Ada Lab Manual - bcl404

DEPARTMENT OF INFORMATION SCIENCE & ENGINEERING
Course Material
2022 SCHEME
ANALYSIS & DESIGN OF ALGORITHMS LAB
(BCSL404)
4th SEMESTER
by
Mr. Anand M
Assistant Professor, ISE, GSSSIETW, Mysuru
anandm@gsss.edu.in
Vision:
To be recognized nationally as an accomplished department producing global leaders and
researchers as committed, enthusiastic and excellent ethical women technical graduate for the
better tomorrow.
Mission:
M1: To provide Students with Innovative and research skills, which is need of the hour for
technical students.
M2: To impart knowledge to the students of Information Science and Engineering with
relevant core and practical knowledge inculcating real time experience in the promising field
of computing.
M3: To prepare the graduates to meet information technology challenges with a blend of
social, human, ethical and value-based education.
Program Educational Objectives (PEO's):

PEO1: To produce graduates who have the ability to demonstrate technical competence in the
fields of information Technology and develop solutions to the problems.
PEO2: Gain the ability to analyze and solve Information science and engineering problems
through application of fundamental knowledge of maths, science, and engineering.
PEO3: To endeavour innovations with a substantial fundamental technical skill aiming the
career in research and development.
PEO4: Understand and deal with ethical, societal and global issues associated with the
computing profession. PEO5: Exhibit professionalism, leadership roles, team work in meeting
the growing demands of IT industry.
Program Outcomes (PO’s)
PO1: Engineering knowledge: Apply the knowledge of mathematics, science, engineering fundamentals, and an
engineering specialization to the solution of complex engineering problems.
PO2:Problem analysis: Identify, formulate, review research literature, and analyze complex engineering
problems reaching substantiated conclusions using first principles of mathematics, natural sciences, and
engineering sciences.
PO3:Design/development of solutions: Design solutions for complex engineering problems and design system
components or processes that meet the specified needs with appropriate consideration for the public health and
safety, and the cultural, societal, and environmental considerations.
PO4:Conduct investigations of complex problems: Use research-based knowledge and research methods
including design of experiments, analysis and interpretation of data, and synthesis of the information to provide
valid conclusions.
PO5:Modern tool usage: Create, select, and apply appropriate techniques, resources, and modern engineering
and IT tools including prediction and modelling to complex engineering activities with an understanding of the
limitations.
PO6:The engineer and society: Apply reasoning informed by the contextual knowledge to assess societal, health,
safety, legal and cultural issues and the consequent responsibilities relevant to the professional engineering
practice.
PO7:Environment and sustainability: Understand the impact of the professional engineering solutions in societal
and environmental contexts, and demonstrate the knowledge of, and need for sustainable development.
PO8:Ethics: Apply ethical principles and commit to professional ethics and responsibilities and norms of the
engineering practice.
PO9:Individual and team work: Function effectively as an individual, and as a member or leader in diverse
teams, and in multidisciplinary settings.
PO10:Communication: Communicate effectively on complex engineering activities with the engineering
community and with society at large, such as, being able to comprehend and write effective reports and design
documentation, make effective presentations, and give and receive clear instructions.
PO11:Project management and finance: Demonstrate knowledge and understanding of the engineering and
management principles and apply these to one’s own work, as a member and leader in a team, to manage
projects and in multidisciplinary environments.
PO12:Life-long learning: Recognize the need for, and have the preparation and ability to engage in independent
and life-long learning in the broadest context of technological change.
Program Specific Outcomes (PSO)
PSO1: Participate in planning, implementing, evaluating reliable and secure IT Solutions to specific business
problems.
PSO2: Enabling the students with Project Management Skill through Software Development lifecycle.
Template for Practical Course and if AEC is a practical Course Annexure-V
Analysis & Design of Algorithms Lab Semester 4

