CSE101: Design and Analysis of Algorithmscseweb.ucsd.edu/~rajaiswal/Winter2020/cse101/Slides/Week-01/lec-1.pdfBasic graph algorithms Algorithm Design Techniques Divide and Conquer

CSE101: Design and Analysis of Algorithms

Ragesh Jaiswal, CSE, UCSD

Ragesh Jaiswal, CSE, UCSD CSE101: Design and Analysis of Algorithms

Administrative Information


Administrative Information

Course Instructor:

Ragesh JaiswalOffice: 3130, CSEEmail: [email protected]

Course webpage:http://www.cs.ucsd.edu/~rajaiswal/Winter2020/cse101/.

The discussion sections will be held this week. We will recap.material from the previous courses that will be used in this course.

The first lecture is being conducted by Prof. Russell Impagliazzo.

The detailed administrative information will be discussed in thesecond lecture by the course instructor.


http://www.cs.ucsd.edu/~rajaiswal/Winter2020/cse101/

Recap. of Data Structures and Algorithms


Recap.

What is an algorithm?


Recap.


A step-by-step way of solving a problem.

How do we measure the performance of an algorithm?


Recap.




Main ideas for performance measurement:

Worst-case analysis: Largest possible running time over allinput instances of a given size n and then see how thisfunction scales with n.Asymptotic order of growth: The worst-case running time for

large n (e.g., T (n) = 5n3 + 3n2 + 2n + 10)


Recap.



How do we measure the performance of an algorithm?Main ideas for performance measurement:

Worst-case analysis: Largest possible running time over all inputinstances of a given size n and then see how this function scaleswith n.Asymptotic order of growth: The worst-case running time for large

n (e.g., T (n) = 5n3 + 3n2 + 2n + 10)

Figure: Plot of n2 and 2n + 2


Recap.




Main ideas for performance measurement:

Worst-case analysis: Largest possible running time over allinput instances of a given size n and then see how thisfunction scales with n.Asymptotic order of growth: The worst-case running time for

large n (e.g., T (n) = 5n3 + 3n2 + 2n + 10)

Asymptotic order of growth (O,Ω,Θ):

T (n) is O(f (n)) (or T (n) = O(f (n))) iff there exists constantsc > 0, n0 ≥ 0 such that for all n ≥ n0, we have T (n) ≤ c · f (n).


Recap.

Growth rates:Arrange the following functions in ascending order of growthrate:

n2√log n

nlog n

2log n

n/ log nnn


Introduction

Algorithm: A step-by-step way of solving a problem.

Design of Algorithms:

“Algorithm is more of an art than science”However, we will learn some basic tools and techniques thathave evolved over time. These tools and techniques enable youto effectively design and analyse algorithms.

Analysis of Algorithms:

Proof of correctness: An argument that the algorithm workscorrectly for all inputs.Analysis of worst-case running time as a function of the inputsize.


Introduction




Analysis of Algorithms:Proof of correctness: An argument that the algorithm workscorrectly for all inputs.

Proof: A valid argument that establishes the truth of amathematical statement.

Analysis of worst-case running time as a function of the inputsize.


Introduction


The statements used in a proof can include axioms, definitions,the premises, if any, of the theorem, and previously proventheorems and uses rules of inference to draw conclusions.

A proof technique very commonly used when provingcorrectness of Algorithms is Mathematical Induction.

Definition (Strong Induction)

To prove that P(n) is true for all positive integers, where P(n) is apropositional function, we complete two steps:

Basis step: We show that P(1) is true.

Inductive step: We show that for all k , if P(1),P(2), ...,P(k)are true, then P(k + 1) is true.


Introduction

Definition (Strong Induction)

To prove that P(n) is true for all positive integers, where P(n) is apropositional function, we complete two steps:

Basis step: We show that P(1) is true.

Inductive step: We show that for all k , if P(1),P(2), ...,P(k)are true, then P(k + 1) is true.

Question: Show that for all n > 0, 1 + 3 + ... + (2n− 1) = n2.


Introduction

Question: Show that for all n > 0, 1 + 3 + ... + (2n − 1) = n2.

Proof

Let P(n) be the proposition that 1 + 3 + 5 + ... + (2n − 1) equals n2.Basis step: P(1) is true since the summation consists of only a singleterm 1 and 12 = 1.Inductive step: Assume that P(1),P(2), ...,P(k) are true for anyarbitrary integer k . Then we have:

1 + 3 + ... + (2(k + 1) − 1) = 1 + 3 + ... + (2k − 1) + (2k + 1)

= k2 + 2k + 1 (since P(k) is true)

= (k + 1)2

This shows that P(k + 1) is true.Using the principle of Induction, we conclude that P(n) is true for alln > 0.


Introduction




Analysis of Algorithms:Proof of correctness: An argument that the algorithm workscorrectly for all inputs.


Analysis of worst-case running time as a function of the inputsize.


Introduction

Algorithm Design Techniques

Divide and ConquerGreedy AlgorithmsDynamic ProgrammingNetwork Flows


Introduction

Material that will be covered in the course:

Basic graph algorithmsAlgorithm Design Techniques

Divide and ConquerGreedy AlgorithmsDynamic ProgrammingNetwork Flows

Computational intractability


IntroductionDivide and Conquer

Some examples of Divide and Conquer Algorithms:

Binary SearchMedian findingMultiplying numbersMerge sort, quick sort.


IntroductionGreedy Algorithms

Problem

Interval scheduling: You have a lecture room and you get nrequests for scheduling lectures. Each request has a start time andan end time. The goal is to maximise the number of lectures.


IntroductionDynamic Programming

Problem

Interval scheduling: You have a lecture room and you get nrequests for scheduling lectures. Each request has a start time, anend time, and a price (that you will get in case the lecture isscheduled). The goal is to maximise your earnings.


IntroductionNetwork Flows

Problem

Job assignment: There are n people and n jobs. Each person has alist of jobs he/she could possibly do. Find a job assignment sothat:

1 each job is assigned to a different person, and

2 each person is assigned a job from his/her list.


IntroductionComputational Intractability

Is it always possible to find a fast algorithm for any problem?

Problem

Given a social network, find the largest subset of people such thatno two people in the subset are friends.


IntroductionComputational Intractability

The problem in the previous slide is called the IndependentSet problem and no one knows if it can be solved inpolynomial time (quickly).

There is a whole class of problems to which Independent Setbelongs.

If you solve one problem in this class quickly, then you cansolve all the problems in this class quickly.

You can also win a million dollars!!

We will see techniques of how to show that a new problembelongs to this class:

Why: because then you can say to your boss that the newproblem belongs to the difficult class of problems and even themost brilliant people in the world have not been able to solvethe problem so do not expect me to do it. Also, if I can solvethe problem there is no reason for me to work for you!


End


Documents

CSE101: Design and Analysis of Algorithmscseweb.ucsd.edu/~rajaiswal/Winter2020/cse101/Slides/Week-01/lec-1.pdfBasic graph algorithms Algorithm Design Techniques Divide and Conquer