1. Chapter 22: Elementary Graph Algorithms
Overview:
Definition of a graph
Representation of graphs
adjacency list
matrix
Elementary search algorithms
breadth-first search (BFS)
depth-first search(DFS)
topological sort
strongly connected components

2. Notation Graph G(V,E) is a data structure defined by
a set of vertices V
a set of eges E
|V| = # of vertices
|E| = # of edges
Usually omit | | in asymptotic notation
Q(V, E) means Q(|V|, |E|)

3. Adj = adjacency-list representation of G(V,E) is an array of |V| lists
Adj[u] contains all vertices adjacent to vertex u
For a directed graph, edge (uv) represented by v in Adj[u].
 sum of lengths of adjacency lists = |E|
For an undirected graph, edge (u,v) appears as v in Adj[u]
and as u in Adj[v].
sum of lengths of adjacency lists = 2|E|
Memory requirement is Q(V+E)
A weighted graph has a function w(u,v) defined on the domain E.
The weight of edge (u,v) can be stored as property of vertex v in Adj[u].

4. adjacency-matrix representation:
Number the vertices 1,2,...|V|
Define |V| x |V| matrix A such that aij = 1 if (i, j)  E,
aij = 0 otherwise
For a weighted graph, set aij = w(i,j) if (i, j)  E
Disadvantage is requires Q(V2) memory regardless of |E|
Advantage is speed of determining if edge (u,v) is in graph
For an undirected graph A is symmetric
A sparse graphs mean |E| << |V|2
adjacency-list representation preferred (saves space)
A dense graph means |E| ~ |V|2
adjacency-matrix representation preferred (speed)

5. Graph representations1 1

6. Breadth-first search: Given G(V,E) and a source vertex s,
Breadth-first search does the following:
(1) finds every vertex v reachable from s
(2) calculates distance (minimum number of edges)
between s and v
(3) produces a Breadth-first tree with s as the root and
every reachable vertex as a node
The path from s to v in the Breadth-first tree corresponds to the
shortest path in G.
“Breadth-first” finds all vertices at distance k from s before
searching for any vertices at distance k+1.

7. BFS(G,s) pseudo code for each u  V(G) – {s}
do color[u]  white, d[u]  , p[u]  NIL
(Initialization: color all vertices white for “undiscovered”
set distance from s as infinite,
set predecessor in Breadth-first tree to NIL)
color[s]  gray, d[s]  0, p[s]  NIL
(initialize s as gray for “discovered” i.e. reachable from s
A gray vertex has been discovered but its adjacency list
has not been searched for links to other vertices.
After adjacency list is searched, color is changed to black)
Q  0, Enqueue(Q,s)
(A “first-in first-out” queue holds a list of the current gray vertices
Gray vertices with the smallest distance from s are processed first
Resolve ambiguity by some rule like alphabetical order)

8. BFS(G,s) algorithm continued
while Q  0
do u  Dequeue(Q)
for each v  Adj[u]
do if color[v] = white then
color[v] = gray
d[v]d[u] + 1
p[v] u
Enqueue(Q,v)
color[u]  black (adj list searched)
Runtime analysis:
initialization requires O(V).
each vertex is discovered at most once
 queue operations require O(V).
searching adjacency lists requires O(E)
 total runtime O(V+E)

9. Example of BFS p596

10. Predecessor sub-graph Gp(Vp,Ep)
Predecessor of v:
p(v) is the vertex in whose
adjacency list v was discovered
Predecessor sub-graph Gp(Vp,Ep)
Vp = {v  V : p[v]  NIL}  {s}
Ep = {(p[v],v) : v  Vp - {s}}
Predecessor sub-graph of BFS is a
single tree

11. Finding shortest path is most common application of
Breadth-first search
d(s,u) = shortest path between s and any u  V
d(s,u) =  if u is not reachable from s
Weight of a path is the sum of the weights of its edges
For an unweighted graph,
weight of path = number of edges = length of path
For weighted graph, least-weight is not necessarily shortest path

12. Theorem 22.5: On termination of BFS(G,s),
all reachable vertices have been found
(2) d[v] = d(s,v) for v  V (some may be infinite)
(3) for any reachable v  s, a shortest path includes
p[v] followed by edge (p[v],v).

