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Online Linear Discrepancy Mitchel T. Keller , Noah Streib, and William T. Trotter School of Mathematics Georgia Institute of Technology SIAM DM10 16 June 2010 Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 1 / 16 Outline

SLIDE 1

Online Linear Discrepancy

Mitchel T. Keller, Noah Streib, and William T. Trotter

School of Mathematics Georgia Institute of Technology

SIAM DM10 16 June 2010

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 1 / 16

SLIDE 2

Outline

1

Linear Discrepancy

2

Online Algorithms

3

Online Linear Discrepancy Lower Bounds Upper Bounds Representations

4

Open Problems

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 2 / 16

SLIDE 3

Linear Discrepancy Defined

Definition

The linear discrepancy of a linear extension L of a poset P is ld(P, L) = max

xPy |hL(y) − hL(x)|.

Definition (Tanenbaum, Trenk, and Fishburn 2001)

The linear discrepancy of a poset P is ld(P) = min

L∈E(P) ld(P, L) if P is not

a chain. If P is a chain, ld(P) = 0.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 3 / 16

SLIDE 4

Facts About Linear Discrepancy

Theorem (Fishburn, Tanenbaum, and Trenk 2001)

If P is a poset with incomparability graph G, then ld(P) = bw(G).

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 4 / 16

SLIDE 5

Facts About Linear Discrepancy

Theorem (Fishburn, Tanenbaum, and Trenk 2001)

If P is a poset with incomparability graph G, then ld(P) = bw(G).

Theorem (Kloks, Kratsch, Müller 1999)

ld(P, L) ≤ 3 ld(P)

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 4 / 16

SLIDE 6

Facts About Linear Discrepancy

Theorem (Fishburn, Tanenbaum, and Trenk 2001)

If P is a poset with incomparability graph G, then ld(P) = bw(G).

Theorem (Kloks, Kratsch, Müller 1999)

ld(P, L) ≤ 3 ld(P)

Lemma (TTF 2001, Rautenbach 2005)

(1) Linear discrepancy is monotonic. (2) ld(P) ≥ width(P) − 1 (3) ∆(P)/2 ≤ ld(P) ≤ 2∆(P) − 2

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 4 / 16

SLIDE 7

Online Algorithm Framework

(1) Builder presents poset one point at a time (2) Assigner makes irrevocable decision about where to insert it into a linear extension.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 5 / 16

SLIDE 8

Online Algorithm Framework

(1) Builder presents poset one point at a time (2) Assigner makes irrevocable decision about where to insert it into a linear extension. (3) Variant: Builder is required to provide a representation (interval, unit interval, etc.)

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 5 / 16

SLIDE 9

Lower Bound for General Posets

Lemma (K, Streib, Trotter)

For each k ≥ 1, Builder can construct an interval order P with ld(P) = k so that Assigner must form a linear extension L with ld(P, L) = 3k − 1.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 6 / 16

SLIDE 10

Lower Bound for General Posets

k + 1 k + 1

L z D x A

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 11

Lower Bound for General Posets

L z D x A z x D A u u

k k

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 12

Lower Bound for General Posets

L z D x A z x D A u u

k k

y y

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 13

Lower Bound for General Posets

L z D x A z x D A u u

k k

y y C C

k − 2

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 14

Lower Bound for General Posets

L z D x A z x D A u u

k k

y y C C

k − 2

B B

k − 1

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 15

Lower Bound for General Posets

L z D x A z x D A u u

k k

y y C C

k − 2

B B

k − 1

Lopt D z u C y B x A

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 7 / 16

SLIDE 16

Lower Bound for Semiorders

Lemma (KST)

For each k ≥ 1, Builder can construct a semiorder P with ld(P) = k so that Assigner must form a linear extension L with ld(P, L) = 2k.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 8 / 16

SLIDE 17

Upper Bounds

Theorem (KST)

If Builder constructs a poset P with ld(P) = k ≥ 1 and Assigner uses algorithm A to assemble a linear extension L of P, then (1) ld(P, L) ≤ 2k if P is a semiorder and (2) ld(P, L) ≤ 3k − 1 otherwise.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 9 / 16

SLIDE 18

First Attempts at Algorithms

M: Insert point in middle of its allowable range in existing linear extension. G: Insert point to minimize ld(P, L) for updated linear extension.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 10 / 16

