1. On the Graph Decomposition
Yangjun Chen and Yibin Chen
Dept. Applied Computer Science,
University of Winnipeg
515 Portage Ave.
Winnipeg, Manitoba, Canada R3B 2E9

2. Outline Motivation Graph stratification Main algorithm
- Bipartite graphs and virtual nodes
- Chain generation
- Resolving virtual nodes
Summary

3. Motivation Graph decomposition into node-disjoint chains
Given a directed acyclic graph G, decompose it into a minimum set of node-disjoint chains, which cover all the nodes of G. For any two nodes u and v on a chain, if u is above v then there is a path from u to v in G.
Application
- Decomposition of a partially ordered sets (posets) into a minimum set of chains
- Compress transitive closures of graphs

4. Motivation a1 a2 a1 a2 a1 a2 a3 a3 a3 a4 a4 a4 a5 a6 a5 a6 a5 a6

5. Motivation Compression of transitive closures O(n2) O(n)
index
Index sequence
1 2 a3
(1, 1) a1
(2, 1) a2
(1, 1)(2, 2)
(1, 2)(2, 1) a4
(1, 1) a3
(2, 2) a4
(1, 2)(2, 2)
(1, 3)(2, 2) a5 a6
(1, 3) a5 a6
(2, 3)
(1, 3)
(2, 3)
O(n2)
O(n)
 - number of chains

6. Main idea of graph decomposition
Stratifying a graph into a series of bipartite graphs: G1, …, Gh.
Finding a maximum matching Mi for each Gi. (In this process, virtual nodes may be introduced.)
Combining all Mi’s to get a set of chains, denoted as M1  …  Gh, which may contain virtual nodes.
Removing the virtual nodes to get the final result.

7. Graph stratification
Definition 1 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.  q r s
V4: q r s n o p m
V3: n o p m j k l t i
V2: j k l t i g h f x
V1: g x h f b y c d e z a b
V0: y c d e z a

8. Graph stratification Dividing a graph into
a series of bipartite graphs:
V4: q r s
V3: n o p m
V3: n o p m e a b y d f g h i j k l m n o p q r s t z x c
V2: j k l t i
V2: j k l t i
V1: g x h f
V1: g f x h
V0: b y c d e z a

9. Virtual nodes Find a maximum matching for each bipartite graph
and then combine them together, by which ‘virtual
node’ may be introduced.
Definition 2 (virtual nodes) Let G(V, E) be a DAG, divided into V0, ..., Vh (i.e., V = V0  ...  Vh). Let Mi be a maximal matching of B(Vi, Vi-1′; Ci) for i = 1, …, h. For each free node v in Vi-1′ with respect to Mi, a virtual node v′ created for v is a new node added to Vi (1  i  h - 1), denoted as v = s(v′). 

10. Virtual Nodes
Definition 2 (virtual nodes) Let G(V, E) be a DAG, divided into V0, ..., Vh (i.e., V = V0  ...  Vh). Let Mi be a maximal matching of B(Vi, Vi-1′; Ci) for i = 1, …, h. For each free node v in Vi-1′ with respect to Mi, a virtual node v′ created for v is a new node added to Vi (1  i  h - 1), denoted as v = s(v′).  q r s
V1:
V0: e a b y d g h z x c
V4:
V3: n o p m
V2: j k l t i
M1: g x h
V1’: g x c’ h e’ z’ f b y c d e z a
label(t  e’) = 0
label(j  e’) = 1
label(t  z’) = 0
label(k  c’) = 1
label(i  e’) = 0
label(k  e’) = 1
label(i  z’) = 0
label(l  c’) = 1
label(j  c’) = 1
label(l  e’) = 1

11. Virtual Arcs
inherited arcs - If there is u  Vj (j > i) such that u  v  E, add u  v′, referred to as an inherited arc.
transitive arcs - If there exist u  Vj (j > i) and w  Vi such that u  w  E and w  v  Ci, add u  v′ if it has not been created as an inherited arc, referred to as a transitive arc.
alternating arcs of the first kind - If there exists a covered node w  Vi-1′ (relative to Mi) such that one of v’s parents is connected to w through an -segment in B(Vi, Vi-1′; Ci), and u  Vj (j > i) such that one 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 been created as an inherited or a transitive arc.

12. Chain Generation V1: V0: e a b y d g h z x c f g h i j k l t c’ e’ z’
M1: e a b y d g h z x c
M2: j k l t i g x c’ h e’ z’ f q r s
V4: q r s
M3: m n o p i j k l t
M4:
V3: n o p m o p
V2: j k l t i
V1’: g x c’ h e’ z’ f

13. Chain Generation q r s M4: o p q r s M3: m n o p i j k l t n o p m M2:x c’ h e’ z’ f g x c’ h e’ z’ f
M1: e a b y d g h z x c b y c d e z a

