1. Graph Representation Adjacency list representation of G = (V, E)
An array of V lists, one for each vertex in V
Each list Adj[u] contains all the vertices v such that there is an edge between u and v
Adj[u] contains the vertices adjacent to u (in arbitrary order)
Can be used for both directed and undirected graphs 1 2 5 / 1 2 5 4 3 2 1 5 3 4 / 3 2 4 4 2 5 3 / 5 4 1 2
Undirected graph

2. Properties of Adjacency-List Representation
Sum of the lengths of all the adjacency lists
Directed graph:
Edge (u, v) appears only once in u’s list
Undirected graph:
u and v appear in each other’s adjacency lists: edge (u, v) appears twice 1 2 3 4
Directed graph
E  1 2 5 4 3
2 E 
Undirected graph

3. Properties of Adjacency-List Representation
Memory required
(V + E)
Preferred when
the graph is sparse: E  << V 2
Disadvantage
no quick way to determine whether there is an edge between node u and v
Time to list all vertices adjacent to u:
(degree(u))
Time to determine if (u, v)  E:
O(degree(u)) 1 2 5 4 3
Undirected graph 1 2 3 4
Directed graph

4. Graph Representation Adjacency matrix representation of G = (V, E)
Assume vertices are numbered 1, 2, … V 
The representation consists of a matrix A V x V :
aij = if (i, j)  E
0 otherwise 1 2 3 4 5
Matrix A is
symmetric:
aij = aji
A = AT 1 1 1 2 5 4 3 2 1 3 1 4 1
Undirected graph 5 1

5. Properties of Adjacency Matrix Representation
Memory required
(V2), independent on the number of edges in G
Preferred when
The graph is dense E is close to V 2
We need to quickly determine if there is an edge between two vertices
Time to list all vertices adjacent to u:
(V)
Time to determine if (u, v)  E:
(1)

6. Weighted Graphs
Weighted graphs = graphs for which each edge has an associated weight w(u, v)
w: E  R, weight function
Storing the weights of a graph
Adjacency list:
Store w(u,v) along with vertex v in u’s adjacency list
Adjacency matrix:
Store w(u, v) at location (u, v) in the matrix

7. Searching in a Graph
Graph searching = systematically follow the edges of the graph so as to visit the vertices of the graph
Two basic graph searching algorithms:
Breadth-first search
Depth-first search
The difference between them is in the order in which they explore the unvisited edges of the graph
Graph algorithms are typically elaborations of the basic graph-searching algorithms

8. Breadth-First Search (BFS)
Input:
A graph G = (V, E) (directed or undirected)
A source vertex s  V
Goal:
Explore the edges of G to “discover” every vertex reachable from s, taking the ones closest to s first
Output:
d[v] = distance (smallest # of edges) from s to v, for all v  V
A “breadth-first tree” rooted at s that contains all reachable vertices

9. Breadth-First Search (cont.)
Discover vertices in increasing order of distance from the source s – search in breadth not depth
Find all vertices at 1 edge from s, then all vertices at 2 edges from s, and so on 1 2 5 4 3 11 12 6 7 9

10. Breadth-First Search (cont.)
Keeping track of progress:
Color each vertex in either white, gray or black
Initially, all vertices are white
When being discovered a vertex becomes gray
After discovering all its adjacent vertices the node becomes black
Use FIFO queue Q to maintain the set of gray vertices
source 1 2 5 4 3 1 2 5 4 3 1 2 5 4 3

11. Breadth-First Tree BFS constructs a breadth-first tree
Initially contains the root (source vertex s)
When vertex v is discovered while scanning the adjacency list of a vertex u  vertex v and edge (u, v) are added to the tree
u is the predecessor (parent) of v in the breadth-first tree
A vertex is discovered only once  it has at most one parent
source 1 2 5 4 3

12. BFS Additional Data Structures
G = (V, E) represented using adjacency lists
color[u] – the color of the vertex for all u  V
[u] – predecessor of u
If u = s (root) or node u has not yet been discovered  [u] = NIL
d[u] – the distance from the source s to vertex u
Use a FIFO queue Q to maintain the set of gray vertices
source
d=1
=1 1 2 5 4 3
d=2
=2
d=1
=1
d=2
=5

13. BFS(G, s) for each u  V[G] - {s} do color[u]  WHITE d[u] ← x y
for each u  V[G] - {s}
do color[u]  WHITE
d[u] ← 
[u] = NIL
color[s]  GRAY
d[s] ← 0
[s] = NIL
Q  
Q ← ENQUEUE(Q, s)  r s t u v w x y  r s t u v w x y
Q: s