SLIDE 19

Critical Pairs

Definition

A pair (x, y) of incomparable points is a critical pair if (1) D(x) ⊆ D(y) (2) U(y) ⊆ U(x)

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 11 / 16

SLIDE 20

Critical Pairs

Definition

A pair (x, y) of incomparable points is a critical pair if (1) D(x) ⊆ D(y) (2) U(y) ⊆ U(x)

Lemma (K, Young 2010)

(1) If x and y are incomparable and hL(y) − hL(x) = ld(P, L), then (x, y) is a critical pair. (2) There exists an optimal linear extension L of P so that if (x, y) is a critical pair reversed by L, then (y, x) is also a critical pair.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 11 / 16

SLIDE 21

An Optimal Online Linear Discrepancy Algorithm

In four words:

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 12 / 16

SLIDE 22

An Optimal Online Linear Discrepancy Algorithm

In four words:

Reverse few critical pairs.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 12 / 16

SLIDE 23

An Optimal Online Linear Discrepancy Algorithm

In four words:

Reverse few critical pairs.

A bit more detail

Builder presents new point x. Assigner uses algorithm A: Identify one-way critical pairs (x, u) and (v, x) u0 the L-smallest u, v0 the L-largest v None of either type: x in any legal position Only (x, u): x below u0 Only (v, x): x above v0 v0 < u0 in L: x any legal position between v0 and u0

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 12 / 16

SLIDE 24

An Optimal Online Linear Discrepancy Algorithm

In four words:

Reverse few critical pairs.

A bit more detail

Builder presents new point x. Assigner uses algorithm A: Identify one-way critical pairs (x, u) and (v, x) u0 the L-smallest u, v0 the L-largest v None of either type: x in any legal position Only (x, u): x below u0 Only (v, x): x above v0 v0 < u0 in L: x any legal position between v0 and u0 u0 < v0 in L?

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 12 / 16

SLIDE 25

An Optimal Online Linear Discrepancy Algorithm

In four words:

Reverse few critical pairs.

A bit more detail

Builder presents new point x. Assigner uses algorithm A: Identify one-way critical pairs (x, u) and (v, x) u0 the L-smallest u, v0 the L-largest v None of either type: x in any legal position Only (x, u): x below u0 Only (v, x): x above v0 v0 < u0 in L: x any legal position between v0 and u0 u0 < v0 in L? x anywhere between u0 and v0

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 12 / 16

SLIDE 26

A Look At Why It Works

C : x w z y

Lemma (KST)

A never assembles a linear extension L with a copy of C such that x < w < z < y in L, all points less than x in L are less than y in P, and all points greater than y in L are greater than x in P. < y in P > x in P x . . . . . . . . . y w z

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 13 / 16

SLIDE 27

Performance of A

Theorem (KST)

If Builder constructs a poset P with ld(P) = k ≥ 1 and Assigner uses algorithm A to assemble a linear extension L of P, then (1) ld(P, L) ≤ 2k if P is a semiorder and (2) ld(P, L) ≤ 3k − 1 otherwise.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 14 / 16

SLIDE 28

Builder Required to Present Representation

Theorem (KST)

If Builder is required to present an interval representation of an interval

rder P with ld(P) = k ≥ 1 and Assigner constructs a linear extension

L ordering by left endpoints, then (1) ld(P, L) = k if P is a semiorder and Builder presents unit intervals and (2) ld(P, L) ≤ 2k otherwise.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 15 / 16

SLIDE 29

Open Problems

Online linear discrepancy

No representation, interval orders without 1 + 4? Proper representation for semiorders instead of unit? Up-growing?

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 16 / 16

SLIDE 30

Open Problems

Online linear discrepancy

No representation, interval orders without 1 + 4? Proper representation for semiorders instead of unit? Up-growing?

Is ld(P) ≤ 3∆(P) − 1 2

Known for width 2, disconnected, interval orders. Best general upper bound is 2∆(P) − 2.

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 16 / 16

SLIDE 31

Open Problems

Online linear discrepancy

No representation, interval orders without 1 + 4? Proper representation for semiorders instead of unit? Up-growing?

Is ld(P) ≤ 3∆(P) − 1 2

Known for width 2, disconnected, interval orders. Best general upper bound is 2∆(P) − 2.

Is ld(P) ≥ dim(P) for dim(P) ≥ 5?

Mitch Keller (Georgia Tech) Online Linear Discrepancy SIAM DM10 16 / 16