Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Some Results on the Online Partitioning of Permutations Benjamin Leroy-Beaulieu 1 Marc Demange 2 1 IMA-ROSO Ecole Polytechnique Fédérale de Lausanne 2 Départment SID ESSEC, Paris Aussois 2006 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Outline Preliminaries 1 The Lattice Representation Isotone Partitioning 2 Performance of the First-Fit algorithm Online Partitoning w.r.t. a Given Direction Upper and Lower Bounds in the General Model Upper Bound Lower Bound Monotone Partitioning 3 The Continuous Case The Discrete Case Relaxations of the problem B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Lattice Representation for a Permutation 6 5 4 3 2 1 0 0 1 2 3 4 5 6 π = [ 4 , 2 , 6 , 3 , 5 , 1 ] y -axis ⇔ value x -axis ⇔ position B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Outline Preliminaries 1 The Lattice Representation Isotone Partitioning 2 Performance of the First-Fit algorithm Online Partitoning w.r.t. a Given Direction Upper and Lower Bounds in the General Model Upper Bound Lower Bound Monotone Partitioning 3 The Continuous Case The Discrete Case Relaxations of the problem B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Partitioning π w.r.t any Direction − → u = ( x , y ) Proposition 1 If − → u = ( x , y ) such that x · y ≤ 0 and ( x , y ) � = ( 0 , 0 ) , then First-Fit partitions π exactly Proposition 2 If x · y > 0, then no algorithm can guarantee an exact partitioning for any arbitrary permutation π , even if the corresponding permutation graph G is a P 4 . B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of Proposition 1 (Illustration) 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of Proposition 1 Suppose x ≥ 0 and y ≤ 0 Element colored with color k → Call this element e k . ⇒ ∃ e k − 1 s.t. e k 1 was presented before e k and e k − 1 and e k form a decreasing sequence. Similarily ∃ e i , 1 ≤ i ≤ k − 1 where e i is of color i and x i ≤ x i + 1 ; y i ≥ y i + 1 . Then, { e i } , i ∈ { 1 , ..., k } constitute a decreasing sequence of size k Similar proof for the case x ≤ 0 and y ≥ 0 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Proof of proposition 2 6 5 4 3 2 1 0 0 1 2 3 4 5 6 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Indigestion L1 1 L2 2 3 4 1 2 1 2 3 u = (1,1) 1 B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Indigestion: Analysis At each step k , add k new elements At each step k , colors 1 , ..., k are used. Last step K : K colors have been used. K ( K + 1 ) elements have been presented. 2 ⇒ Performance of First-Fit is not better than O ( √ n ) . B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Indigestion: Analysis At each step k , add k new elements At each step k , colors 1 , ..., k are used. Last step K : K colors have been used. K ( K + 1 ) elements have been presented. 2 ⇒ Performance of First-Fit is not better than O ( √ n ) . B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Indigestion: Analysis At each step k , add k new elements At each step k , colors 1 , ..., k are used. Last step K : K colors have been used. K ( K + 1 ) elements have been presented. 2 ⇒ Performance of First-Fit is not better than O ( √ n ) . B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion Indigestion: Analysis At each step k , add k new elements At each step k , colors 1 , ..., k are used. Last step K : K colors have been used. K ( K + 1 ) elements have been presented. 2 ⇒ Performance of First-Fit is not better than O ( √ n ) . B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
Preliminaries Isotone Partitioning Performance of the First-Fit algorithm Monotone Partitioning Upper and Lower Bounds in the General Model Conclusion χ -binding Theorem Kirstead, Penrice, Trotter , 1994 The problem of coloring cocomparability graphs is χ -bounded. Permutation graphs are cocomparability graphs. Partitioning a permutation is easier than coloring a permutation graph. ⇒ We want to find a χ -binding function for online permutation partitioning. B. Leroy-Beaulieu, M. Demange Online Partitioning of Permutations
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