subset containing one triangle in the plane Kiyoshi Hosono Department of Mathematics, Tokai University,
Lecture 2: Divide and conquer and Dynamic programming
Chapter 2 Lecture 2: Divide and conquer and Dynamic programming 2.1 Divide and Conquer Idea: - divide the problem into subproblems in linear time - solve subproblems recursively - combine the results in
2. Project management
2. Project management In what follows, we consider production processes where only a single item of a specific product is produced in the planning horizon In this case specific instruments for planning
Making Fast Buffer Insertion Even Faster via Approximation Techniques
Making Fast Buffer Insertion Even Faster via Approximation Techniques Zhuo Li, C. N. Sze, Jiang Hu and Weiping Shi Department of Electrical Engineering Texas A&M University Charles J. Alpert IBM Austin
Neural Networks. Xiaojin Zhu Computer Sciences Department University of Wisconsin, Madison. slide 1
Neural Networks Xiaoin Zhu erryzhu@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison slide 1 Terminator 2 (1991) JOHN: Can you learn? So you can be... you know. More human. Not
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
On empty convex polygons in a planar point set
Journal of Combinatorial Theory, Series A 113 (006 385 419 www.elsevier.com/locate/jcta On empty convex polygons in a planar point set Rom Pinchasi a,1, Radoš Radoičić a, Micha Sharir b,c a Department
Turing Machine Variants
CS311 Computational Structures Turing Machine Variants Lecture 12 Andrew Black Andrew Tolmach 1 The Church-Turing Thesis The problems that can be decided by an algorithm are exactly those that can be decided
Axiomatic Semantics: Verification Conditions. Review of Soundness and Completeness of Axiomatic Semantics. Announcements
Axiomatic Semantics: Verification Conditions Meeting 12, CSCI 5535, Spring 2009 Announcements Homework 4 is due tonight Wed forum: papers on automated testing using symbolic execution 2 Questions? Review
Controlled Perturbation of Sets of Line Segments in IR 2 with Smart Processing Order
Controlled Perturbation of Sets of Line Segments in IR 2 with Smart Processing Order Eli Packer, IBM Research Center, Yorktown, NY, USA Abstract Controlled Perturbation is a framework for perturbing geometric
Brief Introduction of Machine Learning Techniques for Content Analysis
1 Brief Introduction of Machine Learning Techniques for Content Analysis Wei-Ta Chu 2008/11/20 Outline 2 Overview Gaussian Mixture Model (GMM) Hidden Markov Model (HMM) Support Vector Machine (SVM) Overview
CONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS
CONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS Igor V. Konnov Department of Applied Mathematics, Kazan University Kazan 420008, Russia Preprint, March 2002 ISBN 951-42-6687-0 AMS classification:
SDPTools: A High Precision SDP Solver in Maple
MM Research Preprints, 66 73 KLMM, AMSS, Academia Sinica Vol. 28, December 2009 SDPTools: A High Precision SDP Solver in Maple Feng Guo Key Laboratory of Mathematics Mechanization Institute of Systems
Lecture «Robot Dynamics» : Kinematics 3
Lecture «Robot Dynamics» : Kinematics 3 151-0851-00 V lecture: CAB G11 Tuesday 10:15-12:00, every week exercise: HG G1 Wednesday 8:15-10:00, according to schedule (about every 2nd week) office hour: LEE
Equivalence relations
Equivalence relations R A A is an equivalence relation if R is 1. reflexive (a, a) R 2. symmetric, and (a, b) R (b, a) R 3. transitive. (a, b), (b, c) R (a, c) R Example: Let S be a relation on people
Improving the Accuracy of Primality Tests by Enhancing the Miller-Rabin Theorem
Improving the Accuracy of Primality Tests by Enhancing the Miller-Rabin Theorem Shyam Narayanan Fourth Annual MIT-PRIMES Conference Mentor: David Corwin Project Proposed by Stefan Wehmeier and Ben Hinkle
Downloaded from
CLASS VI Mathematics (Knowing our Numbers) Worksheet-01 Choose correct option in questions 1 to 7. 1. Which is greatest? a. 234 b. 543 c. 657 56 2. Which is smallest? a. 4567 b. 3456 c. 2345 d. 1234 3.
