Network Optimization
Depth First Search
James Orlin 2010
Initialize
4
2
1 1
Unmark all
nodes in N;
5
3
8
7
6
9
Mark node 1
and add it to
LIST
LIST
1
2
Select a node i in LIST
4
2
1 1
Select last
node in LIST
LIST
5
3
8
7
6
9
1
3
If node i is incident to an admissible arc…
2 2
1 1
4
5
Scan an arc
out of node 1.
3
8
7
6
9
If endpoint is unmarked, then
mark it and add it to LIST.
LIST
1
2
4
Select the last node on LIST
2 2
1 1
5
Select
Node 2.
LIST
3
1
4
8
7
6
9
2
5
If node i is incident to an admissible arc…
2 2
1 1
4 3
5
Scan arc out
of node 2.
3
8
7
6
9
Mark node 4
and add it to
LIST.
LIST
1
2
4
6
Select
2 2
1 1
4 3
5
8
7
Select node 4
3
LIST
1
2
6
9
4
7
If node i is incident to an admissible arc…
2 2
4 3
1 1
5
Scan arc out
of node 4.
3
8 4
7
6
9
Mark 8 and
add it to LIST.
LIST
1
2
4
8
8
Select
2 2
4 3
1 1
5
8 4
7
Select node 8
3
LIST
1
2
4
6
9
8
9
If node i is not incident to an admissible arc…
2 2
4 3
1 1
5
If there are no
unscanned
arcs out of
node 8, delete
it from LIST.
LIST
1
3
2
4
8 4
7
6
9
8
10
Select
2 2
4 3
1 1
5
8 4
7
Select Node 4
3
LIST
1
2
4
6
9
8
11
If node i is incident to an admissible arc…
2 2
4 3
1 1
5
Scan arc out
of node 4.
3
5
8 4
7
6
9
Mark 5 and
add it to LIST.
LIST
1
2
4
8
5
12
Select
2 2
4 3
1 1
5
5
8 4
7
Select node 5
3
LIST
1
2
4
6
9
8
5
13
If node i is incident to an admissible arc…
2 2
4 3
1 1
5
Scan arc out
of node 5.
3
1
2
4
7
6
6
Mark 6 and
add it to LIST.
LIST
5
8
5
8 4
9
6
14
Select the last node on LIST
2 2
4 3
1 1
5
5
8 4
7
Select node 6
3
LIST
1
2
4
6
6
8
5
9
6
15
If node i is incident to an admissible arc…
2 2
4 3
1 1
5
Scan arc out
of node 6.
5
7
3
6
6
Mark 9 and
add it to LIST.
LIST
1
2
4
8
5
6
8 4
9
7
9
16
Select the last node on LIST
2 2
4 3
1 1
5
5
8 4
7
Select node 9
3
LIST
1
2
4
6
6
8
5
6
9
7
9
17
If node i is incident to an admissible arc…
2 2
4 3
1 1
5
Scan arc out
of node 9.
5
7 8
3
1
2
4
9
7
6
6
Mark 7 and
add it to LIST.
LIST
8
5
6
8 4
9
7
18
Select the last node on LIST
2 2
4 3
1 1
5
5
8 4
7 8
Select node 7
3
LIST
1
2
4
9
7
6
6
8
5
6
9
7
19
Scan arc (7,4)
2 2
4 3
1 1
5
Scan arc out
of node 7
5
7 8
3
1
2
4
8
5
9
7
6
6
If endpoint is marked, then
label arc as “scanned”.
LIST
6
8 4
9
7
20
Scan arc (7,5)
2 2
4 3
1 1
5
Scan arc out
of node 7
5
7 8
3
1
2
4
8
5
9
7
6
6
If endpoint is marked, then
label arc as “scanned”.
LIST
6
8 4
9
7
21
Scan arc (7,8)
2 2
4 3
1 1
5
Scan arc out
of node 7
5
7 8
3
1
2
4
8
5
9
7
6
6
If endpoint is marked, then
label arc as “scanned”.
LIST
6
8 4
9
7
22
If node i is not incident to an admissible arc…
2 2
4 3
1 1
5
There are no
arcs out of 7.
Delete 7 from
LIST
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
23
Select node 9
2 2
4 3
1 1
5
Select node 9.
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
24
Scan (9,4)
2 2
4 3
1 1
5
Scan arc out of
node 9.
5
7 8
3
1
2
4
8
5
9
7
6
6
If endpoint is marked, then
label arc as “scanned”.
LIST
6
8 4
9
7
25
Delete 9 from LIST
2 2
4 3
1 1
5
Node 9 has no
incident arcs.
Delete it from
list.
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
26
Select node 6
2 2
4 3
1 1
5
Select node
6.
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
27
Scan arc (6.7)
2 2
4 3
1 1
5
Select arc out
of node 6.
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
28
Select node 6
2 2
4 3
1 1
5
Delete node 6
from LIST
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
29
Select node 5
2 2
4 3
1 1
5
1
7 8
3
Select Node 5
LIST
5
2
4
9
7
6
6
8
5
6
8 4
9
7
30
Select node 5
2 2
4 3
1 1
5
1
7 8
3
Delete node 5
from LIST
LIST
5
2
4
9
7
6
6
8
5
6
8 4
9
7
31
Select node 4
2 2
4 3
1 1
5
5
8 4
7 8
Select node 4
3
LIST
1
2
4
9
7
6
6
8
5
6
9
7
32
Delete node 4
2 2
4 3
1 1
5
Delete node 4
from LIST
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
33
Select node 2
2 2
4 3
1 1
5
5
8 4
7 8
Select Node 2
3
LIST
1
2
4
9
7
6
6
8
5
6
9
7
34
Scan arc (2,5)
2 2
4 3
1 1
5
Scan arc out
of node 2
LIST
1
5
7 8
3
2
4
9
7
6
6
8
5
6
8 4
9
7
35
Delete node 2
2 2
4 3
1 1
5
1
7 8
3
Delete node 2
from LIST
LIST
5
2
4
9
7
6
6
8
5
6
8 4
9
7
36
Select node 1
2 2
4 3
1 1
5
5
8 4
7 8
Select node 1
3
LIST
1
2
4
9
7
6
6
8
5
6
9
7
37
Scan arc (1,3)
2 2
4 3
1 1
5
Scan arc out
of node 1
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
38
Select node 3
2 2
4 3
1 1
5
5
8 4
7 8
Select node 3
3
9
LIST
1
2
3
4
9
7
6
6
8
5
6
9
7
39
Scan arcs (3,5) and (3,6)
2 2
4 3
1 1
5
Scan arcs out
of node 3
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
40
Delete node 3
2 2
4 3
1 1
5
Delete node 3
from LIST
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
41
Select node 1
2 2
4 3
1 1
5
5
8 4
7 8
Select node 1
3
9
LIST
1
2
3
4
9
7
6
6
8
5
6
9
7
42
Scan arc (1, 5)
2 2
4 3
1 1
5
Scan arc out
of (1,5)
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
43
Delete node 1
2 2
4 3
1 1
5
Delete node 1
from LIST
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
44
LIST is empty
2 2
4 3
1 1
5
The algorithm
ends!
LIST
1
5
7 8
3
9
2
3
4
9
7
6
6
8
5
6
8 4
9
7
45
The depth first search tree
Note that each
induced
subtree has
consecutively
labeled nodes
1 1
22
3 9
4 3
4 8
5 5
6 6
9 7
7 8
46
MITOpenCourseWare
http://ocw.mit.edu
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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