13. Depth-first search: Explores all edges leaving a given vertex, v,
before “backtracking” to explore edges leaving the vertex from
which v was discovered
Continues until all vertices reachable from a given source are discovered
If the graph still contains undiscovered vertices, choose a new source
p[v] = u if v was discovered in a search of the adjacency list of u
As in BFS, vertices initialized to white, colored gray when discovered,
colored back after their adjacency list has been examined
d[v] is the timestamp field when v was discovered
f[v] is the timestamp field when v was blackened
range of timestamps is 1 to 2|V| d[v] < f[v]

14. Classification of edges:
Tree edge connects vertex to its predecessor
Back edge connects vertex to an ancestor in the same tree
(also self loops in directed graph)
Forward edge connects vertex to descendant in the same tree
Cross edge: all others
If in the same tree, then one vertex cannot be an ancestor of the other
Edges (u,v) can be classified by the color v when it is first explored
white v  tree edge
gray v  back edge
black v  forward or cross edge

15. DFS Pseudocodes: DFS(G) for each u  V (initialization)
do color[u]  white, p[u]  NIL
time  0
for each u  V (choose a new source)
do if color[u] = white then DFS-Visit(u)
DFG-Visit(u) (DFS from source u)
color[u]  gray, time  time +1, d[u]  time
(vertex u discovered)
for each v  adj[u]
do if color[v] = white
then p[v]  u, DFG-Visit(v)
color[u]  black, f[u]  time, time  time +1
(finished with vertex u)

16. DFS: Fig 22.4 p605

17. Fig 22.5 p607 Predecessor subgraph Gp = (V,Ep);
Ep = {(p[v],v) ; v  V and p[v]  NIL}
Gp contains all the vertices of G;
hence, may be multiple trees
Predecessor subgraph showing
Non-tree edges
B-edge: head () contains tail ()
F-edge: tail () contains head ()
C-edge: connects disjoint ()
Note C-edges inside trees.
Predecessor subgraph showing
“parenthesis” structure

18. Vertex v is a descendant of u in the depth-first forest of G
Properties of DFS:
Gp of DFS is a forest of trees, each of which reflects the pattern of
recursive calls to DFS-Visit (slide 15)
Pattern of discovery and finishing times has “parenthesis structure”
(slide 18)
For any 2 vertices u and v, exactly one of the following is true
(1) intervals [d[u], f[u]] and [d[v], f[v]] are disjoint and
neither u nor v is a descendent of the other
(2) [d[u], f[u]] is contained entirely in [d[v], f[v]] and
u is a descendent of v
(3) [d[v], f[v]] is contained entirely in [d[u], f[u]] and
v is a descendent of u.
Corollary of the Parenthesis Theorem:
Vertex v is a descendant of u in the depth-first forest of G
if and only if d[u] < d[v] < f[v] < f[u]

19. Theorem 22.10: Only tree and back edges occur in a DFS
of an undirected graph
In DFS of undirected graph, edge type is determined by direction of search when edge is encountered.
Consider edge(u,v) with v in u’s adjacency list.
If edge(u,v) is traversed from u to v, then v is discovered on the traversal and edge(u,v) is a tree edge.
The only remaining option for traversal of edge(u,v) is from v to u = p(v), which we call a “back” edge in an undirected graph.
Predecessor graph is a single tree

20. CptS 450 Spring 2015
[All problems are from Cormen et al, 3rd Edition]
Homework Assignment 11: due 4/15/15
1. ex p 601 Draw Gp Show all edges
2. ex p 610 Draw Gp Show all edges

21. White-Path Theorem:
Vertex v is a descendant of u in the depth-first forest of G (directed or undirected) if and only if at the time d[u] when u is discovered, v can be reached from u by a path consisting entirely of white vertices.

