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BMS Institute of Technology and Mgmt Department of ISE Department of ISE BMS Institute of Technology and Mgmt

Design and Analysis of Algorithm

18CS42

4th Sem

BMS Institute of Technology and Mgmt Department of ISE

Module

Number

Module Title Page

Number

1 Introduction 1-78

2 Divide and Conquer 79-137

3 Greedy Method 138-187

4 Dynamic Programming 188-257

5 Backtracking 258-317

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MODULE ² 1

INTRODUCTION

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Course Outcomes(COs):

CO1 Gain knowledge on various algorithmic concepts to solve problems. CO2 Apply the basic knowledge of mathematical fundamentals for finding time complexity of recursive and non-recursive algorithms. CO3 Analyse various problems and choose appropriate algorithmic technique to use for solving real time problems. CO4 Design algorithms for various real time applications. CO5 Conduct investigation on societal problems and develop code using contemporary computing languages. CO6 Work in team and communicate effectively on various algorithmic techniques. At the end of the course, the students will be able to attain the following skills.

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Agenda

9 9 9 9

9-KZv}šš]}v~KUKuPv}šš]}v~QU

dZšv}šš]}v~:Uv>]ššooh notation (o)

9-Recursive

9 9

Combinatorial Problems.

9

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Learning Outcomes of Module -1

Students will be able to

9Representing real world problem into algorithmic notation.

9Performance analysis of an algorithm.

9Important problem types.

9Fundamental Data structures. 4

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: The sprit of computing tDavid Harel.

Another reason for studying algorithms is their

usefulness in developing analytical skills. can be seen as special kinds of solutions to problems tnot answers but rather precisely defined for getting answers.

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Finiteness

I

Definiteness

I

Clearly specified input

I

Clearly specified/expected output

I

Effectiveness

I

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‡Can be represented in various forms ‡Unambiguity/clearness ‡Effectiveness ‡Finiteness/termination ‡Correctness

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is a sequence of unambiguous instructions legitimate input in a finite amount of time.

³&RPSXWHU´

Input Output

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µo][oP}OE]šZu

Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? trivial.

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= 0, return and stop; otherwise go to Step 2

Divide by r

and the value of to

Step 1.

whilenBì} r і m mі n n і r m

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Step 2 Divide by

otherwise, go to Step 4

Divide by and stop;

otherwise, go to Step 4

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-school procedure

Step 2 Find the prime factorization of

and return it as gcd

How efficient is it?

13

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Integer •2

Output: List of primes less than or equal to

Vîš}'OEš~do

if z0 //Zv[švo]u]vš}v‰OEÀ]}µ‰ V "n Vì V

Output: 2 3 5 7 11 13 17 19

14

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Fundamental steps in solving problems

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Fundamental steps in solving problems

9 9 9 9 9 9 9 9

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±vertices, some of

which are connected by line segments called edges. ‡-life problems

±WOE}išZµo]vPY

±-path algorithms

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data type that are stored contiguously in computer memory and made

OEOEÇ[]vAEX

each containing two kinds of nodes of the linked list. pointers) "Arrays "fixed length (need preliminary reservation of memory) "contiguous memory locations "direct access "Insert/delete "Linked Lists "dynamic length "arbitrary memory locations "access by following links "Insert/delete " an a2

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front.

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" Priority queues (implemented using heaps) " A data structure for maintaining a set of elements, each associated with a key/priority, with the following operations " Finding the element with the highest priority " Deleting the element with the highest priority " Inserting a new element " Scheduling jobs on a shared computer 9 6 8 5 2 3

9 6 5 8 2 3

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±is defined by a pair of two sets: a finite set V of items called vertices and a set E of vertex pairs called edges. ‡and directed graphs (digraphs). ‡ and sparse graphs complete, K 1 2 3 4

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‡-node star shape? 0 0 0 1 0 0 0 1 0 0 0 0 2 3 4 4 4

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1 2 3 4 8 9

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-- Paths and Connectivity

±: All edges of a path are distinct.

±t1.

there is a path from u to v.

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-- Acyclicity a the same vertex.

±(Directed Acyclic Graph)

1 2 3 4

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±free tree) is a connected acyclic graph.

path from one of these vertices to the other. Why? ‡: The above property makes it possible to select an arbitrary vertex in a free tree and consider it as the root of the so called rooted tree. " |E| = |V| - 1 1 3 2 4 5 1 3 2 4 5

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±in a tree

from the root to that vertex are called ancestors. ‡Descendants

±is an ancestor are said to be

descendants of ‡ and siblings ±is the last edge of the simple path from the root to vertex is said to be the parent of and is called a child of ±with all its descendants is called the subtree of

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‡ of a vertex ‡ of a tree 1 3 2 4 5

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two children and each children is designated s either a left child or a ‡¬¼dh dn t1, where h is the height of a binary tree and n the size. 9 6 8 5 2 3 6 3 9 2 5 8

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constant log logarithmic n-n n quadratic 2

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Values of some important functions as n o f

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WL

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-case, average-case, worst-case ‡Ctmaximum over inputs of size ‡Ctminimum over inputs of size input.

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‡no faster than ‡,at same rate as at least as fast as

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f"order of growth of i.e., there exist positive constant and non-negative integer such that ‡5= 1 The Upper Bound indicates that the function will be the worst case that it does not consume more than this computing time.

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-oh

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-omega

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f]]vae~ with the constraint that c1 g(n) " "Ffor every •

Example:

‡ae ‡ g(n) ""Ffor every • ‡n ""for every

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-theta

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Properties of asymptotic order of growth

Note similarity with "

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‡indicating ‡/vš](ÇoP}OE]šZu[ ‡cases for input of size ‡executed

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lim

0 order of growth of Tn) < order of growth of gn

c order of growth of Tn) = order of growth of gn ' order of growth of Tn) > order of growth of gn ‡ nvs. n ‡ nnn n:' BMS Institute of Technology and Mgmt Department of ISE - O(n) Best case - Q t: BMS Institute of Technology and Mgmt Department of ISE BMS Institute of Technology and Mgmt Department of ISE

Comparison

A[i] > max

4.

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BMS Institute of Technology and Mgmt Department of ISE 3. 4. /2Q2)

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Input parameter is input size n

i,j] == C[i,j] + A[i,k] * B[k,j] 3. /2Q3)

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(A[0..n-1]) //Input: An array A[0..n-1] of orderable elements //Output: Array A[0..n-1] sorted in ascending order

Å0 to n t2 do

min Åi for j Åi + 1 to n t1 do if A[j] < A[min]quotesdbs_dbs10.pdfusesText_16

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