Reading and Writing. Mathematical Proofs. Slides by Arthur van Goetham
Reading and Writing Mathematical Proofs Slides by Arthur van Goetham What is a proof? Why explanations are not proofs What is a proof? A method for establishing truth What establishes truth depends on
Pairwise Away Steps for the Frank-Wolfe Algorithm
Pairwise Away Steps for the Frank-Wolfe Algorithm Héctor Allende Department of Informatics Universidad Federico Santa María, Chile hallende@inf.utfsm.cl Ricardo Ñanculef Department of Informatics Universidad
Lecture 2 September 4, 2014
CS 224: Advanced Algorithms Fall 2014 Prof. Jelani Nelson Lecture 2 September 4, 2014 Scribe: David Liu 1 Overview In the last lecture we introduced the word RAM model and covered veb trees to solve the
CS 372: Computational Geometry Lecture 4 Lower Bounds for Computational Geometry Problems
CS 372: Computational Geometry Lecture 4 Lower Bounds for Computational Geometry Problems Antoine Vigneron King Abdullah University of Science and Technology September 20, 2012 Antoine Vigneron (KAUST)
586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
Scheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev, m.uetz}@ke.unimaas.nl
Лекция 26. Оценки функциональной сложности двумерной задачи о метрической близости для метрики городских кварталов.
Лекция 26. Оценки функциональной сложности двумерной задачи о метрической близости для метрики городских кварталов. 1 On Functional Complexity of Two-dimensional Manhattan Metrics Closeness Problem We
Linear Classification: Linear Programming
Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong Recall the definition of linear classification. Definition 1. Let R d denote the d-dimensional space where the domain
- 1 - Items related to expected use of technology appear in bold italics.
- 1 - Items related to expected use of technology appear in bold italics. Operating with Geometric and Cartesian Vectors Determining Intersections of Lines and Planes in Three- Space Similar content as
the subset partial order Paul Pritchard Technical Report CIT School of Computing and Information Technology
A simple sub-quadratic algorithm for computing the subset partial order Paul Pritchard P.Pritchard@cit.gu.edu.au Technical Report CIT-95-04 School of Computing and Information Technology Grith University
Region Description for Recognition
Region Description for Recognition For object recognition, descriptions of regions in an image have to be compared with descriptions of regions of meaningful objects (models). The general problem of object
arxiv:cs/ v1 [cs.cg] 7 Feb 2006
![arxiv:cs/ v1 [cs.cg] 7 Feb 2006](/thumbs/90/102634153.jpg "arxiv:cs/ v1 [cs.cg] 7 Feb 2006")
Approximate Weighted Farthest Neighbors and Minimum Dilation Stars John Augustine, David Eppstein, and Kevin A. Wortman Computer Science Department University of California, Irvine Irvine, CA 92697, USA
Multiprocessor Scheduling I: Partitioned Scheduling. LS 12, TU Dortmund
Multiprocessor Scheduling I: Partitioned Scheduling Prof. Dr. Jian-Jia Chen LS 12, TU Dortmund 22/23, June, 2015 Prof. Dr. Jian-Jia Chen (LS 12, TU Dortmund) 1 / 47 Outline Introduction to Multiprocessor
Integer Programming ISE 418. Lecture 13. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 13 Dr. Ted Ralphs ISE 418 Lecture 13 1 Reading for This Lecture Nemhauser and Wolsey Sections II.1.1-II.1.3, II.1.6 Wolsey Chapter 8 CCZ Chapters 5 and 6 Valid Inequalities
Geometric Spanners With Small Chromatic Number
Geometric Spanners With Small Chromatic Number Prosenjit Bose 1, Paz Carmi 1, Mathieu Couture 1, Anil Maheshwari 1, Michiel Smid 1, and Norbert Zeh 2 1 School of Computer Science, Carleton University 2
1. Introduction Bottom-Up-Heapsort is a variant of the classical Heapsort algorithm due to Williams ([Wi64]) and Floyd ([F64]) and was rst presented i
![1. Introduction Bottom-Up-Heapsort is a variant of the classical Heapsort algorithm due to Williams ([Wi64]) and Floyd ([F64]) and was rst presented i](/thumbs/93/112938972.jpg "1. Introduction Bottom-Up-Heapsort is a variant of the classical Heapsort algorithm due to Williams ([Wi64]) and Floyd ([F64]) and was rst presented i")
A Tight Lower Bound for the Worst Case of Bottom-Up-Heapsort 1 by Rudolf Fleischer 2 Keywords : heapsort, bottom-up-heapsort, tight lower bound ABSTRACT Bottom-Up-Heapsort is a variant of Heapsort. Its
A Humble Introduction to DIJKSTRA S A A DISCIPLINE OF PROGRAMMING
A Humble Introduction to DIJKSTRA S A A DISCIPLINE OF PROGRAMMING Do-Hyung Kim School of Computer Science and Engineering Sungshin Women s s University CONTENTS Bibliographic Information and Organization
sset(x, Y ) n = sset(x [n], Y ).
