Some Results on the Online Partitioning of Permutations Benjamin - - PowerPoint PPT Presentation

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion

Some Results on the Online Partitioning of Permutations

Benjamin Leroy-Beaulieu1 Marc Demange2

1IMA-ROSO

Ecole Polytechnique Fédérale de Lausanne

2Départment SID

ESSEC, Paris

Aussois 2006

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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

1 2 3 4 5 6 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

1 2 3 4 5 6 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

1 2 3 4 5 6 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

1 2 3 4 5 6 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

1 2 3 4 5 6 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

1 2 3 4 5 6 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 Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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 Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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 P4.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of Proposition 1 (Illustration)

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of Proposition 1

Suppose x ≥ 0 and y ≤ 0 Element colored with color k → Call this element ek. ⇒ ∃ek−1 s.t. ek1 was presented before ek and ek−1 and ek form a decreasing sequence. Similarily ∃ei, 1 ≤ i ≤ k − 1 where ei is of color i and xi ≤ xi+1; yi ≥ yi+1. Then, {ei}, i ∈ {1, ..., k} constitute a decreasing sequence

• f size k

Similar proof for the case x ≤ 0 and y ≥ 0

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Proof of proposition 2

1 2 3 4 5 6 1 2 3 4 5 6

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Indigestion

L1 L2 1 1 2 2 3 1 3 1 2 4 u = (1,1)

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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) 2

elements have been presented.

⇒ Performance of First-Fit is not better than O(√n).

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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) 2

elements have been presented.

⇒ Performance of First-Fit is not better than O(√n).

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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) 2

elements have been presented.

⇒ Performance of First-Fit is not better than O(√n).

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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) 2

elements have been presented.

⇒ Performance of First-Fit is not better than O(√n).

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

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

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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 Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

OPP-IS

Online Permutation Partitioner - Integer Sequences Works on any online model Uses integer intervals as colors Uses renaming of colors.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

The Coloring System (1)

1 2 3 4 5 6 1 2 3 4 5 6

Colors are intervals of integers.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

The Coloring System (1)

1 2 3 4 5 6 1 2 3 4 5 6

Colors are intervals of integers.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

The Coloring System (1)

1 2 3 4 5 6 1 2 3 4 5 6

[2,3,4] Colors are intervals of integers.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

The Coloring System (1)

1 2 3 4 5 6 1 2 3 4 5 6

234 Colors are intervals of integers.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

The Coloring System (2)

Definitions Let C(e) be the color of an element e. C(e) is an interval. Cl(e) is the left-most integer of C(e). Cr(e) is the right-most integer of C(e). Example For a particular e, we have C(e) = [2, 3, 4] Cl(e) = 2 Cr(e) = 4

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Colors might be renamed

10 a b c d f e i (ordering) v (value) 5 15 v 12 12 1 1 2 2

Renaming the colors is not a change of a previous decision

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Colors might be renamed

10 a b c d f e i (ordering) v (value) 5 15 v 23 23 123 12 12 123 2

Renaming the colors is not a change of a previous decision

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Analysis of OPP-IS

Lemma 1 For every element e with color C, there exists a decreasing sequence with length χ(π′) + Cl − Cr, where e is exactly at position Cl. Proof This is obviously true when e is first assigned a color. Everytime χ(π′) increases by one, Cr also increases by one by the renaming of the colors. So the sequence which was used to assign a color to e will always verify this property.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Analysis of OPP-IS

Lemma 2 The color that OPP-IS assigns to any element e never causes any conflict in the coloration.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Analysis of OPP-IS

Proof of Lemma 2 (Sketch) Let e be a newly introduced element, colored with color C ∃ a sequence s of maximal length χ(π′) + Cl − Cr where e has position Cl. Suppose ∃ an element e∗ with the same color C Suppose they form a decreasing sequence and e < e∗. By lemma 1, ∃ a sequence s∗ where e∗ has position Cl. Concatenate the "left part" of s∗ and the "right part" of s ⇒ sequence containing e of size χ(π′) + Cl − Cr + 1, which constitutes a contradiction. A similar argument holds if e is to the left of e∗.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Analysis of OPP-IS

Lemma 3 The number of colors used by OPP-IS is no greater than χ(χ+1)

2

. Proof The colors used by the algorithm are all subintervals of [1, . . . , χ]. There are exactly χ(χ + 1)/2 such subintervals.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Upper bound on the performance ratio.

