Unit 1. Sorting and Divide and Conquer

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Presentation transcript:

Unit 1. Sorting and Divide and Conquer

Lecture 1 Introduction to Algorithm and Sorting

What is an Algorithm? 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

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.

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

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

Insertion Sort

key In the key outside of array.

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

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.

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

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.

Merge Sort

Procedure

Example

Procedure

Symmetry

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

Recursion Tree

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.

We start to work from here.

Where it comes from?

Where comes from?

Conclusion

Master Theorem

Relationship

Summary

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.