Course Code BCSL404 CIE Marks 50
Teaching Hours/Week (L:T:P: S) 0:0:2:0 SEE Marks 50
Credits 01 Exam Hours 2
Examination type (SEE) Practical
Course objectives:
● To design and implement various algorithms in C/C++ programming using suitable development tools to
address different computational challenges.
● To apply diverse design strategies for effective problem-solving.
● To Measure and compare the performance of different algorithms to determine their efficiency and suitability
for specific tasks.
Sl.No Experiments
1 Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a given connected
undirected graph using Kruskal's algorithm.
2 Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a given connected
undirected graph using Prim's algorithm.
3 a. Design and implement C/C++ Program to solve All-Pairs Shortest Paths problem using Floyd's
algorithm.
b. Design and implement C/C++ Program to find the transitive closure using Warshal's
algorithm.
4 Design and implement C/C++ Program to find shortest paths from a given vertex in a weighted
connected graph to other vertices using Dijkstra's algorithm.
5 Design and implement C/C++ Program to obtain the Topological ordering of vertices in a given
digraph.
6 Design and implement C/C++ Program to solve 0/1 Knapsack problem using Dynamic
Programming method.
7 Design and implement C/C++ Program to solve discrete Knapsack and continuous Knapsack
problems using greedy approximation method.
8 Design and implement C/C++ Program to find a subset of a given set S = {sl , s2,.....,sn} of n
positive integers whose sum is equal to a given positive integer d.
9 Design and implement C/C++ Program to sort a given set of n integer elements using Selection
Sort method and compute its time complexity. Run the program for varied values of n> 5000 and
record the time taken to sort. Plot a graph of the time taken versus n. The elements can be read
from a file or can be generated using the random number generator.
10 Design and implement C/C++ Program to sort a given set of n integer elements using Quick Sort
method and compute its time complexity. Run the program for varied values of n> 5000 and
11 Design and implement C/C++ Program to sort a given set of n integer elements using Merge Sort
method and compute its time complexity. Run the program for varied values of n> 5000, and
12 Design and implement C/C++ Program for N Queen's problem using Backtracking.
@# 16032024
Course outcomes (Course Skill Set):

At the end of the course the student will be able to:
1. Develop programs to solve computational problems using suitable algorithm design strategy.
2. Compare algorithm design strategies by developing equivalent programs and observing running
times for analysis (Empirical).
3. Make use of suitable integrated development tools to develop programs
4. Choose appropriate algorithm design techniques to develop solution to the computational and
complex problems.
5. Demonstrate and present the development of program, its execution and running time(s) and
record the results/inferences.
Assessment Details (both CIE and SEE)
The weightage of Continuous Internal Evaluation (CIE) is 50% and for Semester End Exam (SEE) is 50%.
The minimum passing mark for the CIE is 40% of the maximum marks (20 marks out of 50) and for the
SEE minimum passing mark is 35% of the maximum marks (18 out of 50 marks). A student shall be
deemed to have satisfied the academic requirements and earned the credits allotted to each subject/
course if the student secures a minimum of 40% (40 marks out of 100) in the sum total of the CIE
(Continuous Internal Evaluation) and SEE (Semester End Examination) taken together
Continuous Internal Evaluation (CIE):

CIE marks for the practical course are 50 Marks.
The split-up of CIE marks for record/ journal and test are in the ratio 60:40.
● Each experiment is to be evaluated for conduction with an observation sheet and record write-up.
Rubrics for the evaluation of the journal/write-up for hardware/software experiments are
designed by the faculty who is handling the laboratory session and are made known to students at
the beginning of the practical session.
● Record should contain all the specified experiments in the syllabus and each experiment write-up
will be evaluated for 10 marks.
● Total marks scored by the students are scaled down to 30 marks (60% of maximum marks).
● Weightage to be given for neatness and submission of record/write-up on time.
● Department shall conduct a test of 100 marks after the completion of all the experiments listed in
the syllabus.
● In a test, test write-up, conduction of experiment, acceptable result, and procedural knowledge will
carry a weightage of 60% and the rest 40% for viva-voce.
● The suitable rubrics can be designed to evaluate each student’s performance and learning ability.
● The marks scored shall be scaled down to 20 marks (40% of the maximum marks).
The Sum of scaled-down marks scored in the report write-up/journal and marks of a test is the total CIE
marks scored by the student.
Semester End Evaluation (SEE):

