On the Decomposition of Posets

On the Decomposition of Posets

Prof. Dr. Yangjun Chen

Dept. Applied Computer Science,

University of Winnipeg

515 Portage Ave.

Winnipeg, Manitoba, Canada R3B 2E9

Outline

• Poset and poset decomposition• Applications of poset decomposition

- Evaluation of reachability queries

• Algorithm for poset decomposition- Graph stratification and bipartite graphs- Virtual nodes and virtual edges- Resolution of virtual nodes

• Conclusion

Poset and poset decomposition

A partially ordered set (poset for short) S isa pair (S, ) of a set S and a binary relation such that for each a, b, and c in S:

1.a a is true ( is reflexive),

2.a b and b c imply a c ( is transitive),

3.a b and b a imply a = b ( is antisymmetric).

Example


S = (S, )

S = {a, b, c, d, e, f, g, h, i, j, k, l }

d c, d e

g e

l h, l k

b c, b f, b i,

e c, e f, e j,

h a, h k

Any identity such asa = a and transitiverelationship are not shown.

d c, d eg e

l h, l k

b c, b f, b i,

e c, e f, e j,

h a, h k

If we have a poset S = (S, ), a chain in S is a non-empty subset C = {a1, a2, ..., ak} S such that a1 a2 ... ak.


Example

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c f i j a k

Example

• Two elements of S are called comparable if they can be put together in some chain in S.

• Elements which are not comparable are called incomparable.

• A non-empty set, in which every pair of elements is not comparable, is called an anti chain.


A anti-chain:

{a, c, f, i, j, k}

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l

c f i j a k

Theorem (Dilworth, 1950) The minimal number of decomposed chains of a poset equals the size of its maximum anti chain.


Example

R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math.51 (1950), pp. 161-170.

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c f i j a k


“Less clear is how to decompose S into asfew chains as possible.”

A.S. Asratian, T. Denley, and R. Haggkvist,Bipartite Graphs and their Applications,Cambridge University, 1998. (Page 190)

How to decompose S into as few chains as possible?

Applications of poset decompo-tion

Efficient method to evaluate graph reachability queries

- Given a directed acyclic graph (DAG) G, check whether a node v is reachable from another node u through a path in G.- If we can decompose a DAG into a minimum

set of chains, the reachability queries can be efficiently evaluated.

- A DAG can be considered as a poset.

A simple method- store a transitive closure as a matrix

cb

a

d e

G:

cb

a

d e

G*:

M =

abcde

a b c d e00000

10000

10100

00100

10000

M* =

abcde

a b c d e00000

10000

10100

10100

10000

O(n2) spacequery time: O(1)


Reachability based on poset decomposition


cb

a

d e

G:

eb

a (1, 1)

(1, 2)

(1, 1)(2, 1)(3, 1)

(1, 2) d

c (2, 1)

(2, 2)

(2, 1)(3, 1)

(2, 2) (3, 1) (3, 1)

• Decompose G into a minimum set of chains.• Each node in G will be assigned an index (i, j) to show that it is

the jth node on the ith chain.• In addition, each node v on the ith chain will be associated with

an index sequence of length : (1, j1) … (i, ji) … (, j) suchthat any node with index (x, y) is a descendant of v if x = iand y j or x i but y jx, where is the minimal number ofthe node-disjoint chains, referred to the as width of G.


• We can also store all the index sequences as a matrix M, inwhich each entry M(v, j) is the jth element in the indexsequence associated with node v.

• So, a node u with index(u) = (i, j) is a descendant of anothernode v iff M(v, i) j. Thus, using M, a reachability checkingcan be done in O(1) time.

abcde

1 2 3

12___

1_12_

1_1_1

The size of the matrix is O(n).The query time is O(1).

Algorithm for poset decomposition

Graph stratification

Virtual nodes

Resolution of virtual nodes



• Let G(V, E) be a DAG. We decompose V into subsets V0, V1

..., Vh such that V = V0 V1 ... Vh and each node in Vi has

its children appearing only in Vi-1, ..., V0 (i = 1, ..., h), where

h is the height of G, i.e., the length of the longest path in G.

