Unit 1 sorting and divide and conquer
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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 .

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Unit 1. Sorting and Divide and Conquer

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Unit 1.Sorting and Divide and Conquer


Lecture 1 Introduction to Algorithm and Sorting


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


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

  • Substitution method.

  • Recursion-tree method.

  • Master method.


Substitution Method

  • Guess the form of the solution.

  • Use math induction to find the constants and also show the solution works.


Remark


Recursion Tree


Master Theorem


Relationship


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.


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