● SEE marks for the practical course are 50 Marks.
@# 16032024
● SEE shall be conducted jointly by the two examiners of the same institute, examiners are
appointed by the Head of the Institute.
● The examination schedule and names of examiners are informed to the university before the
conduction of the examination. These practical examinations are to be conducted between the
schedule mentioned in the academic calendar of the University.
● All laboratory experiments are to be included for practical examination.
● (Rubrics) Breakup of marks and the instructions printed on the cover page of the answer script
to be strictly adhered to by the examiners. OR based on the course requirement evaluation
rubrics shall be decided jointly by examiners.
● Students can pick one question (experiment) from the questions lot prepared by the examiners
jointly.
● Evaluation of test write-up/ conduction procedure and result/viva will be conducted jointly by
examiners.
● General rubrics suggested for SEE are mentioned here, writeup-20%, Conduction procedure and
result in -60%, Viva-voce 20% of maximum marks. SEE for practical shall be evaluated for 100 marks
and scored marks shall be scaled down to 50 marks (however, based on course type, rubrics shall be
decided by the examiners)
● Change of experiment is allowed only once and 15% of Marks allotted to the procedure part are to be
made zero.
The minimum duration of SEE is 02 hours
Suggested Learning Resources:

● Virtual Labs (CSE): http://cse01-iiith.vlabs.ac.in/
@# 16032024
Year of Study: 2023-24

LESSON PLAN
Semester & Year: 4th Semester, 2023-24 Faculty Name: Mr. Anand M
Subject with Code: Analysis & Design of Algorithms Lab (BCSL404)
COURSE OUTCOME: On completion of this subject, students will be expected to:
CO1: Develop programs to solve computational problems using suitable algorithm design strategy.
CO2: Compare algorithm design strategies by developing equivalent programs and observing running times for
analysis
CO3: Make use of suitable integrated development tools to develop programs
CO4: Choose appropriate algorithm design techniques to develop solution to the computational and complex
problems.
CO5: Demonstrate and present the development of program, its execution and running time(s) and record the
results/inferences.
Sl Program COs
Experiments
No. No.
Practice INTRODUCTION: CO1
1
programs Sample Exercise programs
Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a CO1,CO2
2 1
given connected undirected graph using Kruskal's algorithm.
Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a CO1,CO2
3 2
given connected undirected graph using Prim's algorithm
a. Design and implement C/C++ Program to solve All-Pairs Shortest Paths problem
using Floyd's algorithm. CO1, CO2
4 3
b. Design and implement C/C++ Program to find the transitive closure using Warshal's
algorithm.
Design and implement C/C++ Program to find shortest paths from a given vertex in a CO3
5 4
weighted connected graph to other vertices using Dijkstra's algorithm.
Design and implement C/C++ Program to obtain the Topological ordering of vertices in CO3
6 5
a given digraph
Design and implement C/C++ Program to solve 0/1 Knapsack problem using Dynamic CO4
7 6 Programming method.
Design and implement C/C++ Program to solve discrete Knapsack and continuous CO4
8 7
Knapsack problems using greedy approximation method.
Design and implement C/C++ Program to find a subset of a given set S = {sl , CO4
9 8
s2,.....,sn} of n positive integers whose sum is equal to a given positive integer d.
Design and implement C/C++ Program to sort a given set of n integer elements using
Selection Sort method and compute its time complexity. Run the program for varied
CO5
10 9 values of n> 5000 and record the time taken to sort. Plot a graph of the time taken
versus n. The elements can be read from a file or can be generated using the random
number generator.
Quick Sort method and compute its time complexity. Run the program for varied
11 10 values of n> 5000 and record the time taken to sort. Plot a graph of the time taken CO5
number generator.
Merge Sort method and compute its time complexity. Run the program for varied
12 11 values of n> 5000, and record the time taken to sort. Plot a graph of the time taken CO5
number generator.
Design and implement C/C++ Program for N Queen's problem using Backtracking. CO5
13 12
Evaluation Criteria: Assessment Details (both CIE and SEE):
The weightage of Continuous Internal Evaluation (CIE) is 50% and for Semester End Exam (SEE) is
50%. The minimum passing mark for the CIE is 40% of the maximum marks (20 marks out of 50) and for
the SEE minimum passing mark is 35% of the maximum marks (18 out of 50 marks). A student shall be
deemed to have satisfied the academic requirements and earned the credits allotted to each subject/ course
if the student secures a minimum of 40% (40 marks out of 100) in the sum total of the CIE (Continuous
Internal Evaluation) and SEE (Semester End Examination) taken together.
CIE: 50 Marks
Out of the 50 marks allotted for the lab assessment-Lab internals for 20 marks will be conducted at the end of the
lab cycle. Continuous evaluation will considered for 30 marks.
 The Lab internals will be evaluated for 20 marks.
Procedure Writing 4M
Execution 8M
Viva 6M
 The marks split up continuous evaluation
Procedure Writing 10M
Execution 10M
Record 5M
Viva 5M
Faculty PAC HoD, ISE

