Data Structures

Data Structures and Algorithms

Instructor InformationArash Vaezi

Lab: 712Telegram group: datastructure40254

Telegram Channel: @ DS_40254Email: vaezi.ce@gmail.com

https://vc.sharif.edu/ch/avaezi

Course InformationCS 40254 Data Structure

Times: Sunday - Tuesday 1.5 p.m. - 3 p.m.Online

Overview

Concept (Use Efficiently, Novelty)

Course objectives

Be familiar with problem solving

Be able to develop (and implement) algorithms

Be able to trace algorithms

Be able to select appropriate data structures and algorithms for given problems

Algorithmic Thinking

Be able to provide novel data structures

Previous Knowledge:

Be familiar with: Java, C++, or C#.

References

• Lecture Notes / Slides

Grading/Evaluation

Course Assessment Tools Number Grade

1 Mid term 1 6

2 Final 1 8

3 Homework 3 3

4 Research Project 1 +1

5 Implementation 3 3

6 Quizzes/Class Works ! +1-2

Total 22

What is data?o Data

A collection of raw facts from which conclusion may be drawn

e.g. Data: Temperature 35°C; Conclusion: It is hot.

o Types of data

Textual: For example, your name (Arash)

Numeric: For example, your ID (40254)

Audio: For example, your voice

Video: For example, your voice and picture

(...)

• DS: is representation of the logical relationship existing between individual elements of data

• DS: is a specialized format for organizing and storing data in memory that considers not only

the elements stored but also their relationship to each other.

• A particular way of storing and organizing data in a computer so that it can be used efficiently

and effectively.

• Data structure is the logical or mathematical model of a particular organization of data.

• A group of data elements grouped together under one name. For example, an array of integers

What is data structure?

The Need for Data Structures

o Goal: to organize data

o Criteria: to facilitate efficient storage of data

retrieval of data

manipulation of data

o Design Issue: select and design appropriate data types

(This is the main motivation to learn and understand data structures)

Data Structure Operations

Traversing

Accessing each data element exactly once so that

certain items in the data may be processed

Searching

Finding the location of the data element (key) in the

structure

Insertion

Adding a new data element to the structure

Data Structure Operations

Deletion

Removing a data element from the structure

Sorting

Arrange the data elements in a logical order

(ascending/descending)

Merging

Combining data elements from two or more data

structures into one

Array

Linked ListTree

Queue

Stack

Types of data structures

In detail, when we talk about data structures and algorithms, we are talking queues, stacks, heaps, binary

search, dynamic programming and so on.

These are patterns for abstraction and solving problems, which invented by computer pioneers. We can

effectively learn them and use them to solve many practical development problems.

There are many, but we named a few. We’ll learn these data structures in great detail!

What is algorithm?

• A finite set of instructions which accomplish a particular task

• A method or process to solve a problem

• Transforms input of a problem to output

• The definition of an algorithm is: The methods for solving problems, which could

be implemented in programming.

Algorithm development is an art and it needs practice!

• It must be correct

• It must be finite (in terms of time and size)

• It must terminate

• It must be unambiguous Which step is next?

• It must be space and time efficient

A problem can be solved in many

different ways, which solution,

amongst the different possible

solutions is optimal?

Notes:

*Efficiency is always a concern and as the inout gets larger it becomes more of a concern.

*Scalability

*Efficient procedures for solving large-scale problems

Concept of Abstraction

We use abstraction everyday!!!

Class iPhone: Phone

{

}

Class Samsung: Phone

{

}

Class Huawei: Phone

{

}

.

.

.

Class Repair-Phone

{

public void repair (iPhone ip)

{

\\code

}

public void repair (Samsung s)

{

\\code

}

public void repair (Huawei h)

{

\\code

}

}

Concept of Abstraction

We use abstraction everyday!!!

Abstract class Phone

{

}

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Class iPhone: Phone

{

}

Class Samsung: Phone

{

}

Class Huawei: Phone

{

}

.

.

.

Class Repair-Phone

{

public void repair (Phone p)

{

\\code

}

}

Peak FindingOne-Dimensional

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]

1 2 . n/2 n-1

Peak: Position i is a peak if and only if A[i-1] <= A[i] >= A[i+1].

A[0] >= A[1]

A[n-1] >= A[n-2]

Problem: Find a Peak if it exists.

. n

A[i] >= 0

Peak FindingOne-Dimensional

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]

1 2 . n/2 n-1

Peak: Position i is a peak if and only if A[i-1] <= A[i] >= A[i+1].

A[0] >= A[1]

A[n-1] >= A[n-2]

Problem: Find a Peak (if it exists).

. n

A[i] >= 0

Straight-Forward-Algorithm

Might be Peak

Algorithm: Start from one side, say left, and search the array for a peak.

Worst-case complexity: in the worst-case we have to look at all the elements.

*This is a linear time algorithm.

Analysis of Algorithms

تحلیل برنامه ها

حافظه, زمان اجرا: پارامترهای مهم در تحلیل برنامه ها و الگوریتم ها

, (اری میانیسیستم نرم افز)سیستم عامل , کامپایلر, پیچیدگی سخت افزار, نوع پردازنده...اندازه یا پیچیدگی ورودی و , پیچیدگی الگوریتم

عت در یک دوره ی زمانی طراحی سخت افزار طوری پیشرفت می کند که سر: قانون مور.در واقع سرعت سخت افزار خطی است. آن نسبت به قبل دو برابرمی شود

.یک الگوریتم برحسب اندازه می تواند نمائی باشد( رشد)در حالی که رفتار*

Peak FindingOne-Dimensional

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]

1 2 . n/2 n-1

Peak: Position i is a peak if and only if A[i-1] <= A[i] >= A[i+1].

A[0] >= A[1]

A[n-1] >= A[n-2]

Problem: Find a Peak if it exists.

. n

A[i] >= 0

Peak FindingOne-Dimensional

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]

1 2 . n/2 n-1

Peak: Position i is a peak if and only if A[i-1] <= A[i] >= A[i+1].

A[0] >= A[1]

A[n-1] >= A[n-2]

Problem: Find a Peak if it exists.

. n

A[i] >= 0

A Better Algorithm

Look at the n/2 position

Algorithm: (Recursive)

1- If A[n/2] < A[n/2 -1] then only look at the left half [1 .. n/2] to look for a peak.

2- Else if A[n/2] < A[n/2 + 1] then only look at [n/2 .. n] for a peak.

3- Else n/2 position is a peak.

Peak FindingOne-Dimensional

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]

1 2 . n/2 n-1

Peak: Position i is a peak if and only if A[i-1] <= A[i] >= A[i+1].

A[0] >= A[1]

A[n-1] >= A[n-2]

Problem: Find a Peak if it exists.

. n

A[i] >= 0

Look at the n/2 position

Algorithm: (Recusive)

1- If A[n/2] < A[n/2 -1] then only look at the left half [1 .. n/2] to look for a peak.

2- Else if A[n/2] < A[n/2 + 1] then only look at [n/2 .. n] for a peak.

3- Else n/2 position is a peak.

Complexity of the Algorithm:

T(n) = “work” algorithm does on input of size n

T(n) = T(n/2) + 2

We can prove that this version takes only log n time complexity.

?