The document discusses theoretical foundations of computer science with a focus on graph algorithms, including network flow problems and spanning trees. It introduces algorithms such as Prim's and Kruskal's for finding minimum spanning trees and highlights applications in real-time scenarios like traffic flow and GPS navigation. It also covers NP-complete problems in graphs, such as k-clique and vertex cover, illustrating their complexities and applications in various fields.

1. Theoretical Foundations of
Computer Science
(Graphs)
J.J. ADRI JOVIN
Assistant Professor (Sr. Gr.)/IT
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2. Network Flow Problems
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3. Real-time Applications
• Water Flow through a pipe
• Amount of traffic that can flow on a street between two
intersections
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4. Graph and its Maximum flow
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
3 2
0
2 2
1
2 3
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5. Initial Stage
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
0 0
0
0 0
0
0 0
s
t
a b
c d
3 2
1
3 2
4
2 3
Graph Flow Graph Residual Graph
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6. After 2 units of flow added along s,b,d,t
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
0 2
0
0 2
0
0 2
s
t
a b
c d
3
1
3
4
2 1
Graph Flow Graph Residual Graph
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7. After 2 units of flow added along s,a,c,t
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
2 2
0
2 2
0
2 2
s
t
a b
c d
1
1
1
4
1
Graph Flow Graph Residual Graph
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8. After 1 units of flow added along s,a,d,t –
algorithm terminates
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
2 2
0
2 2
1
2 3
s
t
a b
c d
1
1
3
Graph Flow Graph Residual Graph
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9. If 3 units of flow along s,a,d,t
Sub-optimal Solution
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
3 0
0
0 0
3
0 3
s
t
a b
c d
2
1
3 2
1
2
Graph Flow Graph Residual Graph
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10. 3 units flow added along s,a,d,t using
correct algorithm
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
3 0
0
0 0
3
0 3
s
t
a b
c d
3 2
1
3 2
2 3
Graph Flow Graph Residual Graph
1
3
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11. 2 units flow added along s,b,d,a,c,t using
correct algorithm
s
t
a b
c d
3 2
1
3 2
4
2 3
s
t
a b
c d
3 2
0
2 0
1
2 3
s
t
a b
c d
3 2
1
2
2 3
Graph Flow Graph Residual Graph
3
1
1
2
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12. Spanning Trees
• Prim’s Algorithm
• Kruskal’s Algorithm
Real-time Applications
• Wiring
• GPS Navigation
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13. v1 v2
v6 v7
v5v4v3
2
1
6
4
1
2
v1 v2
v6 v7
v5v4v3
2
1 3 10
6
7
4
1
2
4
5
8
Minimum Spanning Tree
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14. Prim’s Algorithm
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15. v1 v2
v6 v7
v5v4v3
V Known Dv Pv
V1 0 0 0
V2 0 0
V3 0 0
V4 0 0
V5 0 0
V6 0 0
v7 0 0







Initial Configuration
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16. v1 v2
v6 v7
v5v4v3
1
V Known Dv Pv
V1 1 0 0
V2 0 2 V1
V3 0 4 V1
V4 0 1 V1
V5 0 0
V6 0 0
v7 0 0



After v1 is declared known
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17. v1 v2
v6 v7
v5v4v3
2
1
V Known Dv Pv
V1 1 0 0
V2 0 2 V1
V3 0 4 V14
V4 1 1 V1
V5 0 7 V4
V6 0 8 V4
v7 0 4 V4
After v4 is declared known
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18. v1 v2
v6 v7
v5v4v3
2
1
2
V Known Dv Pv
V1 1 0 0
V2 1 2 V1
V3 1 2 V14
V4 1 1 V1
V5 0 7 V4
V6 0 5 V3
v7 0 4 V4
v1 v2
v6 v7
v5v4v3
2
1
2
4
After v2 and v3 are declared known
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19. v1 v2
v6 v7
v5v4v3
2
1
4
1
2
V Known Dv Pv
V1 1 0 0
V2 1 2 V1
V3 1 2 V14
V4 1 1 V1
V5 0 6 V7
V6 0 1 V7
v7 1 4 V4
After v7 is declared known
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20. v1 v2
v6 v7
v5v4v3
2
1
6
4
1
2
V Known Dv Pv
V1 1 0 0
V2 1 2 V1
V3 1 2 V14
V4 1 1 V1
V5 1 6 V7
V6 1 1 V7
v7 1 4 V4
After v6 and v5 are selected
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21. v1 v2
v6 v7
v5v4v3
Kruskal’s Algorithm
v1 v2
v6 v7
v5v4v3
1
v1 v2
v6 v7
v5v4v3
1
1
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22. v1 v2
v6 v7
v5v4v3
2
1
Kruskal’s Algorithm (Contd..)
v1 v2
v6 v7
v5v4v3
2
1
1
2
v1 v2
v6 v7
v5v4v3
2
1
4
1
2
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23. Kruskal’s Algorithm(Contd..)
v1 v2
v6 v7
v5v4v3
2
1
6
4
1
2
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24. NP-Complete Problems in Graphs
• K-Clique
• Vertex Cover
• Independent Set
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25. Cliques
• K-clique = complete subgraph of k nodes and are pair-wise adjacent
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b
a
e
c
d f
g

26. Real-time Applications
• Social Networks
• Computing Protein Similarity
• Finding Error Correcting Codes
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27. K-Cliques
• A K-clique is a set of K nodes with all K(K-1)/2 possible edges between
them
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This graph contains a 4-clique

28. Independent Set
• An independent set is a set of nodes with no edges between them
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This graph contains an independent set of size 3

29. Independent Set
• Given a graph G and k, is there a size k independent set?
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30. Vertex Cover
• vertex cover = set of nodes that cover all edges
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b
a
e
c
d
b
a
e
c
d

31. Vertex Cover is NP Complete
• Given a Graph G(V,E), decide if there is k vertex such that every edge
is covered by one of them?
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32. References
• “Data Structures and Algorithms”, 2nd Ed., Mark Allen Weiss
• “Data Structures”, Seymour Lipschutz
• “Data Structures with C++”, John R. Hubbard
• “Formal Languages, Automata, and Computation (FLAC)”,
http://www.cs.cmu.edu/~emc/flac09/lectures/ by Edmund M.
Clarke
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