Year of Study: 2023-24

CO-PO Mapping
Subject Code / Course code: BCSL404 Faculty Name: Mr. Anand M

Subject Name: Analysis & Design of Algorithms Lab
Course Outcomes
Develop programs to solve computational problems using suitable algorithm
C204.1
design strategy.
Compare algorithm design strategies by developing equivalent programs and
C204.2
observing running times for analysis
C204.3 Make use of suitable integrated development tools to develop programs
Choose appropriate algorithm design techniques to develop solution to the
C204.4
computational and complex problems.
Demonstrate and present the development of program, its execution and running
C204.5
time(s) and record the results/inferences.
CO/PO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 P010 PO11 PO12 PSO1 PSO2
C204.1 2 2 2 2 - - - - - - - - 1 -
C204.2 2 2 2 2 - - - - - - - - 1 -
C204.3 2 2 2 2 - - - - - - - - 1 -
C204.4 2 2 2 2 - - - - - - - - 1 -
C204.5 2 2 2 2 - - - - - - - - 1 -
C204 2 2 2 2 - - - - - - - - 1 -
1: Slight 2: Moderate 3: Substantial - : No Correlation
Faculty PAC HOD-ISE

Analysis & Design of Algorithms Lab (BCSL404)
1. Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a
given connected undirected graph using Kruskal's algorithm.
#include<stdio.h>
int ne=1,min_cost=0;
void main()
{
int n,i,j,min,a,u,b,v,cost[20][20],parent[20];
printf("Enter the no. of vertices:");
scanf("%d",&n);
printf("\nEnter the cost matrix:\n");
for(i=1;i<=n;i++)
for(j=1;j<=n;j++)
scanf("%d",&cost[i][j]);
for(i=1;i<=n;i++)
parent[i]=0;
printf("\nThe edges of spanning tree are\n");
while(ne<n)
{
min=999;
for(i=1;i<=n;i++)
{
for(j=1;j<=n;j++)
{
if(cost[i][j]<min)
{
min=cost[i][j];
a=u=i;
b=v=j;
}
}
}
while(parent[u])
u=parent[u];
while(parent[v])
v=parent[v];
if(u!=v)
{
printf("Edge %d\t(%d->%d)=%d\n",ne++,a,b,min);
min_cost=min_cost+min;
parent[v]=u;
}
cost[a][b]=cost[a][b]=999;
}
printf("\nMinimum cost=%d\n",min_cost);
}
Dept. of ISE, GSSSIETW, Mysuru Page 1

Output:

2. Design and implement C/C++ Program to find Minimum Cost Spanning Tree of a
given connected undirected graph using Prim's algorithm.
#include<stdio.h>
int a,b,u,v,n,i,j,ne=1;
int visited[10]={0},min,mincost=0,cost[10][10];
void main()
{
printf("\n Enter the number of nodes:");

scanf("%d",&n);
printf("\n Enter the adjacency matrix:\n");
for(i=1;i<=n;i++)
for(j=1;j<=n;j++)
{
scanf("%d",&cost[i][j]);
if(cost[i][j]==0)
cost[i][j]=999;
}
visited[1]=1;
printf("\n");
while(ne<n)
{
for(i=1,min=999;i<=n;i++)
for(j=1;j<=n;j++)
if(cost[i][j]<min)
if(visited[i]!=0)
{
min=cost[i][j];
a=u=i;
b=v=j;
}
if(visited[u]==0 || visited[v]==0)
{
printf("\n Edge %d:(%d %d) cost:%d",ne++,a,b,min);
mincost+=min;
visited[b]=1;
}
cost[a][b]=cost[b][a]=999;
}
printf("\n Minimun cost=%d",mincost);
}

Output:

3.a Design and implement C/C++ Program to solve All-Pairs Shortest Paths problem using
Floyd's algorithm.
#include <stdio.h>
#include <limits.h>
#define V 4
void floydWarshall(int graph[V][V]) {

int dist[V][V];
for (int i = 0; i < V; i++)

for (int j = 0; j < V; j++)
dist[i][j] = graph[i][j];
for (int k = 0; k < V; k++)

for (int i = 0; i < V; i++)
for (int j = 0; j < V; j++)
if (dist[i][k] != INT_MAX && dist[k][j] != INT_MAX && dist[i][k] + dist[k][j] <
dist[i][j])
dist[i][j] = dist[i][k] + dist[k][j];
printf("Shortest distances between every pair of vertices:\n");

for (int i = 0; i < V; i++) {
for (int j = 0; j < V; j++) {
if (dist[i][j] == INT_MAX)
printf("INF\t");
else
printf("%d\t", dist[i][j]);
}
printf("\n");
}
}
int main() {
int graph[V][V] = {{0, INT_MAX, 3, INT_MAX},
{2, 0, INT_MAX, INT_MAX},
{INT_MAX, 7, 0, 1},
{6, INT_MAX, INT_MAX, 0}};
floydWarshall(graph);
return 0;
}

Output:

3.b Design and implement C/C++ Program to find the transitive closure using Warshal's
algorithm.
# include <stdio.h>
int n,a[10][10],p[10][10];
void path()
{
int i,j,k;
for(i=0;i<n;i++)
for(j=0;j<n;j++)
p[i][j]=a[i][j];
for(k=0;k<n;k++)
for(i=0;i<n;i++)
for(j=0;j<n;j++)
if(p[i][k]==1&&p[k][j]==1) p[i][j]=1;
}
void main()
{
int i,j;
printf("Enter the number of nodes:");
scanf("%d",&n);
printf("\nEnter the adjacency matrix:\n");
for(i=0;i<n;i++)
for(j=0;j<n;j++)
scanf("%d",&a[i][j]);
path();
printf("\nThe path matrix is showm below\n");
for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
printf("%d ",p[i][j]);
printf("\n");
}
}

Output:

4. Design and implement C/C++ Program to find shortest paths from a given vertex in a
weighted connected graph to other vertices using Dijkstra's algorithm.
#include<stdio.h>
void dij(int, int [20][20], int [20], int [20], int);
void main() {
int i, j, n, visited[20], source, cost[20][20], d[20];
printf("Enter no. of vertices: ");
scanf("%d", &n);
printf("Enter the cost adjacency matrix\n");
for (i = 1; i <= n; i++) {
for (j = 1; j <= n; j++) {
scanf("%d", &cost[i][j]);
}
}
printf("\nEnter the source node: ");
scanf("%d", &source);
dij(source, cost, visited, d, n);
for (i = 1; i <= n; i++) {
if (i != source)
printf("\nShortest path from %d to %d is %d", source, i, d[i]);
}
}
void dij(int source, int cost[20][20], int visited[20], int d[20], int n) {
int i, j, min, u, w;
for (i = 1; i <= n; i++) {
visited[i] = 0;
d[i] = cost[source][i];
}
visited[source] = 1;
d[source] = 0;
for (j = 2; j <= n; j++) {
min = 999;
for (i = 1; i <= n; i++) {
if (!visited[i]) {
if (d[i] < min) {
min = d[i];
u = i;
}
}
}
visited[u] = 1;
for (w = 1; w <= n; w++) {
if (cost[u][w] != 999 && visited[w] == 0) {
if (d[w] > cost[u][w] + d[u])
d[w] = cost[u][w] + d[u];

