Data Structures and Algorithms

Lecture 1: Asymptotic Notations

Basic Terminologies

• Algorithm– Outline– Essence of a computational procedure– Step by step instructions

• Program– Implementation of an Algorithm in some programming

language• Data Structure– Organization of Data needed to solve the problem

effectively– Data Structure you already know: Array

Algorithmic Problem

Specification of Input

?Specification

of Output as a function of

Input

• Infinite number of input instances satisfying the specification

• E.g.: Specification of Input:• A sorted non-decreasing sequence of natural numbers

of non-zero, finite length:• 1, 20,908,909,1000,10000,1000000• 3

Algorithmic Solution

• Algorithm describes the actions on the input instances

• Infinitely many correct algorithms may exist for the same algorithmic problem

Input instance adhering to

Specification

Algorithm

Output related to the

Input as required

Characteristics of a Good Algorithm

• Efficient– Small Running Time– Less Memory Usage

• Efficiency as a function of input size– The number of bits in a data number– The number of data elements

Measuring the running time

• Experimental Study– Write a program that implements the algorithm– Run the program with data sets of varying size and

composition– Use system defined functions like clock() to get an

measurement of the actual running time

Measuring the running time

• Limitations of Experimental Study– It is necessary to implement and test the

algorithm in order to determine its running time.– Experiments can be done on a limited set of

inputs, and may not be indicative of the running time on other inputs not included in the experiment

– In order to compare 2 algorithms same hardware and software environments must be used.

Beyond Experimental Study

• We will develop a general methodology for analyzing the 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.

Pseudo - Code

• A mixture of natural language and high level programming concepts that describes the main ideas behind a generic implementation of a data structure or algorithm.

• E.g. Algorithm arrayMax(A,n)– Input: An array A storing n integers– Output: The maximum element in A. currMax A[0] for i 1 to n-1 do if currMax < A[i] then currMax A[i] return currMax

Pseudo - Code

• It is more structured that usual prose but less formal than a programming language

• Expressions:– Use standard mathematical symbols to describe

numeric and Boolean expressions– Use ‘ ’ for assignment instead of “=” – Use ‘=’ for equality relationship instead of “==”

• Function Declarations:– Algorithm name (param1 , param2)

Pseudo - Code

• 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]

• Functions:– Calls: return_type function_name(param1 ,param2)– Returns: return value

Analysis of Algorithms

• Primitive Operation: – Low level operation independent of programming

language. – Can be identified in a pseudo code.

• For e.g. – Data Movement (assign)– Control (branch, subroutine call, return)– Arithmetic and Logical operations (e.g. addition

comparison)– By inspecting the pseudo code, we can count the number

of primitive operations executed by an algorithm.

Example: Sorting

Sort

INPUTSequence of numbers

OUPUTPermutation of the sequence of numbers in non-decreasing order

a1,a2,a3,a4,….an b1,b2,b3,b4,….bn

2, 5 , 4, 10, 7 2, 4 , 5, 7, 10Correctness (Requirements for the output)For any given input the algorithm halts with the output:• b1 < b2 < b3 < b4 < ….. < bn

• b1, b2, b3, b4,…. bn is a permutation of a1, a2, a3, a4, …. an

Running Time depends on• No. of elements (n)• How sorted (partial) the numbers are? • Algorithm

Insertion Sort• Used while playing cards

3 4 6 8 9 7 2 5 1A1 n

i jStrategy• Start “empty handed”• Insert a card in the right position of the already sorted hand• Continue until all cards are inserted or sorted

INPUT: An array of Integers A[1..n] OUTPUT: A permutation of A such that A[1] <= A[2] <= A[3]<= ..<=A[n]For j 2 to n do

key A[j]Insert A[j] into sorted sequence A[1…j-1]i j-1while i >0 and A[i] >key

do A[i+1] A[i]i- -

A[i+1] key

Total Time = n(c1 + c2 + c3 + c7) + (c4 + c5 + c6)–( c2 +c3 +c5+c6+c7)

tj counts the number of times the values have to be shifted in one iteration

Analysis of Insertion Sort

• For j 2 to n do• key A[j]• Insert A[j] into sorted

sequence A[1…j-1]• i j-1• while i >0 and A[i]

>key• do A[i+1] A[i]• i- - • A[i+1] key

Cost Times c1 n c2 n-1

c3 n-1

c4 c5 c6 c7 n-1

Best / Average/ Worst Case: Difference made by tj

• Total Time = n(c1 + c2 + c3 + c7) + (c4 + c5 + c6)–( c2 +c3 +c5+c6+c7)

• 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(n2) i.e. Quadratic Time

• Average Case– tj = j/2– Running Time = f(n2) i.e. Quadratic Time

Best / Average/ Worst Case

• For a Specific input size say n

Input instance

Runn

ing

Tim

e (s

ec)

A B C D E F G H I J K L M N

5

4

3

2

1

Best Case

Worst Case

Average Case

Best / Average/ Worst Case

• Varying input size

Input Size

Runn

ing

Tim

e (s

ec)

50

40

30

20

10

10 20 30 40 50 60

Worst case

Average Case

Best Case

Best / Average/ Worst Case

• Worst Case:– Mostly used– It is an Upper bound of how bad a system can be.– In cases like surgery or air traffic control knowing

the worst case complexity is of crucial importance.• Average case is often as bad as worst case.• Finding an average case can be very difficult.

