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Home , Interval (mathematics)

On the Interplay b etween Interval



and Dimension

y z x

S Felsner M Habib RH Mohring

Abstract

This pap er investigates a transformation P Q b etween partial orders P Q that

transforms the interval dimension of P to the dimension of Q ie idimP

dimQ Such a construction has b een shown b efore in the context of Ferrers

dimension by Cogis Our construction can b e shown to b e equivalent to

his but it has the advantage of b eing purely ordertheoretic providing

a geometric interpretation of interval dimension similar to that of Ore for

dimension and revealing several somewhat surprising connections to other

ordertheoretic results

For instance the transformation P Q can b e seen as almost an inverse

of the wellknown split op eration it provides a theoretical background for the

inuence of edge sub division on dimension eg the results of Spinrad and

interval dimension and it turns out to b e invariant with resp ect to changes of P

that do not alter its comparability graph thus providing also a simple new pro of

for the comparability invariance of interval dimension

This work was partially supp orted by the DFG and by the DAAD Pro ject PROCOPE

y

TUBerlin Berlin Germany

z

LRIMM Montp ellier France

x

TUBerlin Berlin Germany

Intro duction

An extension Q of a partial order P is an order on the same elements that

contains all the ordered pairs of P ie x y in P implies x y in Q A family

fQ Q g of extensions of P is said to realize P or to b e a realizer of P i

1 k

P Q Q ie x y in P i x y in Q for each i i k If we

1 k i

restrict the Q to b elong to a sp ecial class of orders and seek for a minimum size

i

realizer we come up with a concept of dimension with resp ect to the sp ecial class

A linear extension of P is an extension which is a chain Dushnik and Miller

dened the dimension of a partial order P denoted dimP as the smallest

k for which there exist k linear extensions realizing P Let the real

izer fL L g of an order P with jP j n b e made by the linear extensions

1 k

L x x x With every x P we then asso ciate the vector

i i1 i2 in

1 k k i

x x R where x gives the p osition co ordinate of x in L This map

i

k

ping of the p oints of P to p oints of R emb eds P into the comp onentwise ordering

k k

of R Ore dened dimP as the minimum k such that P emb eds into R

in this way The pro jections of such an emb edding on each co ordinate yield a

realizer of P The extensions of this realizer need not b e linear extensions of P

but it is straightforward to transform them into linear extensions Therefore Ore

dimension and DushnikMiller dimension are equivalent concepts

A partial order P is an interval order if it can b e represented by assigning to

each element x P an op en interval I a b of the such that x y

x x x

in P i b a Such a collection of intervals is called an interval representation of

x y

P Fishburn characterized interval orders as those orders that do not contain a

p oint forming two disjoint chains ie that contain no Interval

dimension denoted idimP is dened by using interval extensions instead of

linear extensions Since linear orders are interval orders we obtain the trivial

idimP dimP

It is well known that interval orders of large dimension exist see

Hence the gap b etween idimP and dimP may b e arbitrarily large

Let I fI I g b e an interval realizer of P and x an op en interval

1 k

j j

representation of each interval order I Let a b b e the interval corresp onding

j

x x

to x P in the representation of I We now dene a box embedding of P

j

Q

k j j k

to R With x P we asso ciate the b ox a b R Each of these

j

x x

k 1

and b b oxes is uniquely determined by its upper extreme corner u b

x

x x

1 k

its lower extreme corner l a a Obviously x y in P i u l

x x y

x x

comp onentwise The pro jections of a b ox emb edding onto each co ordinate yield

an interval realizer so the concepts of b ox emb eddings and interval realizers are

equivalent For interval dimension the b ox emb eddings thus play the role of the

k

ab ovementioned p oint emb eddings into R intro duced by Ore for dimension

A b ox emb edding dep ends not only on the realizer I of P but also on the

representations of the I Now we dene the partial order B I of extreme corners

j

asso ciated with a b ox emb edding or equivalently with an interval realizer I of P