14. BFS(V, E, s) while Q   do u ← DEQUEUE(Q) for each v  Adj[u] r s t u v w x y
while Q  
do u ← DEQUEUE(Q)
for each v  Adj[u]
do if color[v] = WHITE
then color[v] ← GRAY
d[v] ← d[u] + 1
[v] = u
ENQUEUE(Q, v)
color[u]  BLACK
Q: s  1 r s t u v w x y
Q: w 1  r s t u v w x y
Q: w, r

15. Example  r s t u v w x y 1  r s t u v w x y v w x y 1 2  r s t ur s t u v w x y 1  r s t u v w x y v w x y 1 2  r s t u
Q: s
Q: w, r
Q: r, t, x 1 2  r s t u v w x y 1 2 3  r s t u v w x y 1 2 3 r s t u v w x y
Q: t, x, v
Q: x, v, u
Q: v, u, y 1 2 3 r s t u v w x y 1 2 3 r s t u v w x y r s t u 1 2 3 v w x y
Q: u, y
Q: y
Q: 
COSC3101A

16. Analysis of BFS for each u  V - {s} do color[u]  WHITE d[u] ← 
[u] = NIL
color[s]  GRAY
d[s] ← 0
[s] = NIL
Q  
Q ← ENQUEUE(Q, s)
O(V)
(1)

17. Analysis of BFS Total running time for BFS = O(V + E) while Q  
do u ← DEQUEUE(Q)
for each v  Adj[u]
do if color[v] = WHITE
then color[v] = GRAY
d[v] ← d[u] + 1
[v] = u
ENQUEUE(Q, v)
color[u]  BLACK
Scan Adj[u] for all vertices in the graph
Each vertex is scanned only once, when the vertex is dequeued
Sum of lengths of all adjacency lists = (E)
Scanning operations: O(E)
(1)
(1)
Total running time for BFS = O(V + E)

18. Shortest Paths Property
BFS finds the shortest-path distance from the source vertex s  V to each node in the graph
Shortest-path distance = (s, u)
Minimum number of edges in any path from s to u
source r s t u 1 2 3 v w x y

19. Depth-First Search Input: Goal: Output:
G = (V, E) (No source vertex given!)
Goal:
Explore the edges of G to “discover” every vertex in V starting at the most current visited node
Search may be repeated from multiple sources
Output:
2 timestamps on each vertex:
d[v] = discovery time
f[v] = finishing time (done with examining v’s adjacency list)
Depth-first forest 1 2 5 4 3

20. Depth-First Search Search “deeper” in the graph whenever possible
Edges are explored out of the most recently discovered vertex v that still has unexplored edges 1 2 5 4 3
After all edges of v have been explored, the search “backtracks” from the parent of v
The process continues until all vertices reachable from the original source have been discovered
If undiscovered vertices remain, choose one of them as a new source and repeat the search from that vertex
DFS creates a “depth-first forest”

21. DFS Additional Data Structures
Global variable: time-step
Incremented when nodes are discovered/finished
color[u] – similar to BFS
White before discovery, gray while processing and black when finished processing
[u] – predecessor of u
d[u], f[u] – discovery and finish times
GRAY
WHITE
BLACK 2V
d[u]
f[u]
1 ≤ d[u] < f [u] ≤ 2 |V|

22. DFS(G) for each u  V[G] do color[u] ← WHITE [u] ← NIL time ← 0
do if color[u] = WHITE
then DFS-VISIT(u)
Every time DFS-VISIT(u) is called, u becomes the root of a new tree in the depth-first forest u v w x y z

23. DFS-VISIT(u) color[u] ← GRAY time ← time+1 d[u] ← time
for each v  Adj[u]
do if color[v] = WHITE
then [v] ← u
DFS-VISIT(v)
color[u] ← BLACK
time ← time + 1
f[u] ← time u v w x y z
time = 1 1/ u v w x y z 1/ 2/ u v w x y z

24. Example u v w x y z u v w x y z u v w x y z u v w x y z u v w x y z u1/ u v w x y z 1/ 2/ u v w x y z 1/ 2/ 3/ u v w x y z 1/ 2/ 4/ 3/ u v w x y z 1/ 2/ 4/ 3/ u v w x y z B 1/ 2/
4/5 3/ u v w x y z B 1/ 2/
4/5
3/6 u v w x y z B 1/
2/7
4/5
3/6 u v w x y z B 1/
2/7
4/5
3/6 u v w x y z B F

25. Example (cont.) The results of DFS may depend on:
1/8
2/7
4/5
3/6 u v w x y z B F
1/8
2/7 9/
4/5
3/6 u v w x y z B F
1/8
2/7 9/
4/5
3/6 u v w x y z B F C
1/8
2/7 9/
4/5
3/6
10/ u v w x y z B F C
1/8
2/7 9/
4/5
3/6
10/ u v w x y z B F C
1/8
2/7 9/
4/5
3/6
10/11 u v w x y z B F C
1/8
2/7
9/12
4/5
3/6
10/11 u v w x y z B F C
The results of DFS may depend on:
The order in which nodes are explored in procedure DFS
The order in which the neighbors of a vertex are visited in DFS-VISIT

