TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Content:
• Basics, ADT Graph, Representations
.
[GT 13.1-2, LD 12.1, CL 22.1]
• Graph Searching
TTIT33 Algorithms and optimization
[GT 13.3, LD 12.2, CL 22.2-3]
– Depth-First Search
– Breadth-First Search
Algorithms Lecture 4
– Example graph problems solved with search
Graphs
Graphs HT 2006
• Directed Graphs
4.1
TTIT33 Alorithms and Optimization – DALG Lecture 4
Graphs HT 2006
4.2
TTIT33 Alorithms and Optimization – DALG Lecture 4
Basics
Some Terminology
• Graph G = (V,E) :
– V a set of vertices;
– E a set of pairs of vertices, called edges
• Directed graph: the edges are ordered pairs
• Undirected graph: the edges are unordered pairs
– Edges and vertices may be labelled by additional info
Example: A graph with
- vertices representing airports; info airport code
- edges representing flight routes; info mileage
(see [G&T])
Graphs HT 2006
- G is connected
iff there is a path between any two vertices
- G is a (free) tree
iff there is a unique simple path between any two
vertices; notice: root is not distinguished
- G is complete
iff any two vertices are connected by an edge
- G´ = (V´, E´) is a subgraph of G = (V,E)
iff V´⊆ V and E´⊆ E,
- A connected component of G is any maximal
subgraph of G which is connected
4.3
TTIT33 Alorithms and Optimization – DALG Lecture 4
Graphs HT 2006
ADT Graph
Representation of graphs (rough idea)
Graph (V,E) with vertices {v1,…,vn}
[Goodrich & Tamassia]:
endVertices(e): an array of the two endvertices of e
opposite(v, e): the vertex opposite of v on e
areAdjacent(v, w): true iff v and w are adjacent
replace(v, x): replace element at vertex v with x
replace(e, x): replace element at edge e with x
insertVertex(o): insert a vertex storing element o
insertEdge(v, w, o): insert an edge (v,w) storing element o
removeVertex(v): remove vertex v (and its incident edges)
removeEdge(e): remove edge e
incidentEdges(v): edges incident to v
vertices(): all vertices in the graph; edges(): all edges in the graph
represented as:
• Adjacency matrix
M[i,j] = 1 if {vi,vj} in E, and 0 otherwise
• Adjacency list
for each vi store a list of neighbors
2
4.5
1 2 3 4 5
1
4
Graphs HT 2006
4.4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Defined differently by different authors.
•
•
•
•
•
•
•
•
•
•
•
[GT 13.4, LD 12.2, CL 22.4]
3
1
2
3
4
5
0
0
0
1
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
5
Graphs HT 2006
0
1
1
0
0
1
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2
3
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1
3
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5
4.6
1
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Adjacency List Structure [G&T]
Adjacency Matrix Structure [G&T]
Example graph
___________________
a
v
b
u
v
a
Example graph
w
b
u
w
List of vertices
Vertex objects
u
Vertex objects augmented
with integer keys
w
v
0
u
1
Lists of incident edges
0
2D-array adjacency array
0
1
2
0
1
0
1
Edge objects linked to
respective vertices
b
a
a
2
v
w
0
2
0
b
0
List of edges
Graphs HT 2006
4.7
TTIT33 Alorithms and Optimization – DALG Lecture 4
Graphs HT 2006
TTIT33 Alorithms and Optimization – DALG Lecture 4
Graph Searching
DFS Algorithm
The problem : systematically visit the vertices of
a graph reachable by edges from a given vertex.
Numerous applications:
- robotics: routing, motion planning
- solving optimization problems (see OPT part)
Graph Searching Techniques:
- Depth First Search (DFS)
- Breadth First Search (BFS)
Graphs HT 2006
• Input: a graph G and a vertex s
• Visits all vertices connected with s in time O(|V|+|E|)
procedure DepthFirstSearch(G =(V,E), s):
for each v in V do explored(v) ← false;
RDFS(G,s)
procedure RDFS(G,s):
explored(s) ← true;
previsit(s)
{some operation on s before visiting its neighbors}
for each neighbor t of s
if not explored(t) then RDFS(G,t)
postvisit(s)
{some operation on s after visiting its neighbors}
4.9
Graphs HT 2006
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Example (notation of [G&T])
Example (cont.)
unexplored vertex
visited vertex
unexplored edge
discovery edge
back edge
A
A
A
B
A
B
D
4.8
E
C
A
D
E
B
A
D
E
B
C
C
C
A
A
A
B
D
E
C
Graphs HT 2006
4.10
B
D
E
B
C
4.11
D
E
D
E
C
Graphs HT 2006
4.12
2
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Some applications of DFS
Breadth First Search (BFS)
• Input: a graph G and a vertex s
• Visits all vertices connected with s in time O(|V|+|E|)
in order of increasing distance to s
• Apply Queue!!
