Strongly Connected Component

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CS 304: Design and Analysis of Algorithm

Strongly Connected Component



L ast Class¶s Topic

DF ST opological Sort

P roblems:D etect cycle in an undirected graphD etect cycle in a directed graphHow many paths are there from ³s´ to ³t´ in adirected acyclic graph?

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Connectivity

Connected GraphIn an undirected graph G, two vertices u and v are calledconnected if G contains a path from u to v. Otherwise, they

are called disconnected.A directed graph is called connected if every pair of distinct vertices in the graph is connected.

Connected Components

A connected component is a maximal connected subgraphof G. Each vertex belongs to exactly one connectedcomponent, as does each edge.



Connectivity (cont.)

W eakly Connected GraphA directed graph is called w eakly connected if replacing all of its directed edges with undirectededges produces a connected (undirected) graph.

Strongly Connected GraphIt is strongly connected or strong if it contains a

directed path from u to v for every pair of verticesu, v. T he strong components are the maximalstrongly connected subgraphs



Connected Components

Strongly connected graphA directed graph is called strongly connected if for every

pair of vertices u and v there is a path from u to v and a

path from v to u.Strongly Connected Components (SCC)

T he s trongly connected component s (SCC ) of a directedgraph are its maximal strongly connected subgraphs.

Here, we work withD irected unweighted graph



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Strongly Connected Components

G is strongly connected if every pair ( u, v) of vertices in G is reachable from one another.

Astrongly connected component (

SCC ) of G is a maximal set of vertices C V such that

for all u, v C , both u v and v u exist.



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DFS - Strongly ConnectedComponents

A

D

EC

FB

G

H



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DFS - Strongly ConnectedComponents

A

D

EC

FB

G

H



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Component Graph

GSCC = (V SCC , E SCC ).V SCC has one vertex for each SCC in G.

E SCC

has an edge if there¶s an edge betweenthe corresponding SCC¶s in G.GSCC for the example considered:



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Strongly Connected Components

T he tran s po s e MT of an NxN matrix M is the matrix obtainedwhen the rows become columns and the column become rows:

a b c da

bc

d

a b c da

bc

d

a

b c

d da

b c

M M T

G TG

Edge s have rever s edirection!

1

1 1

1

1

1

11



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Transpose of a Directed Graph

GT = tran s po s e of directed G.GT = (V , E T ), E T = { (u, v) : (v, u) E }.GT is G with all edges reversed.

Can create GT in (V + E ) time if usingadjacency lists.G and GT have the same SCC¶s . (u and v arereachable from each other in G if and only if reachable from each other in GT .)





Ex ample

a b c d

e f g h

1/

2/ 3/4 6 5/ 7

8/ 11/

12/

13/ 910 14

15

16

f 4

h6

g 7

d 9

c 10

a14

e15

b16

DFS on t he initi al graph G

DFS on G T:

st ar t a t b: visit a , e

st ar t a t c : visit d st ar t a t g : visit f st ar t a t h

S t r on gly c onn ec t ed c om pon e nts: C 1

= { a , b, e}, C 2

= { c , d}, C 3

= { f , g}, C 4

= { h}

a b c d

e f g h



Component Graph

T he component graph GSCC = (V SCC , ESCC ):

VSCC = {v1, v2, «, v k}, where vi corresponds to each

strongly connected component CiT here is an edge ( vi, v j ) ESCC if G contains a directededge ( x , y) for some x Ci and y C j

T he component graph is a D AG

a b c d

e f g h

a b e

c d

f g h



L emma 1

Let C and C¶ be distinct SCCs in GLet u, v C, and u·, v· C ·Suppose there is a path u @ u· in G

T hen there cannot also be a path v· @ v in G.

u

v u¶

v ¶

Proof

S u pp os e t here is a pa t h v· @ v

T here ex ists u @ u · @ v·

T here ex ists v· @ v @ u

� u a nd v· are reachable fr om each ot her , so t hey are not in s epara t e S CC ¶ s: c ont rad i c tion!

C C ¶



N otations

Extend notation for d (starting time) and f (finishingtime) to sets of vertices U V:

d(U) = min u U { d[ u] } (earliest discovery time)

f(U) = max u U { f[ u] } (latest finishing time)

a b c d

e f g h

1/

2/ 3/4 6 5/ 7

8/ 11/

12/

13/ 910 14

15

16

C 1 C 2

C 3 C 4

d( C 1 )f( C 1 )

d( C 2 )f( C 2 )

d( C 3 )

f( C 3 )

d( C 4 )

f( C 4 )

=11=16

=1=10

=2

=7

=5

=6



L emma 2

Let C and C¶ be distinct SCCs in a directed graph G =(V, E). If there is an edge (u, v) E, where u Cand v C¶ then f(C) > f(C¶).

Consider C 1 and C 2, connected by edge (b , c)a b c d

e f g h

1/

2/ 3/4 6 5/ 7

8/ 11/

12/

13/ 910 14

15

16

C 1 C 2

C 3 C 4

d( C 1 )f( C 1 )

d( C 2 )f( C 2 )

d( C 3 )

f( C 3 )

d( C 4 )

f( C 4 )

=11=16

=1=10

=2

=7

=5

=6



Corollary

Let C and C¶ be distinct SCCs in a directed graph G =(V, E). If there is an edge (u, v) ET , where u Cand v C¶ then f(C) < f(C¶).

Consider C 2 and C 1, connected by edge (c, b)C 1 = C ¶ C 2 = C

C 3 C 4

a b c d

e f g h

S ince (c, b) E T ( b , c) E

Fr om pre viou slemm a :f( C 1 ) > f( C 2 )f( C ¶ ) > f( C )f( C ) < f( C ¶ )



Corollary

Each edge in G T that goes between differentcomponents goes from a component with an earlier finish time (in the DF S) to one with a later finish time

C 1 = C ¶ C 2 = C

C 3 C 4

a b c d

e f g h



W hy does SCC W ork?

W hen we do the second DF S, on G T , we start with a component C such thatf(C) is maximum (b, in our case)W e start from b and visit all vertices in C 1

F rom corollary: f(C) > f(C¶) in G for all C { C¶ there are no edges from

C to any other SCCs in GT

DF S will visit only vertices in C 1

T he depth-first tree rooted at b contains exactly the vertices of C 1

C 1

C 2

C 3 C 4

a b c d

e f g h

f 4

h6

g 7

d 9

c 10

a14

e15

b16



W hy does SCC W ork? (cont.)

T he next root chosen in the second DF S is in SCC C 2 such that f(C) ismaximum over all SCC¶s other than C 1

DF S visits all vertices in C 2

the only edges out of C 2 go to C 1, which we¶ve already visited

T he only tree edges will be to vertices in C 2

Each time we choose a new root it can reach only:vertices in its own componentvertices in components already visited

C 1 C 2

C 3 C 4

a b c d

e f g h

f 4

h6

g 7

d 9

c 10

a14

e15

b16



R eference

B ook: Cormen ± Chapter 22 ± Section 22.5Exercise:

22.5- 1: Number of componets change?22.5-6: Minimize edge list22.5-7: Semiconnected graph

Strongly Connected Component

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