This document provides lecture material on operations research and algorithms analysis. It covers topics like topological sorting, shortest path algorithms for unweighted and weighted graphs using Dijkstra's algorithm. Pseudocode and examples are given to explain topological sorting and computing shortest paths step-by-step. Complexity analysis is also presented for the different algorithms discussed.

1. Operations Research
CS-302
(3+0)
Prerequisites:
Probability and Statistics
SYED MUHAMMAD RAFI
LECTURER, DEPARTMENT OF SOFTWARE ENGINEERING
FACULTY OF ENGINEERING SCIENCE AND TECHNOLOGY,
ZIAUDDIN UNIVERSITY

2. GRAPHS
Lecture # 06
Operations Research
2

3. Analysis Of Algorithms 3

4. Analysis Of Algorithms 4

5. Analysis Of Algorithms 5

6. Analysis Of Algorithms 6

7. Topological Sort
Indegree Before Dequeue
Vertex 1 2 3 4 5
0 0 0 0 0
1 0 0 0 0
2 1 1 1 0
3 2 1 0 0
1 1 0 0 0
3 3 3 3 2
2 2 2 1 0
enqueue
V1
V2
V3
V4
V5
V6
V7
V1
V2
V5
V4
V3
, V7
Analysis Of Algorithms 7

8. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
Step by Step
Step 1
Analysis Of Algorithms 8

9. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
Step by Step
Step 2
Analysis Of Algorithms 9

10. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
Step by Step
Step 3
Analysis Of Algorithms 10

11. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
0
0
1
0
0
3
1
V4
V4
Step by Step
Step 4
Analysis Of Algorithms 11

12. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
0
0
1
0
0
3
1
V4
V4
0
0
0
0
0
2
0
V3, V7
V3
Step by Step
Step 5
Analysis Of Algorithms 12

13. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
0
0
1
0
0
3
1
V4
V4
0
0
0
0
0
2
0
V3, V7
V3
0
0
0
0
0
1
0
V7
Step by Step
Step 6
Analysis Of Algorithms 13

14. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
0
0
1
0
0
3
1
V4
V4
0
0
0
0
0
2
0
V3, V7
V3
0
0
0
0
0
1
0
V7
0
0
0
0
0
0
0
V6
V6
Step by Step
Step 7
Analysis Of Algorithms 14

15. Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7
V1 0
V2 1
V3 2
V4 3
V5 1
V6 3
V7 2
enqueue V1
dequeue V1
Topological Sort
0
0
1
2
1
3
2
V2
V2
0
0
1
1
0
3
2
V5
V5
0
0
1
0
0
3
1
V4
V4
0
0
0
0
0
2
0
V3, V7
V3
0
0
0
0
0
1
0
V7
0
0
0
0
0
0
0
V6
V6
Analysis Of Algorithms 15

16. Analysis Of Algorithms 16
Complexit
y Analysis
Topological Sort

17. Shortest Path Algorithm
Unweighted Graphs
Analysis Of Algorithms 17

18. Shortest Path Algorithm
Unweighted Graphs
Initial State
v Known dv pv
v1 0  0
v2 0  0
v3 0 0 0
v4 0  0
v5 0  0
v6 0  0
v7 0  0
Q v3
Step by Step
Step 1
Analysis Of Algorithms 18

19. Shortest Path Algorithm
Unweighted Graphs
Initial State
v Known dv pv
v1 0  0
v2 0  0
v3 0 0 0
v4 0  0
v5 0  0
v6 0  0
v7 0  0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0  0
1 0 0
0  0
0  0
0 1 v3
0  0
v1, v6
Step by Step
Step 2
Analysis Of Algorithms 19

20. Shortest Path Algorithm
Unweighted Graphs
Initial State
v Known dv pv
v1 0  0
v2 0  0
v3 0 0 0
v4 0  0
v5 0  0
v6 0  0
v7 0  0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0  0
1 0 0
0  0
0  0
0 1 v3
0  0
v1, v6
v1 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
0 1 v3
0  0
v6, v2, v4
Step by Step
Step 3
Analysis Of Algorithms 20

21. Shortest Path Algorithm
Unweighted Graphs
Initial State
v Known dv pv
v1 0  0
v2 0  0
v3 0 0 0
v4 0  0
v5 0  0
v6 0  0
v7 0  0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0  0
1 0 0
0  0
0  0
0 1 v3
0  0
v1, v6
v1 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
0 1 v3
0  0
v6, v2, v4
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
Step by Step
Step 4
Analysis Of Algorithms 21

22. Shortest Path Algorithm
Unweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0  0
Q v4, v5
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
Step by Step
Step 5
Analysis Of Algorithms 22

23. Shortest Path Algorithm
Unweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0  0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
Step by Step
Step 6
Analysis Of Algorithms 23

24. Shortest Path Algorithm
Unweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0  0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
Step by Step
Step 7
Analysis Of Algorithms 24