•Checking if G is connected
•Finding connected components of G
•Finding a path between vertices
•Checking acyclicity/finding a cycle
•Finding a spanning forest of G
Graphs HT 2006
procedure BFS (G=(V,E), s):
for each v in V do explored(v) ← false;
S ← MakeEmptyQueue();
Enqueue(S,s); explored(s) ← true;
while not IsEmpty(S) do
t ← Dequeue(S)
visit(t)
for each neighbor v of t do
if not explored(v) then
explored(v) ← true;
Enqueue(S,v)
4.13
Graphs HT 2006
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Example
Example (cont.)
L0
unexplored vertex
visited vertex
unexplored edge
discovery edge
cross edge
A
A
L0
L1
L1
A
C
E
L1
D
L1
C
D
E
L1
4.15
D
L1
F
L0
C
E
© 2004 Goodrich, Tamassia
D
L1
Graphs HT 2006
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Example (cont.)
Some applications of BFS
L0
L1
B
L2
L0
L1
C
E
D
L1
B
L2
F
A
C
E
D
F
A
B
L2
L0
A
C
E
© 2004 Goodrich, Tamassia
D
C
E
D
F
A
B
L2
F
A
B
L2
A
B
L2
F
Graphs HT 2006
C
E
L0
B
L0
A
B
F
A
F
© 2004 Goodrich, Tamassia
D
E
L0
C
L0
B
A
B
4.14
C
E
D
F
4.16
•Checking if G is connected
•Finding connected components of G
•Finding a path of minimal length
between vertices
•Checking acyclicity/finding a cycle
•Finding a spanning forest of G
Compare with DFS !
F
Graphs HT 2006
4.17
Graphs HT 2006
4.18
3
TTIT33 Alorithms and Optimization – DALG Lecture 4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Directed Graphs
Transitive Closure
• Digraphs: edges are ordered pairs.
• Digraphs have many applications, like
– Task scheduling
– Route planning, …..
• Search algorithms apply, follow directions.
• DFS applications for digraphs:
– Transitive closure but in O(n(m+n))
– Checking strong connectivity
– Topological sort of a directed acyclic graph DAG
(scheduling)
Graphs HT 2006
4.19
© 2004 Goodrich, Tamassia
TTIT33 Alorithms and Optimization – DALG Lecture 4
C
G
A
D
E
B
C
A
G*
Graphs HT 2006
4.20
DAGs and Topological Ordering
g
c
– If there’s a w not visited, print “no”.
d
e
b
f
– If there’s a w not visited, print “no”.
– Else, print “yes”.
a
G’:
g
c
d
• Running time: O(n+m).
e
b
f
Graphs HT 2006
4.21
TTIT33 Alorithms and Optimization – DALG Lecture 4
A directed acyclic graph (DAG) is a
digraph that has no directed cycles
• A topological ordering of a digraph
is a numbering
v1 , …, vn
of the vertices such that for every
edge (vi , vj), we have i < j
• Example: in a task scheduling
digraph, a topological ordering a
task sequence that satisfies the
v2
precedence constraints
Theorem
A digraph admits a topological
v1
ordering if and only if it is a DAG
D
•
a
G:
• Let G’ be G with edges reversed.
• Perform a DFS from v in G’.
© 2004 Goodrich, Tamassia
E
B
TTIT33 Alorithms and Optimization – DALG Lecture 4
Strong Connectivity
Algorithm
• Pick a vertex v in G.
• Perform a DFS from v in G.
D
• Given a digraph G, the
transitive closure of G is the
digraph G* such that
– G* has the same vertices
as G
– if G has a directed path
from u to v (u ≠ v), G* has
a directed edge from u to
v
• The transitive closure
provides reachability
information about a digraph
© 2004 Goodrich, Tamassia
E
B
C
A
DAG G
D
B
C
A
Graphs HT 2006
v4
E
v5
v3
Topological
ordering of G
4.22
TTIT33 Alorithms and Optimization – DALG Lecture 4
Topological Sorting
Topological Sorting Example
Use modified DFS!
procedure TopologicalSort(G):
nextnumber ← |G|
for each vertex v in G do explored(v) ← false;
for each vertex v in G do
if not explored(v) then RDFS(G,v)
procedure RDFS(G,s):
explored(s) ← true;
for each neighbor t of s
if not explored(t) then RDFS(G,t)
number(s) ←nextnumber;
nextnumber ←nextnumber-1
Graphs HT 2006
4.23
© 2004 Goodrich, Tamassia
Graphs HT 2006
4.24
4
TTIT33 Alorithms and Optimization – DALG Lecture 4
Topological Sorting Example
2
1
3
4
6
5
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Graphs HT 2006
4.25
5