The vertices of B I are the at most n dierent lower resp upp er extreme corners

of elements of P The order relation of B I is given by the comp onentwise order

k

in R

k

By denition we have an emb edding of B I in R so dim B I k

idimP The starting p oint of our investigations was the following question

concerning the interplay b etween dimension and interval dimension

Is dim B I idimP

In Section we dene a transformation P B P such that idimP

dim B P We provide two interp etations of this transformation a combinatorial

one and a geometrical one In the combinatorial interpretation the elements of

B P are of P In the geometrical interpretation B P is the p oset B I

of extreme corners of I Here I is a b ox emb edding obtained from an arbitrary

b ox emb edding I by a normalizing pro cedure From the pro ofs we obtain an

armative answer to the question ab ove

Section investigates several consequences to and relations with other order

theoretic results First we study the transformation P B P on sp ecial partial

orders of height In particular we show that the standard example S of a n

n

dimensional order is an almost xed p oint of the transformation P B P

Therefore dimS idimS

n n

Second we investigate the relationship with the split op eration This has a

2

surprising consequence for the iterated transformation P B P B P

k

B P For every n there are partial orders P such that dimP

k n+1

dim B P for all k n but dimP dim B P m where m is

arbitrary

Third we relate the interval dimension of sub divisions of P to the dimension

of P thus providing a theoretical framework for the examples of Spinrad

Finally we show the comparability invariance of the transformation P

B P which as a consequence gives another pro of that the interval dimension is

a comparability invariant Some remarks concerning the recognitioncomplexity

of sp ecial classes of orders and graphs close the pap er

The main result

In the last section we dened the partial order B I of extreme corners asso ciated

k

with a b ox emb edding of P in R With the next lemmas we show that B I

inherits some structure which is indep endent of the realizer I leading to the b ox

emb edding

Lemma Let B I be the partial order of extreme corners of a box representa

tion of P

a If the lower extreme corners of x and y are comparable in B I eg l

x

l then the predecessor sets of x and y in P are ordered by inclusion ie

y

Pred x Pred y

P P

b If the upper extreme corners of x and y are comparable in B I eg u u

x y

then the successor sets of x and y in P are ordered by reversed inclusion ie

Succ x Succ y

P P

c If the lower extreme corner of x and the upper extreme corner of y are related

by l u then Pred z Pred x for al l z Succ x or equivalently

x y P P P

Pred x Pred z

P P

z Succ (y )

P

d If u l then Pred z Pred y

x y P P

z Succ (x)

P

j j

for all j Therefore in each I x has a Pro of a From l l we obtain a

j x y

y x

less predecessors than y ie Pred x Pred y The claim now follows from

j j

Pred x since the I realize P Pred x

j j P

j

b The pro of of this part is symmetric to part a

j j j j j

c From l u we have a b If z Succ y then necessarily a b a

x y

x y z y x

Hence Pred z Pred x for all j The claim follows

j j

d From u l we immediately obtain x y ie y Succ x therefore

x y

Pred y Pred z 2

z Succ (x)

All statments except the conclusion part of b use only the sets Pred x and

P

Pred z This irregularity is resolved with the next lemma

P

z Succ (x)

P

Lemma Pred z Pred z if and only if Succ x Succ y

z Succ (x) z Succ (y )

Pro of The if direction is trivial We now prove the only if direction Let

Pred z From the assumed inclusion z Succ x and note that y

z Succ (y )

we obtain y Pred z hence z Succ y 2

Denition With each vertex x of a partial order P we associate the lower

Lx Pred x and the upp er set U x Pred z The case x

P P

z Succ (x)