22. Lemma 22:11
A directed graph is acyclic if and only if a DFS yields no back edges
Proof:
If G has a back edge (u,v) then v is an ancestor of u and (u,v)
completes a cycle in G.
If G contains cycle c and v is the first vertex discovered in c, then a
white path exist to vertex u the last vertex in c
By white-path theorem, u is a decendent of v
 (u,v) must be a back edge

23. DAG = directed acyclic graph
Precedence among events: common use of DFS on DAGs
Example: precedence of events getting dressed

24. topological sort: New graph topology: all edges point left-to-right
linear ordering of vertices in a dag such that if G contains directed edge (u,v),
then u appears before v in the ordering
Pseudocode for topological sort:
Topological-Sort(G)
DFS(G) yields f[v] for all vertices
At each f[v], insert the vertex into the front of a linked list
return list
New graph topology: all edges point left-to-right
f(v) decreases left-to-right

25. Proof of the correctness of Topological-Sort(G)
Theorem 22.12:
Proof of the correctness of Topological-Sort(G)
When edge (u,v) is explored in the DFS of a dag, v must be white
or black because
if v is gray, then (u,v) is a back edge (we would be exploring the
adjacency list of v) and DFS of a dag cannot yield a back edge.
If v is white, then v is a descendant of u and f[v] < f[u] (Corollary
of the Parenthesis Theorem in slide #19)
If v is black, then f[v] has already been assigned and, since we are
still exploring the adj[u], f[v] < f[u].
Thus, for any edge(u,v) in the dag, f[v] < f[u]
in topological sort, v will be on the right of u
all edges point to the right

26. Strongly Connected Components (SCCs)
Strongly connected components (SCCs) of directed graph G(V,E)
are a set of vertices VSCC  V such that for every pair of vertices
u and v, u ~> v and v ~> u
(u reachable from v and v reachable from u)
Component graph GSCC = (VSCC, ESCC)
Let C1, C2, ..., Ck denote the SCCs of directed graph G
VSCC = {v1, v2, ..., vk} contains a vertex from each SCC
edge (vi,vj)  ESCC if G contains edge (x,y) for some x  Ci and y  Cj
(i.e. edges the component graph connect the SCCs of G)
Transpose of directed G(V,E) = GT(V,ET) where ET = {(u,v) : (v,u)  E}
(i.e. GT has same vertices a G, but its edges are reversed)

27. SCC(G) Pseudocode: Call DFS(G) to get f[u] for all vertices
Construct the transpose of G
Call DFS(GT) with vertices in order of decreasing f[u] of DFS(G)
The vertices of each tree of the DFS(GT) forest are a SCC of G
Example: Fig 22.9 p616

28. SCC(G) Pseudocode: Call DFS(G) to get f[u] for all vertices
Construct the transpose of G
Call DFS(GT) with vertices in order
of decreasing f[u] of DFS(G)
The vertices of each tree of the
DFS(GT) forest are a SCC of G

29. CptS 450 Spring 2015
[All problems are from Cormen et al, 3rd Edition]
Homework Assignment 12: due 4/22/15
1. ex p 614 Topo sort on DAG of Fig 22.8 p615
2. ex p 620 SCCs on graph in Fig 22.6 p611

30. Theorems behind the SCC pseudo-code
Lemma 22.14
Let C and C’ be distinct SCCs of directed G(V,E).
Let (u,v)  E be an edge with u in C and v in C’,
then f(C) > f(C’)
Case 1: d(C) < d(C’) use white path theorem
Case 2: d(C) > d(C’) use parenthesis structure
Corollary 22.15
Let C and C’ be distinct SCCs of directed G(V,E).
Let (u,v)  ET be an edge with u in C and v in C’,
then f(C) < f(C’)

31. How Strongly-Connected-Components(G) works:
In the DFS of GT, start with a vertex in Cmax, the SCC that has
the largest finish time.
The search will visit all of the vertices of Cmax but will not find an edge
that connects Cmax to any other SCC.
If such an edge were found to C’, then by Corollary of
Lemma f(Cmax) < f(C’), which violates the choice of Cmax
as the SCC with the largest finish time
As the source of the next DFS in GT, chose a vertex in C’, the SCC with
f(C’) larger than all other finishing time except those in Cmax.
The search will visit all the vertices of C’ and may find some edges that
connect it to Cmax but not to any other SCC.
Since the vertices of Cmax have already been discovered, the only new
vertices visited from the second source are those in C’.