}
}
}
}
Output:

5. Design and implement C/C++ Program to obtain the Topological ordering of vertices in
a given digraph.
#include<stdio.h>
void findindegree(int [10][10],int[10],int);
void topological(int,int [10][10]);
void main()
{
int a[10][10],i,j,n;
printf("Enter the number of nodes:");
scanf("%d",&n);
printf("\nEnter the adjacency matrix\n");
for(i=1;i<=n;i++)
for(j=1;j<=n;j++)
scanf("%d",&a[i][j]);
printf("\nThe adjacency matirx is:\n");
for(i=1;i<=n;i++)
{
for(j=1;j<=n;j++)
{
printf("%d\t",a[i][j]);
}
printf("\n");
}
topological(n,a);
}
void findindegree(int a[10][10],int indegree[10],int n)
{
int i,j,sum;
for(j=1;j<=n;j++)
{
sum=0;
for(i=1;i<=n;i++)
{
sum=sum+a[i][j];
}
indegree[j]=sum;
}
}
void topological(int n,int a[10][10])
{
int k,top,t[100],i,stack[20],u,v,indegree[20];
k=1;
top=-1;
findindegree(a,indegree,n);
for(i=1;i<=n;i++)

{
if(indegree[i]==0)
{
stack[++top]=i;
}
}
while(top!=-1)
{
u=stack[top--];
t[k++]=u;
for(v=1;v<=n;v++)
{
if(a[u][v]==1)
{
indegree[v]--;
if(indegree[v]==0)
{
stack[++top]=v;
}}
}}
printf("\nTopological sequence is\n");
for(i=1;i<=n;i++)
printf("%d\t",t[i]);
}
Output:

6. Design and implement C/C++ Program to solve 0/1 Knapsack problem using Dynamic
Programming method.
#include<stdio.h>
#define MAX 50
int p[MAX],w[MAX],n;
int knapsack(int,int);
int max(int,int);
void main()
{
int m,i,optsoln;
printf("Enter no. of objects: ");

scanf("%d",&n);
printf("\nEnter the weights:\n");
for(i=1;i<=n;i++)
scanf("%d",&w[i]);
printf("\nEnter the profits:\n");
for(i=1;i<=n;i++)
scanf("%d",&p[i]);
printf("\nEnter the knapsack capacity:");
scanf("%d",&m);
optsoln=knapsack(1,m);
printf("\nThe optimal soluntion is:%d",optsoln);
}
int knapsack(int i,int m)
{
if(i==n)
return (w[n]>m) ? 0 : p[n];
if(w[i]>m)
return knapsack(i+1,m);
return max(knapsack(i+1,m),knapsack(i+1,m-w[i])+p[i]);
}
int max(int a,int b)
{
if(a>b)
return a;
else
return b;
}

Output:

7. Design and implement C/C++ Program to solve discrete Knapsack and continuous
Knapsack problems using greedy approximation method.
#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
// Structure to represent an item
struct Item {
int weight;
int value;
};
// Function to solve discrete knapsack using greedy approach
int discreteKnapsack(vector<Item>& items, int capacity) {
// Sort items based on their value per unit weight
sort(items.begin(), items.end(), [](const Item& a, const Item& b) {
return (double)a.value / a.weight > (double)b.value / b.weight;
});
int totalValue = 0;
int currentWeight = 0;
// Fill the knapsack with items
for (const Item& item : items) {
if (currentWeight + item.weight <= capacity) {
currentWeight += item.weight;
totalValue += item.value;
}
}
return totalValue;
}
// Function to solve continuous knapsack using greedy approach
double continuousKnapsack(vector<Item>& items, int capacity) {
// Sort items based on their value per unit weight
sort(items.begin(), items.end(), [](const Item& a, const Item& b) {
return (double)a.value / a.weight > (double)b.value / b.weight;
});
double totalValue = 0.0;
int currentWeight = 0;
// Fill the knapsack with items fractionally
for (const Item& item : items) {
if (currentWeight + item.weight <= capacity) {
currentWeight += item.weight;
totalValue += item.value;
} else {
int remainingCapacity = capacity - currentWeight;
totalValue += (double)item.value / item.weight * remainingCapacity;
break;

