Introduction to Algorithms and Sorting: Efficiency and Running Time Calculation

1. Unit 1.Sorting and Divide and Conquer

2. Lecture 1 Introduction to Algorithm and Sorting

3. What is anAlgorithm? • An algorithm is a computational procedure that takes some value, or a set of values, as input and produces some values, or a set of values, as output. • So, it is a sequence of computational steps that transform the input into the output. • Correction: sequence -> combination

4. Sequential and Parallel • In sequential computation, an algorithm is a sequence of computational steps. • However, it is not true in parallel computation. • In this course, we study only algorithms in sequential computation.

5. Algorithms for Sorting e.g., Insertion Sort, Merge Sort

6. Efficiency • Running time from receiving the input to producing the output. Running time Insert Sort Merge Sort

9. Insertion Sort

11. key In the key outside of array.

12. How to calculate running time? • Each “line” of pseudocode requires a constant time. (In RAM model)

13. This loop runs n-1 times and each time runs at most 4+3(j-1) lines. This loop runs at most j-1 times and each time runs at most 3 lines.

14. Remark on Running Time • Running time is a function of input size. • In Turing machine model, the input size of sorting is

15. Divide and Conquer • Divide the problem into subproblems. • Conquer the subproblems by solving them recursively. • Combine the solutions to subproblems into the solution for original problem.

16. Merge Sort

17. Procedure

18. 5,2,7,4,6,8,1,3 5,2,7,4 6,8,1,3 1,3 5,2 7,4 6,8 6 4 8 1 5 2 3 7 2,5 4,7 1,3 6,8 1,3,6,8 2,4,5,7 1,2,3,4,5,6,7,8

21. Example

23. Procedure

24. Symmetry

25. How to Solve Recurrences • Recursion-tree method. (give reasonable estimate) • Substitution method. (mathematical induction and find correct constant) • Master method. (already proved results)

26. Recursion Tree

27. Substitution Method • Guess the form of the solution (e.g., use recursive tree). • Use math induction to find the constants and also show the solution works.

29. Conclusion

30. Master Theorem

31. Relationship

32. Summary

33. What we learnt in this lecture? • How to calculate running time. • How to solve recurrences. • Insertion sort and Merge sort. • Divide and conquer • Lecture Notes give you key points in each lecture. • You must read textbook after lectures in order to study well.

34. We start to work from here.

35. Where it comes from?

36. Where comes from?