26. Edge Classification Tree edge (reaches a WHITE vertex):
(u, v) is a tree edge if v was first discovered by exploring edge (u, v)
Back edge (reaches a GRAY vertex):
(u, v), connecting a vertex u to an ancestor v in a depth first tree
Self loops (in directed graphs) are also back edges 1/ u v w x y z 1/ 2/ 4/ 3/ u v w x y z B

27. Edge Classification
Forward edge (reaches a BLACK vertex & d[u] < d[v]):
Non-tree edges (u, v) that connect a vertex u to a descendant v in a depth first tree
Cross edge (reaches a BLACK vertex & d[u] > d[v]):
Can go between vertices in same depth-first tree (as long as there is no ancestor / descendant relation) or between different depth-first trees 1/
2/7
4/5
3/6 u v w x y z B F
1/8
2/7 9/
4/5
3/6 u v w x y z B F C

28. Analysis of DFS(G) for each u  V[G] do color[u] ← WHITE [u] ← NIL
time ← 0
do if color[u] = WHITE
then DFS-VISIT(u)
(V)
(V) – exclusive of time for
DFS-VISIT

29. Analysis of DFS-VISIT(u)
color[u] ← GRAY
time ← time+1
d[u] ← time
for each v  Adj[u]
do if color[v] = WHITE
then [v] ← u
DFS-VISIT(v)
color[u] ← BLACK
time ← time + 1
f[u] ← time
DFS-VISIT is called exactly once for each vertex
Each loop takes
|Adj[v]|
Total: ΣvV |Adj[v]| + (V) =
(V + E)
(E)

30. Properties of DFS
u = [v]  DFS-VISIT(v) was called during a search of u’s adjacency list
Vertex v is a descendant of vertex u in the depth first forest  v is discovered during the time in which u is gray 1/ 2/ 3/ u v w x y z

31. Parenthesis Theorem
In any DFS of a graph G, for all u, v, exactly one of the following holds:
[d[u], f[u]] and [d[v], f[v]] are disjoint, and neither of u and v is a descendant of the other
[d[v], f[v]] is entirely within [d[u], f[u]] and v is a descendant of u
[d[u], f[u]] is entirely within [d[v], f[v]] and u is a descendant of v y z s t
3/6
2/9
1/10
11/16
4/5
7/8
12/13
14/15 x w v u s t z v u y w x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 (s (z (y (x x) y) (w w) z) s) (t (v u) (u u) t)
Well-formed expression: parenthesis are
properly nested

32. Other Properties of DFS
Corollary
Vertex v is a proper descendant of u
 d[u] < d[v] < f[v] < f[u]
Theorem (White-path Theorem)
In a depth-first forest of a graph G, vertex v is a descendant of u if and only if at time d[u], there is a path u  v consisting of only white vertices.
1/8
2/7
9/12
4/5
3/6
10/11 u v B F C 1/ 2/ u v

33. Topological Sort
Topological sort of a directed acyclic graph G = (V, E): a linear order of vertices such that if there exists an edge (u, v), then u appears before v in the ordering.
Directed acyclic graphs (DAGs)
Used to represent precedence of events or processes that have a partial order
a before b b before c
b before c a before c
What about
a and b?
a before c
Topological sort helps us establish a total order

34. Topological Sort TOPOLOGICAL-SORT(V, E) Running time: (V + E)
Call DFS(V, E) to compute finishing times f[v] for each vertex v
When each vertex is finished, insert it onto the front of a linked list
Return the linked list of vertices
undershorts
11/ 16
17/ 18
socks
12/ 15
pants
shoes
13/ 14
shirt 1/ 8 6/ 7
belt
watch 9/ 10
tie 2/ 5
jacket 3/ 4
socks
undershorts
pants
shoes
watch
shirt
belt
tie
jacket
Running time: (V + E)

35. Topological Sort undershorts 11/ 16 17/ 18 socks Topological sort:
an ordering of vertices along a
horizontal line so that all directed
edges go from left to right.
12/ 15
pants
shoes
13/ 14
shirt 1/ 8 6/ 7
belt
watch 9/ 10
tie 2/ 5
jacket 3/ 4
socks
undershorts
pants
shoes
watch
shirt
belt
tie
jacket

36. Lemma A directed graph is acyclic  a DFS on G yields no back edges.
Proof:
“”: acyclic  no back edge
Assume back edge  prove cycle
Assume there is a back edge (u, v)
v is an ancestor of u
 there is a path from v to u in G (v  u)
 v  u + the back edge (u, v) yield a cycle v u
(u, v)

37. Lemma A directed graph is acyclic  a DFS on G yields no back edges.
Proof:
“”: no back edge  acyclic
Assume cycle  prove back edge
Suppose G contains cycle c
Let v be the first vertex discovered in c, and (u, v) be the preceding edge in c
At time d[v], vertices of c form a white path v  u
u is descendant of v in depth-first forest (by white-path theorem)
 (u, v) is a back edge v
(u, v) u

38. Strongly Connected Components
Given directed graph G = (V, E):
A strongly connected component (SCC) of G is a maximal set of vertices C  V such that for every pair of vertices u, v  C, we have both u  v and v  u.

