Introduction to Data Structure, Lecture notes of Data Structures and Algorithms

Download Introduction to Data Structure and more Lecture notes Data Structures and Algorithms in PDF only on Docsity!

Lecture # 1

Introduction

to

Data Structure and Algorithms

B y : M r. V i j a y K u m a r S o n i

Data Structure & Algorithms

• (^) Algorithms : Outline ,the essence of computational procedures, step-by-step instructions.
• (^) Program: An implementation of an algorithm in some programming language.
• (^) Data Structure: Organization of data needed to solve the problem.

Algorithm Solution

• (^) Algorithm describes action on the input instance.
• (^) Infinitely many correct algorithms for the same algorithmic problem. Input Instance, adhering to the specification Algorithm Output related to the input as required

What is a Good Algorithm?

• (^) Efficient:
  • (^) Running time
  • (^) Space used
• (^) Efficiency as a function of input size:
  • (^) The number of bits in an input number.
  • (^) Number of data element(numbers, etc.).

Limitations of Experimental Studies

• (^) It is necessary to implement and test the algorithm in order to determine its running time.
• (^) Experiment can be done only on a limited set of inputs.
• (^) In order to compare two algorithms, the same hardware and software environment should be used.

Beyond Experimental Studies

• (^) We will develop a general methodology for

analyzing running time of algorithms:

• (^) This approach uses a high-level description of the algorithm instead of testing one of its implementations.
• (^) Takes into account all possible inputs.
• (^) Allows one to evaluate the efficiency of any algorithm in a way that is independent of the hardware and software environment.

P s e u d o - C o d e ( 2 )

• (^) It is more structured then usual prose but less

formal than a programming language.

• (^) Expressions:
  • (^) Use standard mathematical symbols to describe numeric and boolean expressions.
  • (^) Use ← for assignment (“=“ in java)
  • (^) Use = for the equality relationship (“==“ in java)
• (^) Method Declarations:
  • (^) Algorithm name(param1,param2)

P s e u d o - C o d e ( 3 )

• (^) Programming Constructs:
  • (^) Decision structures: if……..then….[else..]
  • (^) While-loops: while…do
  • (^) Repeat-loops: repeat….until …
  • (^) For-loop: for…do
  • (^) Array indexing: A[i],A[i,j]
• (^) Methods
  • (^) Calls: object method(args)
  • (^) Returns: return value

E x a m p l e : S o r ti n g

INPUT OUTPUT

Sequence of numbers a permutation of the sequence of numbers a1,a2,a3,…..an b1,b2,b3………bn 2 5 4 6 8 2 4 5 6 8 Sort Running Time Depends on

• (^) numbers of elements(n)
• How(partially) sorted they are
• algorithm

E x a m p l e : S o r ti n g

INPUT OUTPUT

Sequence of numbers a permutation of the sequence of numbers a1,a2,a3,…..an b1,b2,b3………bn 2 5 4 6 8 2 4 5 6 8 Running Time Depends on

• (^) numbers of elements(n)
• How(partially) sorted they are
• algorithm

Analysis of Insertion Sort

for j=2 to length (A) do key=A[j] “insert A[j] into the sorted sequence A[1..j-1]” i=j- while i>0 and A[i]>key do A[i+1]=A[i] i-- A[i+1]:=key cost c 1 c 2 0 c 3 c 4 c 5 c 6 c 7 times n n- n- n- n- 2 n  j  t j 2 ( 1) n  j  t j  2 ( 1) n  j  t j 

• (^) Time to compute the running time as a function of the input size

Best/Worst/Average Case

• Best case : elements already sorted ® tj=1, running time = f(n), i.e., linear time.
• (^) Worst case : elements are sorted in inverse order ® tj=j , running time = f(n^2 ), i.e., quadratic time
• (^) Average case : tj=j/2, running time = f(n^2 ), i.e., quadratic time

Best/Worst/Average Case (3)

• (^) For inputs of all sizes: 1n 2n 3n 4n 5n 6n Input instance size Running time 1 2 3 4 5 6 7 8 9 10 11 12 ….. best-case average-case worst-case

Best/Worst/Average Case (4)

• (^) Worst case is usually used:
  • (^) It is an upper-bound and in certain application domains (e.g., air traffic control, surgery) knowing the worst-case time complexity is of crucial importance
  • (^) For some algorithms worst case occurs fairly often
  • (^) The average case is often as bad as the worst case
  • (^) Finding the average case can be very difficult