![sset(x, Y ) n = sset(x [n], Y ).](/thumbs/86/93370543.jpg "sset(x, Y ) n = sset(x [n], Y ).")
1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,
TDT4173 Machine Learning
TDT4173 Machine Learning Lecture 3 Bagging & Boosting + SVMs Norwegian University of Science and Technology Helge Langseth IT-VEST 310 helgel@idi.ntnu.no 1 TDT4173 Machine Learning Outline 1 Ensemble-methods
Lecture 8 DEGENERACY AND PERTURBATION
1. A Perturbation Framework Lecture 8 Page 1 A foolish consistency is the hobgoblin of little minds Lecture 8 DEGENERACY AND PERTURBATION Ralph Waldo Emerson The central concern of theoretical algorithms
Spectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
On the minimum gap between sums of square roots of small integers
On the minimum gap between sums of square roots of small integers Qi Cheng a, Yu-Hsin Li a a School of Computer Science The University of Oklahoma Norman, OK 73019, USA. Email: {qcheng,yli}@cs.ou.edu.
Minimalist robots and motion coordination Deployment and Territory Partitioning for Gossiping Robots
Minimalist robots and motion coordination Deployment and Territory Partitioning for Gossiping Robots What kind of systems? Groups of systems with control, sensing, communication and computing Francesco
l 2 b 2 A ( φ) θ/2 b 1 j p k
Computing the Angularity Tolerance Mark de Berg y Henk Meijer z Mark Overmars y Gordon Wilfong x July 30, 1996 Abstract In computational metrology one needs to compute whether an object satises specications
Winter Lecture 10. Convexity and Concavity
Andrew McLennan February 9, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 10 Convexity and Concavity I. Introduction A. We now consider convexity, concavity, and the general
CSE 417. Chapter 4: Greedy Algorithms. Many Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
CSE 417 Chapter 4: Greedy Algorithms Many Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Greed is good. Greed is right. Greed works. Greed clarifies, cuts through,
Only Intervals Preserve the Invertibility of Arithmetic Operations
Only Intervals Preserve the Invertibility of Arithmetic Operations Olga Kosheleva 1 and Vladik Kreinovich 2 1 Department of Electrical and Computer Engineering 2 Department of Computer Science University
2 S. FELSNER, W. T. TROTTER S. S. Kislitsyn [9] made the following conjecture, which remains of the most intriguing problems in the combinatorial theo
![2 S. FELSNER, W. T. TROTTER S. S. Kislitsyn [9] made the following conjecture, which remains of the most intriguing problems in the combinatorial theo](/thumbs/80/82226669.jpg "2 S. FELSNER, W. T. TROTTER S. S. Kislitsyn [9] made the following conjecture, which remains of the most intriguing problems in the combinatorial theo")
COLLOQUIA MATHEMATICA SOCIETATIS J ANOS BOLYA I 64. COMBINATORICS, KESZTHELY (HUNGARY), 1993 Balancing Pairs in Partially Ordered Sets S. FELSNER 1 and W. T. TROTTER Dedicated to Paul Erdos on his eightieth