Theorem It is possible to guarantee a performance ratio of χ+1

2

for the problem of online partitioning of a permutation into increasing sequences. Proof Use OPP-IS, and consider the lemmae 2 and 3.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Tightness of OPP-IS Analysis

1234 123 12 1 234 23 2 34 3 4

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Lower Bound for a 3-Partitionable Permutation

1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion Performance of the First-Fit algorithm Upper and Lower Bounds in the General Model

Back to OPP-IS

Theorem For χ ≤ 3, OPP-IS is optimal. Proof For χ = 1, it is evident. For χ = 2, it follows immediately from proposition 2. For χ = 3 it follows from the previous slide.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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 Continuous Case The Discrete Case Relaxations of the problem

Monotone Partitioning a Permutation over a Continuous Interval

The Model Elements are delivered one by one on the Lattice Representation. The goal is to partition them in as few sequences as possible. Each sequence can be increasing or decreasing. Proposition Partitioning a permutation into monotone sequences on a continous interval can be done with n

2 sequences, and this

number is optimal.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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 Continuous Case The Discrete Case Relaxations of the problem

Monotone Partitioning a Permutation over a Discrete Interval

The Model Elements are delivered one by one. Goal: partition them in as few sequences as possible. Theorem (Di Stefano, Krause, Luebbecke, Zimmermann, 2005) If a permutation can be partitioned into k monotone sequences, with k ≥ 3, no algorithm can guarantee better than O(log2 n) colors for this permutation. Improvement We prove here that this result holds even if k = 2

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof (Monotone Part. Cont. Interval)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Outline

1

Preliminaries

The Lattice Representation

2

Isotone Partitioning 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

3

Monotone Partitioning 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 Continuous Case The Discrete Case Relaxations of the problem

"Split" Permutations

A "Split" Permutation is a permutation which can be partitioned into one increasing sequence and one deceasing sequence. We consider here that the permutation is presented w.r.t. the direction − → u = (1, 0). It is enough to have a delay of 1 in order to partition it exactly. Proof: will be summed-up orally during the talk.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

"Split" Permutations

A "Split" Permutation is a permutation which can be partitioned into one increasing sequence and one deceasing sequence. We consider here that the permutation is presented w.r.t. the direction − → u = (1, 0). It is enough to have a delay of 1 in order to partition it exactly. Proof: will be summed-up orally during the talk.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

"Split" Permutations

A "Split" Permutation is a permutation which can be partitioned into one increasing sequence and one deceasing sequence. We consider here that the permutation is presented w.r.t. the direction − → u = (1, 0). It is enough to have a delay of 1 in order to partition it exactly. Proof: will be summed-up orally during the talk.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

"Split" Permutations

A "Split" Permutation is a permutation which can be partitioned into one increasing sequence and one deceasing sequence. We consider here that the permutation is presented w.r.t. the direction − → u = (1, 0). It is enough to have a delay of 1 in order to partition it exactly. Proof: will be summed-up orally during the talk.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

k-Partitionable Permutations with k ≥ 3.

Proposition Even if an algorithm is allowed to assign an element to a color-class after a delay of n − 5, it is not possible to optimally partition a 3-partitionable permutation.

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion The Continuous Case The Discrete Case Relaxations of the problem

Proof

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion

Conclusion

Results An upper and a lower bound for the problem of isotone partitioning a permutation online in the general model. Analysed monotone partitioning (continuous and part of discrete). Ongoing work Make the upper and lower bounds match for values higher than 3. Online split-coloration of permutations. For now... Thank you for your attention

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations

Preliminaries Isotone Partitioning Monotone Partitioning Conclusion

Conclusion

Results An upper and a lower bound for the problem of isotone partitioning a permutation online in the general model. Analysed monotone partitioning (continuous and part of discrete). Ongoing work Make the upper and lower bounds match for values higher than 3. Online split-coloration of permutations. For now... Thank you for your attention

• B. Leroy-Beaulieu, M. Demange

Online Partitioning of Permutations