25. Shortest Path Algorithm
Unweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0  0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v7 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
1 3 v4
empty
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
Step by Step
Step 8
Analysis Of Algorithms 25

26. Shortest Path Algorithm
Unweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0  0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v7 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
1 3 v4
empty
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0  0
1 1 v3
0  0
v2, v4
n = |V|
T(n) = O(n2) for arrays
T(n) = O(|V| + |E|) for adjacency List
Analysis Of Algorithms 26

27. Shortest Path Algorithm
Unweighted Graphs
Analysis Of Algorithms 27
Complexit
y Analysis

28. Analysis Of Algorithms 28
END OF (Shortest Path Unweighted Graphs)
Algorithm

29. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Analysis Of Algorithms 29

30. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Initial State
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
Q v1
Step by Step
Step 1
Analysis Of Algorithms 30

31. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Initial State
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
Q v1
v1 dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0  0
0  0
0  0
v2, v4
Step by Step
Step 2
Analysis Of Algorithms 31

32. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
v2 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 v4
0 1 v1
0 3 v4
0 9 v4
0 5 v4
v2, v5,v3, v6, v7,
Step by Step
Step 3
v1dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0  0
0  0
0  0
v4, v2
Initial State
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
Q v1 Analysis Of Algorithms 32

33. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
v2 dequeued
Known dv pv
1 0 0
1 2 v1
0 0 0
0 1 v1
0 12 v2
0  0
0  0
v4, v5
Step by Step
Step 4
v1dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0  0
0  0
0  0
v2, v4
Initial State
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
Q v1
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 v4
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
Analysis Of Algorithms 33

34. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Step by Step
Step 5
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 v4
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
Analysis Of Algorithms 34

35. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Step by Step
Step 6
v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
Analysis Of Algorithms 35

36. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
v6 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
1 8 v3
0 5 v4
v7
Step by Step
Step 7
v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
Analysis Of Algorithms 36

37. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
v6 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
1 8 v3
0 5 v4
v7
Step by Step
Step 8
v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
v7 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
1 6 v7
1 5 v4
Empty
Analysis Of Algorithms 37

38. Shortest Path Algorithm
Weighted Graphs
Dijkstra’s Algorithm
Analysis Of Algorithms 38
Complexity
Analysis

39. Analysis Of Algorithms 39
END OF Dijkstra’s Algorithm

40. Minimum Spanning Tree
Weighted and Undirected Graphs
Prim’s Algorithm
Analysis Of Algorithms 40

41. Prim’s
Algorithm
Initial State
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
Step by Step
Step 1
Minimum Spanning Tree
Weighted and Undirected Graphs
Analysis Of Algorithms 41

42. Initial State
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
Minimum Spanning Tree
Weighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0  0
0  0
0  0
Prim’s
Algorithm
Step by Step
Step 2
Analysis Of Algorithms 42

43. Initial State
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
Minimum Spanning Tree
Weighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0  0
0  0
0  0
Prim’s
Algorithm
Step by Step
Step 3
v4 known
Known dv pv
1 0 0
0 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v2 ,v3 , v5 , v6 , v7
Analysis Of Algorithms 43

44. Initial State
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
Minimum Spanning Tree
Weighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0  0
0  0
0  0
Prim’s
Algorithm
Step by Step
Step 4
v4 known
Known dv pv
1 0 0
0 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v2 ,v3 , v5 , v6 , v7
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
Analysis Of Algorithms 44

45. v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning Tree
Weighted and Undirected Graphs
Prim’s
Algorithm
Step by Step
Step 5
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
Analysis Of Algorithms 45

46. v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning Tree
Weighted and Undirected Graphs
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
Prim’s
Algorithm
Step by Step
Step 6
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
Analysis Of Algorithms 46

47. v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning Tree
Weighted and Undirected Graphs
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
Prim’s
Algorithm
Step by Step
Step 7
v6 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
1 1 v7
1 4 v4
v5
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
Analysis Of Algorithms 47

48. Minimum Spanning Tree
Weighted and Undirected Graphs
Prim’s
Algorithm
Step by Step
Step 8
v5 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
1 6 v7
1 1 v7
1 4 v4
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
v6 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
1 1 v7
1 4 v4
v5
Total Cost = 16
Analysis Of Algorithms 48

49. Prim’s Algorithm
Analysis Of Algorithms 49
Complexity
Analysis

50. Analysis Of Algorithms 50
END OF Prim’s Algorithm

51. Minimum Spanning Tree
Weighted and Undirected Graphs
Kruskal’s Algorithm
Analysis Of Algorithms 51

52. Kruskal’s Algorithm
Step by Step
Step 1
Minimum Spanning Tree
Weighted Undirected Graphs
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Analysis Of Algorithms 52

53. Minimum Spanning Tree
Weighted and Undirected Graphs
Kruskal’s Algorithm
Analysis Of Algorithms 53
Complexity
Analysis

54. Analysis Of Algorithms 54
End of Kruskal’s Algorithm

55. End
GRAPHS
Analysis Of Algorithms 55