14. Chain Generation m n o p q r s f g h i j k l t x a y c d e z b q r s n

15. Chain Generation m n o p q r s f g h i j k l t x a y c d e z b q r s n

16. Chain Generation = V( ) = T  S, and
Definition 3 (alternating graph) Let B(T, S; E) be a bipartite graph. Let M be a
matching of B(T, S; E). The alternating graph with respect to M is a directed
graph with the following sets of nodes and arcs:
= V( ) = T  S, and
= E( ) = {u  v | u  S, v  T, and (u, v)  M} 
{v  u | u  S, v  T, and (u, v)  E\M}. 
V1: g x h
V1: g x h
V0:
V0: b y c d e z a b y c d e z a
M1: g x h b y c d e z a

17. Chain Generation
Next, we will combine two consecutive alternating graphs
= (Vi′, Vi-1′; Ci) and = (Vi+1, Vi′; Ci+1), denoted as
 , by connecting each node in Vi+1 to all its reachable
nodes in Vi-1′. j k l t i
V2:
V1: g x h f g x h f
V1:
V0: b y c d e z a

18. Chain Generation f g h i j k l t x e a y c d z b j k l t i V2: V1: g x
A maximum set of node disjoint
paths with each starting from a
free node u relative to Mi+1 in
Vi+1, and ending at a free node
v relative to Mi in Vi-1′,

19. Chain Generation
What we want is to find a maximum set S of node-disjoint
paths in  , each starting from a free node u relative to Mi+1
in Vi+1, and ending at a free node v relative to Mi in Vi-1′. Let P
be such a path with u and v being the starting and ending nodes.
We will create a virtual node v´ for v, connect it u.
However, for each free node (in Vi-1′) not appearing on such a
path, its virtual node will be added to Vi+1, for which only
inherited and transitive arcs, as well as a new kind of virtual
arcs, called alternating arcs of the second kind will be created.

20. Chain Generation
Alternating arc of the second kind – Let v´ be a virtual node
Created for v in Vi-1′ and added to Vi+1. If there exist a free
node w  Vi-1′ (relative to Mi) and a node u  Vj (j > i) such that
one of v’s parents is connected to w through a -segment in
B(Vi′, Vi-1′; Ci), satisfying one of the following two conditions:
- u  w  E, or
there is an alternating path in (Vi′, Vi-1′; Ci), which does not
go through any node in S, but connects w to a node x  Vi-1′
such that x is reachable from u,
add u  v′ if it has not been created as an inherited or a
transitive arc. label(u  v′) is set to be i, same as an
alternating arc incident to a virtual node added to Vi′.

21. Chain Generation M2: j k c′ l t i q r s V1: V0: V4: V2: V3: e a b y dz c f g h x c′ z′ m n o p i j k l t e′
c′′ g x h e′ z′ f n o p m
M3: j k c′ l t i n o p m g x h e′ z′ f j k c′ l t i b y c d e z a
M4: q r s
label(l  e’) = 1
label(m  c’) = 1
label(t  z) = 0
label(q  c’) = 1
label(q  c) = 1 n o p m

22. Chain Generation q r s e a b y d q r s z c f g h x c′ z′ m n o p i j kl t e′
c′′ q r s
V4: n o
c′′ p m n o
c′′
V3: p m j k c′ l t i q r s g h
M4: x e′ z′ f n o
c′′ p m b y c d e z a

23. Virtual Node Resolution
After the chain generation, the next step is to resolve
(or say, to remove) virtual nodes from chains. For this purpose,
we will work top-down along the chains. Two steps will be
carried out:
Remove virtual nodes, and at the same time connect some
nodes according to the connectivity represented by them, and
Establish new connections between free nodes by transferring
edges along alternating paths within a bipartite graph or cross
more than one bipartite graph.

24. Virtual Node Resolution
When we try to remove virtual nodes v with label(u  v) = i,
all the virtual nodes with higher labels must have been eliminated.
In this step, the following operations will be conducted.
Let v be a virtual node in Vi′. If v does not have a parent along
the corresponding chain, it will be simply removed.
If v has a parent u along a chain with label(u  v) = 0,
remove v and connect u to s(v). label(u  s(v)) is set to 0.
Let level(s(v)) = i. Connect u to each reachable node in Vi.

25. Virtual Node Resolution
If v has a parent u along a chain with label(u  v) = i,
remove v and connect u to each reachable node in Vi-1.
Construct a combined graph in a way similar to the chain
generation, involving the corresponding bipartite graphs,
where the direction of each arc corresponding to an edge
belonging to a maximum matching is reversed.

26. Virtual Node Resolutionq r s q r s n o p m n o p m j k l t i j k l t i g x h g x h z′ f e′ f b y c d e z a b y c d e z a

27. Virtual Node Resolutionq r s q r s n o p m n o p m j k l t i j k l t i j k l t i g x h f g x h f b y c d e z a g x h f b y c d e z a

28. Virtual Node Resolutionq r s q r s n o
c′′ p m n o p m j k c′ l t i j k l t i g x h e′ z′ g h f x f b y c d e z a b y c d e z a

29. Summary A efficient algorithm for graph decomposition
- Graph stratification
- Algorithm for finding maximum matching
for bipartite graphs
- Virtual nodes
Computational complexities
time: O(max{ n, m})
space overhead: O(n)

30. Thanks you.