Online Chain Partitioning of Upgrowing Interval Orders Bartlomiej - PowerPoint PPT Presentation

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Online Chain Partitioning of Upgrowing Interval Orders Bartlomiej Bosek Algorithmics Research Group Jagiellonian University, Cracow, Poland Chambery-Krakow-Lyon Workshop on Computational Logic and Applications, 21 June 2005 Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Poset Definition Example We say that P = ( X , � ) is a partially ordered set (poset) if for each x , y , z from X x � x (reflexive) x � y , y � x imply x = y (antisymmetric) x � y , y � z imply x � z (transitive) Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Chain and Antichain Definition Example The subposet C is a chain if for each x , y from C x � y or y � x . (complete) The subposet A is an antichain if for each x , y from A x � y and y � x . (incomparability) Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Chain and Antichain Definition Example The subposet C is a chain if for each x , y from C x � y or y � x . (complete) The subposet A is an antichain if for each x , y from A x � y and y � x . (incomparability) Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Width and Chain Partition Definition Example The width of poset P = ( X , � ) is a maximal size of antichain in P . The chain partition of poset P = ( X , � ) is a family of chains C 1 , . . . , C k so that C 1 ∪ . . . ∪ C k = X . Question Is there a connection between the minimal chain covering of and the width of he poset? Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Width and Chain Partition Definition Example The width of poset P = ( X , � ) is a maximal size of antichain in P . The chain partition of poset P = ( X , � ) is a family of chains C 1 , . . . , C k so that C 1 ∪ . . . ∪ C k = X . Question Is there a connection between the minimal chain covering of and the width of he poset? Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Dilworth’s Theorem Example Theorem (Dilworth) Minimal chain partition of poset is equal to poset’s width. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Online Chain Partitioning of the Posets Online chain partitioning of the posets can be viewed as two-person game . We call the players Algorithm and Spoiler. During each round: Spoiler introduces a new point with a comparability status of previously presented points to the new one. Algorithm covers this new point by some chain . Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c 1 Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c 1 Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . d Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c 1 Spoiler puts b . Algorithm covers b by 2. Spoiler puts c . d 1 Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c 1 Spoiler puts b . Algorithm covers b by 2. e Spoiler puts c . d 1 Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example Spoiler puts a . Algorithm covers a by 1. c 1 Spoiler puts b . Algorithm covers b by 2. e 3 Spoiler puts c . d 1 Algorithm covers c by 1. Spoiler puts d . Algorithm covers d by 1. a 1 b 2 Spoiler puts e . Algorithm covers e by 3. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Example c 1 c 1 e 3 e 1 d 1 d 2 a 1 b 2 a 1 b 2 result of the game optimal off-line solution Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Value of Online Chain Partitioning Game Definition The value of the chain partitioning game for width w posets is the largest integer CP ( w ) so that there is a strategy of presenting points that forces any algorithm to use CP ( w ) chains. Note that we may as well define CP ( w ) as the least integer so that there is an algorithm that never uses more chains. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Theorem (Kierstead, Szemeredi) � w + 1 � CP ( w ) � 5 w − 1 � 2 4 This is already a complicated result, and no progress has been made for the last 20 years. Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders

Introduction Online Chain Partitioning of the Posets Upgrowing Version Interval Orders Upgrowing version We suppose that Spoiler introduces points which are always maximal at the moment of their arrival. Online posets with this property are called upgrowing posets . We define also the value of this game for width w posets and we denote it CPU ( w ) . Bartlomiej Bosek Online Chain Partitioning of Upgrowing Interval Orders