P

Max P is settled by the convention U x P

Dene B P fLx U x x P g ordered by setinclusion

Note that this construction is in fact equivalent with Cogis construction in the

context of Ferrers dimension Cogis also uses Lx but replaces U x by the

equivalent set fz P Succ x Succ z g He also proves Theorem but in

a dierent way and without the geometrical interpretation that our approach is

based on

The preceding lemmas prove that l Lx and u U x together form

x x

an order preserving mapping from B I to B P hence

idimP dim B I dim B P

To get more structure into interval realizers we now intro duce a pro cedure

that transforms a interval extension I f a b x P g of P into its P

x x

normalization I f a b x P g

x x

In the rst step of the P normalization we up date left endp oints

a max fb z Pred xg if x is not minimal

z

x

a minfa z MinP g if x is minimal

z

x

In the second step we up date right endp oints

b minfa z Succ xg if x is not maximal

x z

b max fb z Max P g if x is maximal

z

x

Note that the interval order I need not b e isomorphic to I In general I is a

sub order of I and a minimal interval extension of P if all the a b were dierent

x x

If P is realized by I fI I g then I fI I g realize P as well

1 k

1 k

We call the b ox emb edding corresp onding to I the normalized b ox emb edding

of I For an example see Figure

I I

2

6 6

2

r

r

c

r

c

r r

r

d

sc sd

d

r P

a

a

r r

r sa sb

b b

r

r

- -

I I

1

1

Figure P an interval realizer of P and its normalization

After normalizing we have a realizer I fI I g of P interval repre

1 k

j j

and an asso ciated partial order of extreme corners B I b sentations a

x x

x P g The next theorem shows that the geometrically dened B I u fl

x x

and the combinatorially dened B P are isomorphic

Theorem If I is a normalized realizer of P then B I B P

Pro of First observe that b oth partial orders have a least element generated by

and Lx resp ectively and a greatest element generated by x MinP as l

x

x Max P as u and U x resp ectively

x

Moreover l Lx and u U x denes an order preserving mapping by

x x

the remarks preceeding To show the converse we distinguish four cases

U x Ly We know that x U x so x Pred y Since I is an interval

j

j j

extension of P we obtain b a for all j Hence u l

x y x y

j j j

Lx Ly Rememb er that a max fb z Pred xg and a

x z y

j j j

max fb z Pred y g By assumption Pred x Pred y so a a and

z x y

l l

x y

U x U y By Lemma this is equivalent to Succ x Succ y Now

u u follows symmetrically with the second case

x y

is normalized there are z Pred x and z Succ y Lx U y Since I

0 1

j

j j j j j j

with a b and b a The hyp othesis provides z z hence a b

0 1

x z y z x z

j j j

b a b The validity of this inequality for all j again gives l u 2

z z y x y

We are ready now to prove our main theorem ab out interval dimension and

dimension

Theorem dim B P idimP

Pro of As Inequality we allready have obtained dim B P idimP For

the converse we need two arguments We rst show that a linear extension L of

B P induces an interval extension I of P Secondly we prove that if L L

L 1 k

I form an is a realizer of B P then the induced interval extensions I

L L

k

interval realizer of P

Let L M M M b e a linear extension of B P For each x P there

1 2 r

are i j f r g such that M Lx and M U x From Lx U x and

i j

x Lx x U x we obtain that i j So we can asso ciate with x a unique

interval a b which is dened to b e i j We now show that the interval order

x x

I induced by the interval representation f a b x P g is an extension of

L x x

P If x y in P then U x Ly and thus with M U x and M Ly

i j

b i j a which implies x y in I

x y L

Let fL L g b e a realizer for B P and let I I for j k

1 k j L

j

The family fI I g of interval extensions of P is an interval realizer i all

1 k

incomparabilities xjjy of P are realized If xjjy in P then U x Ly since

x U x but x Ly Therefore Ly precedes U x in some L which gives

j

j j i i

a b The symmetric argument yields an i with a b Both inequalities

y x x y

I 2 together give xjjy in

j

j

This theorem together with Inequality shows that dim B I is indep endent

of the interval realizer I

Consequences

In the previous section we intro duced the op eration P B P mapping partial

orders to partial orders We will now investigate several connections to other

ordertheoretical topics and results Note that B P always has a greatest and

a least element We adopt the convention of calling orders with this prop erty

b b

bounded and denote by Q Q the b ounding of a partial order Q ie Q is the

order resulting from Q by adjoining a new greatest and a new least element

We rst lo ok at the eect of the op erator B applied to sp ecial classes of orders

Transformation rules for sp ecial p osets

Let S denote the standard p oset of dimension n ie the set of all element

n

and nsubsets of an n element set ordered by setinclusion Then

c

B S S

n n

Let C denote the r cycle The r cycle is the dimensional p oset on r

r

elements fx y x y x y g with comparabilities

1 1 2 2 r r

x y y x x y x y y x

1 1 1 2 2 2 r r r 1

c

B C C

r r

Let the Hasse diagram of T b e a tree The truncation of T denoted by tr T

is the induced tree on the nonleaf vertices of T Then

d

B T tr T

In particular shows that the standard example S of a ndimensional order

n

is up to closures a xed p oint of the op eration P B P thus showing again

that for every n there is are orders P with dimP idimP n

B as an inverse of the split op eration

We now turn to the natural question whether for every b ounded partial order

Q there is some P with Q B P The next theorem answers this question

armatively Moreover it turns out that the op eration P B P is an almost

left inverse of the split op eration S which has applications in dierent branches of

p oset theory see eg The split S P of a partial order P is the order

of height one with minimal elements fx x P g maximal elements fx x P g

and ordered pairs x y i x y in P

b

Theorem B S P P

Pro of For x P let Pred x Pred x fxg Obviously P is isomorphic to the

setsystem fPred x x P g ordered by inclusion We will show that the elements

of B S P are just the primed sets Pred x ie Pred x fx x Pred xg

together with the greatest element fx x x P g and the least element We

have

Lx and

U x Pred z Pred z Since x Succ x U x Pred x

00 0

z Succ (x ) z Succ [x]