Asymptotic Analysis

• Goal: Simplify analysis of running time by getting rid of details which may be affected by specific implementation and hardware– Like rounding 1000001 to 1000000– 3n2 to n2

• Capturing the essence: How the running time of an algorithm increases with the size of the input in the limit– Asymptotically more efficient algorithms are best for

all but small inputs.

Asymptotic Notation

• The “big Oh” (O) Notation– Asymptotic Upper Bound– f(n) = O(g(n)), if there exists

constants c and n0 such that f(n) ≤ cg(n) for n ≥ n0

– f(n) and g(n) are functions over non negative non-decreasing integers

• Used for worst case analysis

n0f(n) = O(g(n))

n

f(n)

c. g(n)

Examples• E.g. 1:

– f(n) = 2n + 6– g(n) = n– c = 4– n0 = 3– Thus, f(n) = O(n)

• E.g. 2:– f(n) = n2

– g(n) = n– f(n) is not O(n) because there exists no

constants c and n0 such that f(n) ≤ cg(n) for n ≥ n0

• E.g. 3:– f(n) = n2 + 5– g(n) = n2

– c = 2– n0 = 2– Thus, f(n) = O(n2)

Asymptotic Notation

• Simple Rules:– Drop lower order terms and constant factors• 50 nlogn is O(nlogn)• 7n – 3 is O(n)• 8n2logn + 5n2 + n is O(n2logn)

– Note that 50nlogn is also O(n5) but we will consider it to be O(nlogn) it is expected that the approximation should be of as lower value as possible.

Comparing Asymptotic Analysis of Running Time

• Hierarchy of f(n)– Log n < n < n2 < n3 < 2n

• Caution!!!!!– An algorithm with running time of 1,000,000 n is

still O(n) but might be less efficient than one running in time 2n2, which is O(n2) when n is not very large.

Examples of Asymptotic Analysis

• Algorithm: prefixAvg1(X)• Input: An n element array X of numbers• Output: An n element array A of numbers such that

A[i] is the average of elements X[0]…X[i].for i 0 to n-1 do

a 0 for j 0 to i do a a+X[j] A[i] a/(i+1)return A

• Analysis: Running Time O(n2) roughly.

1 Stepi iterations i = 0 , 1, …., i-1

n iterations

Examples of Asymptotic Analysis

• Algorithm: prefixAvg2(X)• Input: An n element array X of numbers• Output: An n element array A of numbers such that A[i]

is the average of elements X[0]…X[i].s 0

for i 0 to n do s s + X[i]

A[i] s/(i+1) return A• Analysis: Running time is O(n)

Asymptotic Notation (Terminologies)

• Special cases of Algorithms– Logarithmic : O(log n)– Linear: O(n)– Quadratic: O(n2)– Polynomial: O(nk), k ≥ 1– Exponential: O(an), a>1

• “Relatives” of Big Oh– Big Omega (Ω(f(n)) : Asymptotic Lower Bound– Big Theta (Θ(f(n)): Asymptotic Tight Bound

Asymptotic Notation

• The big Omega (Ω) Notation– Asymptotic Lower bound– f(n) = Ω(g(n), if there exists

constants c and n0 such that c.g(n) ≤ f(n) for n > n0

• Used to describe best case running times or lower bounds of asymptotic problems– E.g: Lower bound of searching

in an unsorted array is Ω(n).

Asymptotic Notation

• The Big Theta (Θ) notation– Asymptotically tight bound– f(n) = Θ(g(n)), if there exists

constants c1, c2 and n0 such that c1.g(n) ≤ f(n) ≤ c2.g(n) for n > n0

– f(n) = Θ(g(n)), iff f(n) = O(g(n)) and f(n) = Ω(g(n),

– O(f(n)) is often misused instead of Θ(f(n))

Asymptotic Notation

• There are 2 more notations– Little oh notation (o), f(n) = o(g(n))

• For every c there should exist a no. n0 such that f(n) ≤ o(g(n) for n > n0.

– Little Omega notation (ω)• Analogy with real numbers

– f(n) = O(g(n)) , f≤g– f(n) = Ω(g(n)), f≥g– f(n) = Θ(g(n)), f = g– f(n) = o(g(n)), f<g– f(n) = ω(g(n)), f>g

• Abuse of Notation: – f(n) = O(g(n)) actually means f(n) є O(g(n))

Comparison of Running times

Assignment

• Write a C program to implement i) Insertion sort and ii) Bubble Sort.

• Perform an Experimental Study and graphically show the time required for both the program to run.

• Consider varied size of inputs and best case and worst case for each of the input sets.