1. Graph Theory and Algorithm 02
COP 6726: New Directions in Database Systems
Graph Theory and Algorithm 02

2. A* algorithm

3. A* algorithm f(n) = g(n) + h(n)
g(n) = “cost from the starting node to reach n”
h(n) = “estimate of the cost of the cheapest path from n to the goal node”
A* generates an optimal solution if h(n) is an admissible heuristic and the search space is a tree:
h(n) is admissible if it never overestimates the cost to reach the destination node
A heuristic is admissible if it is too optimistic, estimating the cost to be smaller than it actually is.
Example: h(n) = “Euclidean distance to destination”

4. A* algorithm
f(n) = g(n) + h(n)

5. Clustering Graph

6. Markov Chains
0.6
0.4
0.8 = A B
0.2 = A B .6 .2 .4 .8 = =

7. Clustering on Graphs A B C D E F G A 0, 1, 1, 1, 0, 0, 0 B
0, 1, 1, 1, 0, 0, 0
1, 0, 1, 1, 1, 0, 0
1, 1, 0, 1, 0, 0, 0
1, 1, 1, 0, 0, 0, 0
0, 1, 0, 0, 0, 1, 1
0, 0, 0, 0, 1, 0, 1
0, 0, 0, 0, 1, 1, 0 A B E G C D F

8. Random Walk A B C D E F G

9. Random Walk
A B C D E F G A B C D E F G A B C D E F G

10. Random Walk
A B C D E F G A B C D E F G A B C D E F G

11. Random Walk
A B C D E F G A B C D E F G A B C D E F G

12. Random Walk
A B C D E F G A B C D E F G A B C D E F G

13. Random Walk
A B C D E F G A B C D E F G A B C D E F G

14. Random Walk
A B C D E F G A B C D E F G A B C D E F G

15. Random Walk
A B C D E F G A B C D E F G A B C D E F G

16. Random Walk A B C D E F G A B C D E F G Node B has the highest value.
Node A, C, D, E has the second highest value.

17. Markov Cluster Algorithm (MCL)
Normalization
Inflation: the inflation operation is responsible for both strengthening and weakening of current status.
square
normalization
1/2
1/6
1/3
1/4
1/36
1/9
9/14
1/14
4/14

18. Random Walk A B C D A B C D 1

19. MCL A B C D A B C D 1

20. MCL
A B C D E F G A B C D E F G
0, 1, 1, 1, 0, 0, 0
1, 0, 1, 1, 1, 0, 0
1, 1, 0, 1, 0, 0, 0
1, 1, 1, 0, 0, 0, 0
0, 1, 0, 0, 0, 1, 1
0, 0, 0, 0, 1, 0, 1
0, 0, 0, 0, 1, 1, 0 A B E G C D F

21. MCL A B E G C D F

22. Graph Partitioning

23. Graph Partitioning

24. Local Search

25. Suffix Tree

26. Suffix Tree
Suffix trees can be used to solve the exact matching problem in linear time.
The Exact matching problem:
Given a pattern P of length n, and a text T of length m, find all occurrences of P in T in O(n + m) time.
{aeef, ad, bbfe, bbfg, c, aeef}

27. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} a e e f

28. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} a e d e f

29. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} a b e b d e f f e

30. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} a b e b d e f f e g

31. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} c a b e b d e f f e g

32. Suffix Tree
{aeef, ad, bbfe, bbfg, c, aeef} c a b e b d e f f e g

33. Max Flow

34. Max Flow
capacity A B B 4 5
Source
Sink A 3 C 2 1 D

35. Max Flow
capacity A B B 4 5
Source
Sink A 3 C 2 1 D

36. Max Flow
capacity A B B 4 1
Source 4
Sink A 3 C 2 1 D

37. Max Flow
capacity A B B 4 1
Source 4
Sink A 3 C 2 1 D

38. Max Flow
capacity A B B 4 1
Source 4
Sink A 3 C 2 1 D

39. Max Flow
capacity A B B 4 5
Source
Sink A 2 1 C 1 1 1 D

40. Max Flow
capacity A B B 4 5
Source
Sink A 2 1 C 1 1 1 D

41. Max Flow
capacity A B B 4 5
Source
Sink A 2 1 C 2 1 D

42. Max Flow-Min Cut

43. Max Flow / Min Cut Question: Maximum Flow Source Sink B 10 8 A 1 C 6 7D

44. Max Flow / Min Cut
An (S,T)-cut in a flow network G = (V,E) is a partition of vertices V into two disjoint subsets S and T such that s  S, t  T
The capacity of a cut (S,T) is CAP(S,T) = uS vT c(u,v) B
9/10
8/8
Source
Sink A
1/1 C
6/6
7/10 D

45. Bipartite matching

46. Bipartite Matching b1 a1 b2 a2 b3 a3 b4 a4 b5

47. Bipartite Matching b1 a1 b2 a2 s b3 t a3 b4 a4 b5

48. Bipartite Matching b1 a1 b2 a2 s b3 t a3 b4 a4 b5

49. Bipartite Matching b1 a1 b2 a2 b3 a3 b4 a4 b5

50. Shortest path Algorithm

51. Shortest Path Algorithm
Factory: A, E, K
Warehouse: H, I 1 2 3 1 A B C D E 1 1 1 1 1 1 1 1 1 F G H I J 3 1 3 1 2 3 1 2 1 K L M N O

52. Open Pit Mining

53. Open fit Mining
Closed Set
Not a closed Set

54. Open fit Mining
Maximize the profit -2 1 2 -2 1

55. Open fit Mining
Maximize the profit -2 1 2 -2 1

56. Open fit Mining
Construct a flow graph where the minimum cut identifies a feasible set that maximizes profit. s 2 2 ∞ ∞ ∞ ∞ -2 1 2 -2 1 2 1 1 t
Each edge in E has infinite capacity
Add nodes s, t
Each node is attached to s and t with finite capacity edges.

57. Open fit Mining Minimum cut gives optimal solution. s 2 2 ∞ ∞ ∞ ∞ -2 1

58. Open fit Mining 8 -9 -1 7

59. Open fit Mining t -9 ∞ 8 -9 ∞ -1 ∞ -1 -8 ∞ ∞ 7 -7 s

60. Open fit Mining t -9 ∞ 8 -9 ∞ -1 ∞ -1 -8 ∞ ∞ 7 -7 s

61. NP hard problems

62. NP-hard problem
Class of problems which are at least as hard as the hardest problems in NP. Problems that are NP-hard do not have to be elements of NP; indeed, they may not even be decidable.
A decision problem is NP-complete when it is both in NP and NP-hard.
A decision problem C is NP-complete if:
C is in NP, and
Every problem in NP is reducible to C in polynomial time.
C can be shown to be in NP by demonstrating that a candidate solution to C can be verified in polynomial time.

63. NP-complete examples Boolean satisfiability problem (SAT)
Subgraph isomorphism problem
Independent set problem
Knapsack problem
Subset sum problem
Dominating set problem
Hamiltonian path problem
Clique problem
Graph coloring problem
Travelling salesman problem
Vertex cover problem

64. Take Home Message
Graph Algorithms