Applications of totally monotone matrices

Totally Monotone Matrix k l

i

j

<

<

Mik < Mil Mjk < Mjl

If i<j, k<l,

Totally Monotone Matrix

< <

Totally Monotone Monotone

(Inverse) Monge Matrices k l

i

j

If for every i<j, k<l, Mik + Mjl ≥ Mil + Mjk

Inverse Monge Totally Monotone

k l

i

j

<

<

Mik < Mil Mjk < Mjl Mik + Mjl ≥ Mil + Mjk

Inverse Monge Totally Monotone of the transpose

k l

i

j

<

<

Mik < Mjk Mil < Mjl Mik + Mjl ≥ Mil + Mjk

Row maxima in totally monotone matrices

Given a totally monotone matrix compute the maximum in each row.

Classical results on totally monotone matrices

• SMAWK (Shor, Moran, Aggarwal, Wilber, Klawe): Row maxima/minima in O(n) time

The concave least weight subsequence problem

• Input: w(i,j) defined for the integers 0≤i<j≤n

• w(i0,j0)+w(i1,j1) ≤ w(i0,j1)+w(i1,j0) for 0 ≤ i0 < i1 < j0 < j1 ≤n

• Problem: Find an integer k and a sequence 0=ℓ0< ℓ1<……< ℓk=n that minimizes:

1

1

0

( , )k

i i

i

w

The concave least weight subsequence problem

w(i0,j0)+w(i1,j1) ≤ w(i0,j1)+w(i1,j0) for

0 ≤ i0 < i1 < j0 < j1 ≤n

i0 i1 j0 j1

Motivation: Paragraph packing

From my ibook:

The w matrix is Monge 1 2 3 n

0 1

2

n-1

i0

i1

j0 j1

w

Add f(i) to row i 1 2 3 n

+ f(0)

+ f(1)

+ f(i0)

j0 j1

=G

+ f(i1)

+ f(n-1)

The matrix G: g(i,j) = f(i) + w(i,j) 1 2 3 n

0 1

2

n-1

g(i,j)

G

G Monge, f(i) is the minimum of column i

f(i)

i

Column minima form a stairs shape 1 2 3 n

0 1

2

n-1

G

G Monge, f(i) is the minimum of column i

f(i)

i

Can We Use SMAWK ?

• Min Max, monge inverse Monge by negating

• Column row by transpose

BUT

• We do not have constant access to an entry of G

A Linear Algorithm (invariants)

r

c

c

• F(j) = f(j) for j ≤ c • All minima in columns ≥ (c+1) are in rows ≥ r

r ≤ c

A Linear Algorithm

p = c + (c-r) + 1

r

c

c p

A Linear Algorithm

p = c + (c-r) + 1

r

c

c p

SMAWK

A Linear Algorithm

For (c+1) ≤ j ≤ p Set F(j) = min {G(i,j) | r ≤ i ≤ c}

r

c

c p

SMAWK

A Linear Algorithm

For (c+1) ≤ j ≤ p Set F(j) = min {G(i,j) | r ≤ i ≤ c}

p-1

r

c

c p

SMAWK

SMAWK

A Linear Algorithm

For (c+2) ≤ j ≤ p Set H(j) = min {G(i,j) | c+1 ≤ i ≤ p-1}

p-1

r

c

c p

SMAWK

SMAWK

A Linear Algorithm

If F(c+2) ≤ H(c+2) then f(c+2) = F(c+2) and H(c+3) is correct

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2

A Linear Algorithm

If F(c+3) ≤ H(c+3) then f(c+3) = F(c+3) and H(c+4) is correct

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2

A Linear Algorithm

j0, c+2 ≤ j0 ≤ p such that H(j0) < F(j0) ?

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0

A Linear Algorithm

j0, set c := p

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0

A Linear Algorithm

j0, set c := p

c

r

c

A Linear Algorithm

j0, set F(j0) = H(j0), c := j0, r := c+1

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0

A Linear Algorithm

j0, set F(j0) = H(j0), r := c+1, c := j0

c

r

c=j0

Analysis

Work per iteration proportional to c-r+1

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0

Potential function

Φ(i) = r+c Always increasing, and ≤ 2n

Potential increase

j0, r := c+1, c := j0 : r’ = r + (c-r+1), c’ ≥ c+2 Φ’ ≥ Φ + c-r+3

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0

Potential increase

j0, set c := p : c’ = c + c-r+1 Φ’ ≥ Φ + c-r+1 (unless p=n but then it’s the last iteration)

p-1

r

c

c p

SMAWK

SMAWK

c+1 c+2 j0