This document discusses several algorithms including Monte Carlo algorithms, Prim's algorithm, Kruskal's algorithm, backtracking algorithms like the n-Queens problem, and divide and conquer algorithms like binary search. It provides examples of how each algorithm works on sample problems and data structures. The key steps are outlined for solving problems using randomized, greedy, and divide-and-conquer approaches.

1. Mini Project
PREPARED BY: VIKAS TIWARI
ROLL NO: 30

2. Randomized
algorithms

3. 1.Numerical Algorithm
For some problems, finding the exact solution requires a
long time, randomness can be used to find an approximate
numerical solution of such problems.
All these algorithms are called numerical probabilistic
algorithms.
These algorithms produce approximate results and the
quality of the solution can be improved if we run the
algorithm for a longer time

4. 2. Monte Carlo Algorithm
The randomized algorithm runs for a fixed number of steps for each input and produces an answer that
is correct with a bounded probability. That is it may produces incorrect solution.
It has a fixed deterministic running time
For decision problems: Two Kinds
1. one-sided error
2. Two-sided error

5. 1. One-sided error
• It is said to have one-sided error if there is a non-zero probability that it errors
is zero for at least one of the possible outputs (YES / NO) that it produces.
• Example: Min-Cut algorithm, which is better, Monte Carlo or Las Vegas.

6. Two-sided error
• A Monte Carlo algorithms is said to have tow-sided error if there is a non-zero
probability that it errors when it outputs either YES or NO.

7. Greedy Method

8. Prim’s Algorithm
4
37
1 2
5
6
25
14
11
20
12
22
23
10

9. Prim’s Algorithm
4
37
1 2
5
6
Total Wait =
0
Step 1:

10. Prim’s Algorithm
4
37
1 2
5
6
10
Total Weight = 10
Step 2:

11. Prim’s Algorithm
4
37
1 2
5
6
23
10
Step 3:
Total Weight = 33

12. Prim’s Algorithm
4
37
1 2
5
6
20
23
10
Step 4:
Total Weight = 53

13. Prim’s Algorithm
4
37
1 2
5
6
11
20
23
10
Step 5:
Total Weight: 64

14. Prim’s Algorithm
4
37
1 2
5
6
14
11
20
23
10
Step 6:
Total Weight = 78

15. Prim’s Algorithm
4
37
1 2
5
6
14
11
20
23
10
Step 7:
Total Weight = 90
12

16. Kruskal’s Algorithm
4
37
1 2
5
6
25
14
11
20
12
22
23
10
First we select all the vertices. Then an edge with optimum weight is selected from heap, even
though it is not adjacent to previously selected edge.

17. Kruskal’s Algorithm
4
37
1 2
5
6
Total Wait =
0
Step 1:

18. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 10
Step 2:

19. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 21
Step 3:
11

20. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 33
Step 4:
11
12

21. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 47
Step 5:
11
12 14

22. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 67
Step 6:
11
12 14
20

23. Kruskal’s Algorithm
4
37
1 2
5
6
10
Total Weight = 90
Step 7:
11
12 14
20
23

24. Backtracking

25. n-Queen’s Problem
Let us take 4-queens and 4 x 4 chessboard.
Now we start with chessboard.

26. n-Queen’s Problem
Q
2 𝑛𝑑 row and 3 𝑟𝑑 column.

27. n-Queen’s Problem
Q
Q
2 𝑛𝑑
row and 3 𝑟𝑑
column.

28. n-Queen’s Problem
Q
2 𝑛𝑑
row and 4 𝑡ℎ
column.
Q

29. n-Queen’s Problem
Q
3 𝑛𝑑
row and 1 𝑠𝑡
column.
Q
Q

30. Divide and Conquer

31. Binary Search
• Consider a list of elements stored in array A as
10 20 30 40 50 60 70
0 1 2 3 4 5 6
Low HighMid
Left sublist Right sublist
Middle element m = (low + high)/2