On the other hand Pred x Pred z for z Succ x Together this gives

U x Pred x Similarly we obtain

Lx Pred x Pred x Finally

U x fx x x P g by denition since Succ x 2

It is easy to verify that

d

b

B P B P

So we may generalize the theorem to

b

g n b oundings

n n

b

P B S P

Investigations on the eect of iterated splitting to the dimension lead to the

inequality

n

dim P dim S P dim P for all n

As a consequence of and we obtain that for every n there is are partial

orders P such that

k

dim P dim B P for all k n

n

Just take P S Q for some order Q If we cho ose Q however to b e an

m dimensional interval order we obtain a large dierence in dimension with the

next iteration ie

n+1

dim P dim B P m

The interval dimension of sub divisions

With the next theorem we relate the interval dimension of sub divisions of P to

the dimension of P Spinrad showed that the dimension of a sub division

of a partial order can b e an arbitrary multiple of its dimension thus answering

Trotters Problem in With our result we establish a theoretical framework

for his examples

In this context partial orders and their diagrams are regarded as directed

graphs whose edges x y corresp ond to ordered pairs and pairs x y

of P resp ectively An edge x y is subdivided by placing a new vertex z in

the middle of the edge ie x y is replaced by x z and z y In the case

of partial orders we then have to ensure transitivity ie all edges a z with

a Pred x and z b with b Succ y are also added

The complete diagram subdivision DS P is the sub division of all edges of the

diagram of P The complete subdivision CS P is the transitive of the

sub division of all the edges of P Let E b e any set of edges of P we denote

the order obtained by sub dividing the edges of E with Sub P E Since P is

an induced sub order of each sub division Sub P E and since Sub P E is an

induced sub order of CS P we obtain

dimP dim Sub P E dim CS P

With the next theorem we give an upp er b ound for idim Sub P E

Theorem idim Sub P E idim DS P idim CS P dimP

k

Pro of Take any emb edding of P into R with k dimP and grow the p oints

to obtain an emb edding by minib oxes An interval emb edding of a sub division

Sub P E is then obtained by adding the b ox with lower extreme corner u and

x

upp er extreme corner l for the p oint z sub dividing the edge x y E see

y

Figure This gives idim Sub P E dimP indep endent of the choice of E

r r

r r

6 6

r r

r r

r r

r r

r r

P

r r

r r r

r r r r

r r

- -

Figure P a minib ox emb edding of P and the b ox emb edding of DS P

To prove that idim DS P idim CS P dimP we will show that P can

b e emb edded in B DS P This will give dimP dim B DS P From The

orem we know idim DS P dim B DS P Together we obtain dimP

idim DS P

To show that P can b e emb edded in B DS P we apply the normalizing

pro cedure to the b ox emb edding I of DS P constructed in the rst paragraph

of the pro of Because of the construction of the b ox emb edding of DS P the only

changes that o ccur in normalization are shifts of the left endp oints of intervals

corresp onding to elements in MinP and the right endp oints of elements in

Max P We then emb ed P into the lower extreme corners of the minib oxes of the

normalized representation I This gives dimP k idimI idimDS P

2

Note that we obtained in fact a slightly stronger result If a set E of edges of

P contains the edges of the diagram then B Sub P E VS P where VS P

denotes the vertical split of P ie the order obtained from P by substituting

each vertex by a chain In a distinct pro of for dimP idim V S P has

b een given

Comparability invariance

For the denition and basic facts on comparability invariance see Let

Comp P b e the comparability graph of P We will show that Comp B P

is a comparability invariant of P in the sense that if CompP Comp Q then

Comp B P Comp B Q Together with Theorem and the known fact that

dimension is a comparability invariant this gives an alternative pro of of the com

parability invariance of interval dimension in the nite case The comparability

invariance of interval dimension was rst shown in

A subset of elements A of P is called autonomous if the relation of elements

in A to an element outside A is indep endent of the element of A More formally

if for any x A whenever x a x a or xjja holds for some a A then

it holds for all a A If A is an autonomous subset of P then Pred A shall

denote the predecessors outside A of any and hence every element of A

Theorem Comp B P is a comparability invariant of P

Pro of Let A b e an autonomous subset of P It is enough to show see eg

A A

that Comp B P Comp B P where P denotes the order resulting from

d d

A A

d

substituting A by its dual A in P

Note rst that B A fLa U a a Ag is a closed sub order of B P

g

Let B A b e B A without its greatest element and its least element

B (A) B (A)