• For each node v in Vi, we say, its level is i, denoted l(v) = i.

We also use Cj(v) (j < i) to represent a set of links with each

pointing to one of v’s children, which appears in Vj.

• For each v in Vi, there exist i1, ..., ik (il < i, l = 1, ..., k) such

that the set of its children equals Ci1 ... Cik . Let Vi = {v1,

v2, ..., vl}. We use (j < i) Cji to represent Cj(v1) ... Cj(vl).

C0(b) = {c, f, i} C0(e) = {c, f, j} C0(h) = {a, k}

C1(d) = {e}

C0(d) = {c}

C1(l) = {h}

C0(l) = {k}

C1(g) = {e}

c f i a kj

b e h

d g l

V0:

V1:

V2:



b e h

d g l

c f i j a k



Algorithm graph-stratification(G)begin1. V0 := all the nodes with no outgoing arcs;2. for i = 0 to h - 1 do3. { W := all the nodes that have at least one child in Vi;4. for each node v in W do5. { let v1, ..., vk be v’s children appearing in Vi;6. Ci(v) := {v1, ..., vk};7. if dout(v) > k then remove v from W;8. G := G\{v v1, ..., v vk};9. dout(v) := dout(v) - k; }10. Vi+1 := W;11. }end


Bipartite graphs

Definition 1 (bipartite graph) An undirected graphG(V, E) is bipartite if the node set V can bepartitioned into two sets T and S in such a waythat no two nodes from the same set are adjacent.We also denote such a graph as G(T, S; E).

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c f i j a k


Definition 2 (matching [2]) Let G(V, E) be abipartite graph. A subset of edges E’ E is called amatching if no two edges have a common end node.A matching with the largest possible number ofedges is called a maximal matching, denoted as MG.

M1

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c f i j a k

bb e h

c f i j a k

covered node free node

• A graph stratification can be considered as series ofbipartite graphs.

• We can find a maximum matching for each bipartitegraph, and combine all the maximum matchings toform a set of chains.



b e h

c f i j a kV0:

V1:

b e h

d g l

V1:

V2:

b e h

d g lM2:

b e h

d g l

c f i j a k

Combine M1 and M2 to form a set of7 chains:

M1: e h

c f i j a k

b

It is not minimum.

Virtual nodes


V0’ = V0.Vi’ = Vi {virtual nodes added to Vi} for 1 i h - 1.}Ci = {all the new arcs from the nodes in Vi to the virtualnodes added to Vi-1’} for 1 i h - 1.}G(Vi, Vi-1’; Ci) - the bipartite graph containing Vi and Vi-1’.

Definition 3 (virtual nodes) Let G(V, E) be a DAG, dividedinto V0, ..., Vh (i.e., V = V0 ... Vh). Let Mi be a maximummatching of the bipartite graph G(Vi, Vi-1’; Ci). For each freenode v in Vi-1’ with respect to Mi, a virtual node v’ created forv is a new node added to Vi (1 i h - 1).

Virtual nodes


M1

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i’ j’ k’

Virtual arcs


• The goal of virtual nodes is to establish theconnection between the free nodes (with respectto a certain maximum matching) and the nodesthat may be several levels apart.

• For this purpose, virtual arcs will be used.

inherited arcs - If there is u Vj (j > i) such thatu v E, add u v’, which is not labeled. However,if u v itself is a virtual arc, u v’ will inherit thelabel of u v. In both cases, u v’ is referred to asan inherited arc.


b e i’ j’ k’h

d g l

V1’:

V2:

l k’ is an inherited arc since in the original graph we have anarc l k.

b e h

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c f i j a k


transitive arcs - If there exist u w E and w v Ci with u Vj (j > i) and w Vi, add u v’ if it has not yet beencreated as an inherited arc. Such an arc is labeled with andreferred to as a transitive arc (-arc for short).

d j’ and g j’ are two -arcs since j is reachable respectively from d and g through e, a node in V1.

