daa unit i notes

7/28/2019 DAA Unit I Notes

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1) What is a Computer Algorithm?

An algorithm is a sequence of unambiguous

instructions for solving a problem, i.e., forobtaining a required output for any legitimate

input in a finite amount of time.

2) What are the features of an algorithm? More precisely, an algorithm is a method or process to solve a problem satisfying

the following properties:

Finiteness terminates after a finite number of steps

Definiteness

Each step must be rigorously and unambiguously specified.

-e.g., stir until lumpy Input

Valid inputs must be clearly specified.

Output

can be proved to produce the correct output given a valid can beproved to produce the correct output given a valid input.

Effectiveness Steps must be sufficiently simple and basic.

3) Show the notion of an algorithm.

4) What are different problem types?

5) What are different algorithm design techniques/strategies?


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Brute force

Divide and conquer

Decrease and conquer

Transform and conquer

Space and time tradeoffs

Greedy approach

Dynamic programming

Backtracking

Branch and bound

6) What are fundamental data structures?

list array

linked list string

stack

queue

priority queue graph

tree

set and dictionary

7) What are the sequence of steps in designing and analyzing the algorithm?

Fundamentals of Algorithmic Problem Solving

Understanding the problem

Asking questions, do a few examples by hand, think about special cases,etc.

Deciding on

Exact vs. approximate problem solving

Appropriate data structure Design an algorithm

Proving correctness

Analyzing an algorithm Time efficiency : how fast the algorithm runs

Space efficiency: how much extra memory the algorithm needs.

Coding an algorithm


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8) What is algorithm analysis framework?

Analysis of algorithms means to investigate an algorithms efficiency with respect to

resources: running time and memory spaceTime efficiency: how fast an algorithm runs.

Space efficiency: the space an algorithm requires.

Typically, algorithms run longer as the size of its input increases

We are interested in how efficiency scales wrt input size

Analysis Framework

Measuring an inputs size

Measuring running time

Orders of growth (of the algorithms efficiency function)

Worst-base, best-case and average-case efficiency

9) How the running time of an algorithm is measured?

Units for Measuring Running Time


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Count the number of times an algorithms basic operation is executed.

Basic operation: the operation that contributes the most to the total

running time.

For example, the basic operation is usually the most time-consuming

operation in the algorithms innermost loop.

10) How time effieciency is analysed? Time efficiency is analyzed by determining the number of repetitions of the basic

operation as a function of input size.

11) What is orders of growth?

Orders of Growth

12) What are Worst-Case, Best-Case, and Average-Case Efficiency ?

Worst case Efficiency

Efficiency (# of times the basic operation will be executed) for the worstcase input of size n.

The algorithm runs the longest among all possible inputs of size n.

Best case

Efficiency (# of times the basic operation will be executed) for the bestcase input of size n.

The algorithm runs the fastest among all possible inputs of size n.

runningtime

execution timefor the basic

operation

Number of times the

basic operation is

executed

input

size

T(n) copC (n)


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Average case:

Efficiency (#of times the basic operation will be executed) for a

typical/random input of size n.

NOT the average of worst and best case

How to find the average case efficiency?

13) What are asymptotic notations? Explain in detail.Asymptotic Growth Rate

Three notations used to compare orders of growth of an algorithms basic

operation count

O(g(n)): class of functionsf(n) that grow no fasterthang(n)

(g(n)): class of functionsf(n) that grow at least as fastas g(n)

(g(n)): class of functionsf(n) that grow at same rate asg(n)

O-notation

Formal definition

A function t(n) is said to be in O(g(n)), denoted t(n) O(g(n)), ift(n) is

bounded above by some constant multiple ofg(n) for all large n, i.e., ifthere exist some positive constant c and some nonnegative integern0 such

that

t(n) cg(n) for all n n0

-notation

Formal definition


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A function t(n) is said to be in (g(n)), denoted t(n) (g(n)), ift(n) is

bounded below by some constant multiple ofg(n) for all large n, i.e., if

there exist some positive constant c and some nonnegative integern0 such

that

t(n) cg(n) for all n n0

-notation

Formal definition

A function t(n) is said to be in (g(n)), denoted t(n) (g(n)), ift(n) is

bounded both above and below by some positive constant multiples ofg(n) for all large n, i.e., if there exist some positive constant c1 and c2 and

some nonnegative integern0 such thatc2 g(n) t(n) c1 g(n) for all n n0


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12) What are basic efficiency classes?

Basic Efficiency classes

1 constant

log n logarithmic

n linear

n log n n log n

n2 Basic Efficiency

classes

quadratic

n3 cubic

2n exponential

n! factorial

13) Give an example for basic operations.

Input size and basic operation examples

Problem Input size measure Basic operation

Searching for key in a

list ofn items

Number of lists items,

i.e. nKey comparison

Multiplication of two

matrices

Matrix dimensions or

total number of elements

Multiplication of two

numbers

Checking primality of agiven integern

nsize = number of digits(in binary representation)

Division

Typical graph problem #vertices and/or edgesVisiting a vertex or

traversing an edge


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14) Write an algorithm for counting binary digits for a decimal number.

15) Write an algorithm for matrix multiplication.

daa unit i notes

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