g

Our claim is that B A is autonomous in B P To see this observe rst that

for each a A we can decomp ose Pred a into Pred a Pred A Pred a

A

g

hence the same is valid for all elements of B A On the other hand the elements

of B P n B A either contain all of A or their intersection with A is empty Now

if M B P n B A contains all of A then it also contains Pred A and M is

g g

ab ove all sets in B A If M Pred A then M is b elow all sets in B A In all

g

the other cases M is unrelated to all of B A This gives the claim

d d

To settle the theorem we need that B A and B A are isomorphic orders

k

To see this consider a normalized b ox emb edding of A in R Its extreme corners

k

are an emb edding of B A into R Flip the emb edding ie reverse the relations

d

this gives an emb edding of A and the extreme corners form an emb edding of

d

B A 2

As we have seen autonomous sets in P induce autonomous sets in B P

The converse however is far from b eing true Take as P a prime interval order

jB (P )j

nontrivial then B P is a chain Hence P has none but B P has

2

autonomous sets

Remarks on related results and computational com

p exity

The transformation P B P obviously can b e carried out in p olynomial time

The degree of the p olynomial of this reduction is not imp ortant if we want to

decide if P has interval dimension since the p oset dimension and interval

dimension problems have b een shown to b e NP complete for k by Yannakakis

It is however a crucial p oint if we want to decide if idimP

Dagan Golumbic and Pinter addressed questions ab out the comparability

invariance of interval dimension and fast recognition of interval dimension at

most As remarked b efore the rst problem has b een settled armatively

rst in The recognition problem has b een solved indep endently by several

authors Habib and Mohring use the subp oset of B P dened by B P

fLx x P g together with an algorithm for transitive orientation with side

a a

constraints to derive an O n n algorithm where O n is the b est known time

for matrix multiplication Currently a is ab out

3

Cheah prop osed an algorithm in complexity O n The same complexity

is claimed by Langley Langley calles the p oset B P of the present pap er

the predecessorsuccessor order of P and shows without the aid of the geometric

construction that nding an interval realizer of P is equivalent to nding a linear

realizer of B P

Spinrad has shown that recognition of dimensional orders only requires

2

O n op erations Therefore the b ottleneck with the P B P approach for

the recognition of interval dimension is the computation of the order B P

a

With a careful implementation B P can b e computed in O n where again

a

O n is the b est known time for matrix multiplication

Ma and Spinrad have found an approach which allows to avoid matrix

2

multiplication and leads to a complexity b ound of O n Since this is the b est

known result we outline the ideas b ehind their algorithm

First they construct the op en split S P of the partial order P for which

they want to decide if idimP The elements of the op en split S P are

the same as of the split S P dened ab ove the ordered pairs of S P are

however given by the irreexive relation dening P ie x y in S P i

x y in P They prove that the cochain covering numb er of S P equals the

interval dimension of P A theorem of Yannakakis shows that the cochain

covering numb er of S P and the interval dimension of S P coincide Hence

idimP idimS P and we can reduce attention to the recognition of interval

dimension for bipartite orders ie to orders of height one

The transformation of the interval dimension problem for bipartite orders

to the dimension problem is done in two steps In the rst step it is check

ued whether the bipartite order Q has a chordal bipartite comparability graph

A chordal bipartite graph is a bipartite graph without any induced cycle C

n

n If Comp Q is not chordal bipartite then Q has to contain a crown and

therefore idimQ Otherwise the second step is started In this step Q is

transformed into an order P which can b e obtained by contracting autonomous

Q

chains in Stack Q and hence has the same dimension as Stack Q The stack

op eration has b een intro duced by Trotter He proves that for bipartite p osets

dim Stack Q idimQ As the construction of B Q the construction of P

Q

and Stack Q requires information ab out the containment relation of neighb or

ho o ds in Q For chordal bipartite graphs this information can b e computed in

2

O n using a technique called doubly lexical ordering Since after passing the

rst step we know that Q is chordal bipartite we can construct P and apply

Q

2

Spinrads dimension algorithm all in O n

Acknowledgement

We are greatful to an anonymous referee for his fruitful suggestions which im

proved the presentation of the pap er

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