b e i’ j’ k’h

d g l

V1’:

V2:

b e h

d g l

c f i j a k

alternating arcs - If there exist w Vi-1’ such that v is connectedto w through an alternating path, and u Vj (j > i) such thatone of the two conditions holds:- u w E, or- there is a node w’ Vi such that u w’ E and w’ w Ci,

add u v’ if it has not yet been created as an inherited or atransitive arc. We label such an arc with and call it analternating arc (-arc for short).


an alternating path: f - e - c - b - i and f is reachable from d through a node e in V1.

b e i’ j’ k’h

d g l

V1’:

V2: d i’ and g i’ are two-arcs. We join d and i’since there is a node f thatis connected to i through

an alternating path: f - e - c - b - i and f is reachable from d through a node e in V1.

b e i’ j’ k’h

d g l

V1’:

V2: d i’ and g i’ are two-arcs. We join d and i’since there is a node f thatis connected to i through


b e h

d g l

c f i j a k

b e h

d g l

c f i j a k

Combine M1 and M2 to form a set of 6 chains:


b e h

d l g

c f a i j k

i’ j’ k’

b e i’ j’h

d g l

V1’:

V2:

b e i’ j’ k’k’

d l gM2:

b e h

c f i j a kV0:

V1: M1: e h

c f i j a k

b

h

By solving the virtual nodes, we get a set of 6 chains, which do not contain virtual nodes.


b e h

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l

c f i j a k

b e h

d l g

c f a i j k

i’ j’ k’

Resolution of virtual nodes


• For resolving virtual nodes, we associate each -arcand -arc with a data structure to facilitate thevirtual node resolution.• The data structure for a -arc e = u v’, denoted

by (e), is a pair of the form <i, {w1, ..., wk}>,where- i is the level number, to which v’ is added,- each wj is connected to v (= s(v’)) through an

alternating path, and- for each wj we have u wj E, or there is a node

w’ Vi such that u w’ E and w’ w Ci.


Structure of (e):The data structure for the -arc d i’ should be

(d i’) = <1, {c, f}>for the following reason:- i’ is a virtual node added to V1;- c is connected to i through an alternating path c - b - i, and f

is also connected to i through an alternating path:f - e - c - b - i; and

- d c E, and d e E, e f E.

b e i’ j’ k’h

d g l

V1’:

V2:

b e h

d g l

c f i j a k(g i’) is also <1, {c, f}> for the same reason.


Structure of α(e)

Let v’ be a virtual node created for v, added to Vi.Let e = u v’ be an -arc. Then, there must existw1, ..., wk (k 1) in Vi such that for each wj

(1 j k) u wj E and wj v Ci with u Vl

(for some l > i).

v’w2

u

Vi:

Vl:

w1

wk

v


For α(e), we distinguish among three cases:

i) There exists wj1, ..., wjp (1 j1 jp k) such that for each wjq (1 q p) wjq v is an arc in the original graph or anunlabeled virtual arc.

ii) Each wj v is neither an arc in the original graph nor anunlabeled virtual arc. However, There exists wj1, ..., wja

(1 j1 ja k) such that for each (1 b a), wjb v isan -arc.

iii) Each wj v is a -arc.

In case (i), the data structure is set to be (e) = <I, >.In case (ii) and (iii), the data structure is set to be (e) = <II,{wj1, ..., wja}> and (e) = <III, {w1, ..., wk}>, respectively.

Example


b e i’ j’ k’h

d g l

V1’:

V2:

b e h

d g l

c f i j a k

(d j’) = <I, >(g j’) = <I, >

Note that e j is an edgein the original graph.

If e j were an -edge, then (d j’)and (g j’) would be <II, e>.If e j were an -edge, then (d j’)and (g j’) would be <III, e>.

Consider G(Vh, Vh-1’; Ch). Relative to the found maximummatching Mh of it, all the virtual nodes in Vh-1’ can be classifiedinto four groups:

• uncovered (free nodes),• unlabeled-covered,• transitive-covered, and• alternating-covered.


