1. Topological Sort (topological order)
Let G = (V, E) be a directed graph
with |V| = n.
A Topological Sort is a sequence v1, v2,.... vn, such that
{ v1, v2,.... vn } = V, and
for every i and j with 1  i < j  n, (vj , vi)  E. g a f h e c b d h g e b
There are more than one such order in this graph!! d c a f
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IT 328, review graph algorithms

2. Algorithm for finding Topological Sortx
Principle: vertex with in-degree 0 should be listed first. x y z a
Algorithm: y b
Input G=(V,E)
Repeat the following steps until no more vertex left:
Find a vertex v with in-degree 0;
If can’t find such v and V is not empty
then stop the algorithm; this graph is not acyclic
2. Remove v from V and update E
3. Add v to the end of the sort
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IT 328, review graph algorithms

3. Topological Sort (topological order) of an acyclic graphb d h g e b d c a f
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IT 328, review graph algorithms

4. IT 328, review graph algorithms
Finding the shortest path between two vertices
Starting Vertex
Starting Vertex
Unweighted Graphs 
Weighted Graphs 2 5 1 2 2 3 3 5 1 1 1 5 3 3 7 7 7 2 2 1 9 10 12 8 8 2
Both are using the
greedy algorithm. w w
Final Vertex
Final Vertex
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IT 328, review graph algorithms

5. IT 328, review graph algorithms
Construct a shortest path Map for weighted graph
: decided
Starting Vertex 
0,-1 2 5   1 1 2
2,0
5,0
3,1 1 5   3 4 7
3,1
7,1 1 9  5
10,2
8,4
w-1,y n
w,x
Final Vertex
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IT 328, review graph algorithms

6. A spanning is a walk such that every vertex is included
Minimum Spanning Tree of an undirected graph:
The lowest cost to connect all vertices
Since a cycle is not necessary, such a connection must be a tree. 2 1 2 1 10 4 3 2 3 4 7 5 8 4 5 6 1 6 7
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IT 328, review graph algorithms

7. grow the minimum spanning tree from a root
Prim’s algorithm:
grow the minimum spanning tree from a root 2 1 2 1 10 4 3 2 3 4 7 5 8 4 5 6 1 6 7 2 1 2 1 2 3 4 5 4 6 1 6 7
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IT 328, review graph algorithms

8. IT 328, review graph algorithms
Prim’s algorithm:
grow the minimum spanning tree from a different root 2 1 2 1 10 4 3 2 3 4 7 5 8 4 5 6 1 6 7 2 1 2 1 2 3 4 5 4 6 1 6 7
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IT 328, review graph algorithms

9. IT 328, review graph algorithms
Kruskal’s Algorithm:
construct the minimum spanning from edges 2 1 2 1 10 4 3 2 3 4 7 5 8 4 5 6 1 6 7 2 1 2 1 2 3 4 5 4 6 1 6 7
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IT 328, review graph algorithms

10. Maximum Flow Problem flow capacity Actual flow  capacity
The source node can produce flow as much as possible
For every node other than s and t, flow out  flow in
Source s
actual flow 3/ 4 2/ 2 1/ 1 a b 2/ 3 2/ 2 1/ 2
our textbook says = d c e e
The target node’s flow-in is called the flow of the graph f g
A node may have different ways to dispatch its flow-out . 1/ 2 2/ 3 t
What is the maximum flow of the graph?
Target
(sink)
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IT 279

11. What makes the maximum flow problem more difficult than the shortest path problem?
Source s
The shortest path of t is based on the shortest path of g. 3/ 4 2/ 2
once a node’s shortest path from s is determined, we will never have to revise the decision 1/ 1 a b 2/ 3 2/ 2 1/ 2 d c e e
The maximum flow to t is not based on the maximum flows to f and g. f g 1/ 2 2/ 3 t
Target
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IT 279

12. local optimum does not imply global optimum
Source
Source
Source s s s
5/5
2/2
4/5
2/2
5/5
2/2
1/1
0/1 a
1/1 a b a b b
4/4
3/3
1/4
2/3
0/3
3/4
3/3
3/3
1/3
0/3 d
3/3 d c
1/3 d c c
Target
Target
2/2
5/5
sc
max flow = 6
sd
max flow = 7 t
Target
st
max flow = 7
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IT 279

13. A naïve approach: adding all possible paths together
Source
Source
Source s s s 1 2/ 2 1/ 1 2/ 3 2 1 1 1 a a a b b b 1/ 4 4 4 1 2/ 2 1 2/ 3 2 d d d c c c 2/ 3 1/ 1 2/ 2 3 t t t
Target
Target
Target
Residual Graph
Residual Graph
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IT 279

14. Not always work. 3/3 2/2 1 1 3 1/4 1 3 2/2 2/ 2 3/3 We are just lucky
Source
Source s s
3/3
2/2 1 a 1 b a b 3
1/4 1 3
2/2 d c d c 2/ 2
3/3 t t
Target
Not always work.
Target
We are just lucky
Residual Graph
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15. A wrong choice: 3/ 3 2 2 1 1 3/ 4 1 3 2 3 2 2 3/ 3 2 s s a b a b d c d
Source
Source s s 3/ 3 2 2 1 a 1 b a b 3/ 4 1 3 2 3 2 d c d c 2 3/ 3 2 t t
Target
Target
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16. Keep a path for changing decision:
Source
Source s s 3/ 3 2 2 3 1 a 1 b a b 3/ 4 1 3 2 3 2 3 d c d c 2 3/ 3 2 3 t t
Target
Target
Augmented Graph
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17. Find another path in the augmented graph
No more path from s to t
Source s s s 2/ 2 3 2 2 3 3 1 a 1 1 b a a b b 2 1 2/ 3 2/ 2 3 1 2 1 2 1 2/ 3 2 2 d 1 1 c d d c c 2/ 2 3 2 2 3 3 t t t
Target
Augmented Graph
new Augmented Graph
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IT 279

18. + = Add flows together 3 3 2 2 3 2 2 2 2 2 1 3 3 2 2 Maximum Flow s s
Source + = s s 3 s 3 2 2 a b a b a b 3 2 2 2 2 2 1 d c d c d c 3 3 2 2 t t
Target t
Target
Target
Maximum Flow
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IT 279

19. An algorithm for finding the maximum flow of graphs
Input: G = (V, E), and s,t  V
Let Gmax = (, ), Gaug = G
Find a path p from s to t in Gaug
If no such p exists, output Gmax and stop the program
Add p into Gmax and update Gaug
Repeat 1, 2, and, if possible, 3
Complexity:
O(c |E|)
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IT 279

20. IT 328, review graph algorithms
A Eulerian cycle (path) is a trail that goes through every edge exactly once. b c a h i j g n k f m e l o d p
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IT 328, review graph algorithms

21. IT 328, review graph algorithms
A Hamiltonian cycle (path) is a cycle that contains every vertex;
A graph that contains a Hamiltonian cycle is called Hamiltonian b c a h i j g n k f m e l o d p
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IT 328, review graph algorithms

22. Back Tracking
Going through a Maze
Try this java jar
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IT 328