39. The Transpose of a Graph
GT = transpose of G
GT is G with all edges reversed
GT = (V, ET), ET = {(u, v) : (v, u)  E}
If using adjacency lists: we can create GT in (V + E) time 1 2 5 4 3 1 2 5 4 3

40. Finding the SCC Observation: G and GT have the same SCC’s
u and v are reachable from each other in G  they are reachable from each other in GT
Idea for computing the SCC of a DAG G = (V, E):
Make two depth first searches: one on G and one on GT 1 2 5 4 3 1 2 5 4 3

41. Example a b c d e f g h DFS on the initial graph G 13/ 14 11/ 16 1/ 108/ 9 b 16 e 15 a 14 c 10 d 9 g 7 h 6 f 4
12/ 15 3/ 4 2/ 7 5/ 6 a b c d e f g h
DFS on GT:
start at b: visit a, e
start at c: visit d
start at g: visit f
start at h
Strongly connected components: C1 = {a, b, e}, C2 = {c, d}, C3 = {f, g}, C4 = {h}

42. Component Graph The component graph is a DAGb c d e f g h
a b e
c d
f g h
The component graph GSCC = (VSCC, ESCC):
VSCC = {v1, v2, …, vk}, where vi corresponds to each strongly connected component Ci
There is an edge (vi, vj)  ESCC if G contains a directed edge (x, y) for some x  Ci and y  Cj
The component graph is a DAG

43. Lemma 1 Let C and C’ be distinct SCC’s in G
Let u, v  C, and u’, v’  C’
Suppose there is a path u  u’ in G
Then there cannot also be a path v’  v in G.
Proof
Suppose there is a path v’  v
There exists u  u’  v’
There exists v’  v  u
u and v’ are reachable from each other, so they are not in separate SCC’s: contradiction! C C’ u u’ v v’

44. Notations
Extend notation for d (starting time) and f (finishing time) to sets of vertices U  V:
d(U) = minuU { d[u] } (earliest discovery time)
f(U) = maxuU { f[u] } (latest finishing time) C1 C2 a b c d
d(C2)
f(C2) =1
=10
d(C1)
f(C1)
=11
=16
13/ 14
11/ 16 1/ 10 8/ 9
12/ 15 3/ 4 2/ 7 5/ 6 e f g h C3 C4
d(C3)
f(C3) =2 =7
d(C4)
f(C4) =5 =6

45. Lemma 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  C and v  C’ then f(C) > f(C’).
Consider C1 and C2, connected by edge (b, c) C1 C2 a b c d
d(C2)
f(C2) =1
=10
d(C1)
f(C1)
=11
=16
13/ 14
11/ 16 1/ 10 8/ 9
12/ 15 3/ 4 2/ 7 5/ 6 e f g h C3 C4
d(C3)
f(C3) =3 =7
d(C4)
f(C4) =5 =6

46. 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  C and v  C’ then f(C) < f(C’).
Consider C2 and C1, connected by edge (c, b)
C1 = C’
C2 = C
Since (c, b)  ET  (b, c)  E
From previous lemma:
f(C1) > f(C2)
f(C’) > f(C) a b c d e f g h C3 C4

47. Discussion f(C) < f(C’)
Each edge in GT that goes between different components goes from a component with an earlier finish time (in the DFS) to one with a later finish time
C1 = C’
C2 = C a b c d e f g h C3 C4

48. Why does SCC Work?
When we do the second DFS, on GT, we start with a component C such that f(C) is maximum (b, in our case)
We start from b and visit all vertices in C1
From corollary: f(C) > f(C’) for all C  C’  there are no edges from C to any other SCCs in GT
 DFS will visit only vertices in C1
 The depth-first tree rooted at b contains exactly the vertices of C1 C1 C2 a b c d e f g h f 4 h 6 g 7 d 9 c 10 a 14 e 15 b 16 C3 C4

49. Why does SCC Work? (cont.)
The next root chosen in the second DFS is in SCC C2 such that f(C) is maximum over all SCC’s other than C1
DFS visits all vertices in C2
the only edges out of C2 go to C1, which we’ve already visited
 The only tree edges will be to vertices in C2
Each time we choose a new root it can reach only:
vertices in its own component
vertices in components already visited C1 C2 a b c d e f g h f 4 h 6 g 7 d 9 c 10 a 14 e 15 b 16 C3 C4