TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Content:
• Basics, ADT Graph, Representations
• Graph Searching
TDDB56 – DALGOPT-D
Algorithms and optimization
– Depth-First Search
– Breadth-First Search
Lecture 11 & 12
– Example graph problems solved with search
Graphs
Graphs HT 2005
• Directed Graphs
11.1
Graphs HT 2005
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Basics
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])
-
End vertices of an edge
Edges incident on a vertex
Adjacent vertices (neighbors)
Degree of a vertex (undirected graph)
Indegree/outdegree (directed graph)
2
Path
1
3
Simple path
Cycle (loop)
4
5
Path:
<1 5 3 2 1 4>
Simple path: <1 4>
Cycle:
<1 5 4 1 2 3 5 1>
Simple cycle: <1 5 4 1> = < 5 4 1 5> =...
2
2
1
4
Graphs HT 2005
11.2
11.3
Graphs HT 2005
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5
1
4
3
5
11.4
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TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
ADT Graph
Terminology cont…
- 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
Graphs HT 2005
11.5
TDDB56 DALGOPT-D – Lecture 11 – Graphs
• Adjacency matrix
M[i,j] = 1 if {vi,vj} in E, and 0 otherwise
• Adjacency list
for each vi store a list of neighbors
1 2 3 4 5
1
4
3
0
0
0
1
0
•
•
•
•
•
•
•
•
•
•
•
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
Graphs HT 2005
11.6
The rough idea may need refinement
represented as:
1
2
3
4
5
[Goodrich & Tamassia]:
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Representation of graphs (rough idea)
Graph (V,E) with vertices {v1,…,vn}
2
Defined differently by different authors.
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
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Graphs HT 2005
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1
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3
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1
3
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11.7
• Vertices and edges stored as cells: may contain
additional information
• V and E are sets: use a set representation (list,
table, dictionary?)
• Incident edges of a vertex should be accessible
very efficiently.
• Neighbors of a vertex should be accessible very
efficiently.
For refined variants of Adjacency list and
Adjacency matrix see [G&T]
Graphs HT 2005
11.8
2
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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
v
w
Vertex objects augmented
with integer keys
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
a
b
a
2
2
v
w
0
0
0
b
List of edges
Graphs HT 2005
11.9
Graphs HT 2005
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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 2005
11.11
11.10
• 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}
Graphs HT 2005
11.12
3
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Example (notation of [G&T])
Example (cont.)
unexplored vertex
visited vertex
unexplored edge
discovery edge
back edge
A
A
A
B
A
B
D
E
C
A
D
E
D
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E
C
C
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A
A
B
D
E
C
Graphs HT 2005
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A
B
D
E
B
C
11.13
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Some applications of DFS
Breadth First Search (BFS)
Graphs HT 2005
11.15
E
D
E
C
Graphs HT 2005
TDDB56 DALGOPT-D – Lecture 11 – Graphs
•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
D
11.14
• 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!!
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)
Graphs HT 2005
11.16
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TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Example
Example (cont.)
L0
unexplored vertex
visited vertex
unexplored edge
discovery edge
cross edge
A
A
L0
L1
L1
C
E
L1
D
L1
C
D
E
L1
11.17
D
L1
F
L0
C
E
© 2004 Goodrich, Tamassia
D
L1
Graphs HT 2005
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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 2005
C
E
L0
B
L0
A
B
F
A
F
© 2004 Goodrich, Tamassia
D
E
L0
C
L0
B
A
B
A
C
E
D
F
11.18
•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 2005
11.19
Graphs HT 2005
11.20
5
TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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 2005
11.21
© 2004 Goodrich, Tamassia
TDDB56 DALGOPT-D – Lecture 11 – Graphs
C
G
A
D
E
B
C
A
G*
Graphs HT 2005
11.22
g
c
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
11.23
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’.
Graphs HT 2005
B
DAGs and Topological Ordering
– If there’s a w not visited, print “no”.
© 2004 Goodrich, Tamassia
E
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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
Graphs HT 2005
E
B
C
A
DAG G
D
B
C
A
v4
E
v5
v3
Topological
ordering of G
11.24
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TDDB56 DALGOPT-D – Lecture 11 – Graphs
TDDB56 DALGOPT-D – Lecture 11 – Graphs
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 2005
11.25
TDDB56 DALGOPT-D – Lecture 11 – Graphs
© 2004 Goodrich, Tamassia
Graphs HT 2005
11.26
TDDB56 DALGOPT-D – Lecture 11 – Graphs
Topological Sorting Example
Topological Sorting Example
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9
© 2004 Goodrich, Tamassia
Graphs HT 2005
9
11.27
© 2004 Goodrich, Tamassia
Graphs HT 2005
11.28
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TDDB56 DALGOPT-D – Lecture 11 – Graphs
Topological Sorting Example
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Graphs HT 2005
11.29
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