A virtual node is unlabeled-covered, transitive-covered, oralternating-covered if it is covered by an edge in Mh, whichcorresponds to an unlabeled, transitive, or alternating arc,respectively.


• Each uncovered virtual node can be simplyremoved.

• But for an unlabeled-covered virtual node v, we willconnect its parent u and its child s(v) (along thecorresponding chain) and then remove v. The newarc u s(v) is handled as unlabeled.

u

v

s(u)

virtual node

unlabeled u

s(u)

unlabeled


• Resolution of transitive-covered virtual nodes

For u s(v), we need to do two tasks:-Determine whether it is an -arc, a -arc, or

unlabeled.- If it is labeled, figure out the data structure for it.To this end, we will do the following operations:

u

v

s(u)

virtual node

u

s(u)

or ?

1. Let a(u v) be <, W>, where {I, II, III} and W isa subset of Vh-1.

2. Remove v.3. If = I, create an unlabeled arc u s(v). 4. If = II, create an -arc u s(v). Let W = {w1, ..., wk}.

Let (wj s(v)) = <j, Wj> (j = 1, ..., k). If there exists anj such that j = I, set (u s(v)) = <I, >.Otherwise, X := j =IIWj, Y := j =IIIWj.i) If X , set (u s(v)) to be <II, X>.ii) If X = , set (u s(v)) to be <III, Y>.

5. If = III, find W’ W such that for each x W’ (x s(v))is of the form <h - 2, Wx>, where Wx is a subset of nodesin Vh-3’.i) If W’ , create a -arc u s(v) and set

(u s(v)) = <h - 2, >.ii) If W’ = , create an -arc u s(v) and set

(u s(v)) = (u v).



• Resolution of alternating-covered virtual nodes

Definition 4 (alternating graph) Let Mi be a maximum matchingof G(Vi, Vi-1’; Ci). The alternating graph with respect to Mi is adirected graph with the following sets of nodes and arcs:

V(Gi) =Vi Vi-1’, and

E(Gi) ={u v | u Vi-1’, v Vi, and (u, v) Mi}

{v u | u Vi-1’, v Vi, and (u, v) Ci\Mi}.

b e h

c f i j a kV0:

V1:

f e c b i

j a h k

In order to resolve all the alternating-covered virtual nodes in Vi’, we combine Gi and Gi+1 by connecting each of these nodes in Vi’ in Gi+1 to some nodes in Vi-1’ in Gi as follows:

Let v1, ..., vk be all those alternating-covered virtual nodes in Vi’ in with b(uj vj) = <i, Wj>, where uj is the parent of vj along the corresponding chain and Wj is a set of nodes in Vi-1’ in Gi.

For each vj, connect vj to every node in Wj (j = 1, ..., k).

We denote such a combined graph by Gi+1 Gi.



b e i’ j’ k’h

d g l

V1’:

V2:i’

j’k’

ge

h

d

l

f e c b i

j a h k

b

f e c b i

i’

f ej

i’

c b i

i’

b e h

c f i j a kV0:

V1:

• Let v1 v2 ... vk be a found path.Transfer the arcs on the path.

• If vk is a node in Gi+1, we simply remove thethe corresponding virtual node v1.

• If vk is a node in Gi, connect the parent ofv1 along the corresponding chain to v2.Remove v1.



Example

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l

c f i j a k

b e h

d l g

c f a i j k

i’ j’ k’

f e c b i

i’

b e h

d g l

c f i j a k

b e h

d g l

c f i j a k

Conclusion

• Effiecent Algorithm for the poset decomposition

- Time complexity: O(n2), where is the size of a maximum anti-chain.

- Space complexity: O(n2).

- Graph stratification and Virtual nodes.

- Hopcropt-Karp’s algorithm for finding a maximum matching in a bipartite graph.

• The algorithm can be easily modified to a 0-1 network flow algorithm by defining a chain to be a path.

It will requires O(n2) time and O(n2) space, better than the state-of-the-art 0-1 flow algorithms.

Thank you.

On the Decomposition of Posets

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On the Decomposition of Posets