} }
return totalValue;
}
int main() {
vector<Item> items;
int n, capacity;
// Input number of items and capacity of knapsack
cout << "Enter the number of items: ";
cin >> n;
cout << "Enter the capacity of knapsack: ";
cin >> capacity;
// Input the weight and value of each item
cout << "Enter the weight and value of each item:" << endl;
for (int i = 0; i < n; i++) {
Item item;
cout << "Item " << i + 1 << ": ";
cin >> item.weight >> item.value;
items.push_back(item);
}
// Solve discrete knapsack problem
int discreteResult = discreteKnapsack(items, capacity);
cout << "Maximum value for discrete knapsack: " << discreteResult << endl;
// Solve continuous knapsack problem
double continuousResult = continuousKnapsack(items, capacity);
cout << "Maximum value for continuous knapsack: " << continuousResult << endl;
return 0;
}
Output:

8. Design and implement C/C++ Program to find a subset of a given set S = {sl , s2,.....,sn} of
n positive integers whose sum is equal to a given positive integer d.
#include<stdio.h>
void subset(int,int,int);
int x[10],w[10],d,count=0;
void main()
{
int i,n,sum=0;
printf("Enter the no. of elements: ");

scanf("%d",&n);
printf("\nEnter the elements in ascending order:\n");
for(i=0;i<n;i++)
scanf("%d",&w[i]);
printf("\nEnter the sum: ");
scanf("%d",&d);
for(i=0;i<n;i++)
sum=sum+w[i];
if(sum<d)
{
printf("No solution\n");
return;
}
subset(0,0,sum);
if(count==0)
{
printf("No solution\n");
return;
}
}
void subset(int cs,int k,int r)
{
int i;
x[k]=1;
if(cs+w[k]==d)
{
printf("\n\nSubset %d\n",++count);
for(i=0;i<=k;i++)
if(x[i]==1)
printf("%d\t",w[i]);
}
else

if(cs+w[k]+w[k+1]<=d)
subset(cs+w[k],k+1,r-w[k]);
if(cs+r-w[k]>=d && cs+w[k]<=d)
{
x[k]=0;
subset(cs,k+1,r-w[k]);
}
}
Output:

9. Design and implement C/C++ Program to sort a given set of n integer elements using
Selection Sort method and compute its time complexity. Run the program for varied values
of n> 5000 and record the time taken to sort. Plot a graph of the time taken versus n. The
elements can be read from a file or can be generated using the random number generator.
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
// Function to perform selection sort
void selectionSort(int arr[], int n) {
int i, j, minIndex, temp;
for (i = 0; i < n - 1; i++) {
minIndex = i;
for (j = i + 1; j < n; j++) {
if (arr[j] < arr[minIndex]) {
minIndex = j;
}
}
// Swap the found minimum element with the first element
temp = arr[minIndex];
arr[minIndex] = arr[i];
arr[i] = temp;
}
}
// Function to generate random numbers between 0 and 999
int generateRandomNumber() {
return rand() % 1000;
}
int main() {
// Set n value
int n = 6000;
// Allocate memory for the array

int* arr = (int*)malloc(n * sizeof(int));
// Generate random elements for the array

srand(time(NULL));
printf("Random numbers for n = %d:\n", n);
for (int i = 0; i < n; i++) {
arr[i] = generateRandomNumber();
printf("%d ", arr[i]);
}
printf("\n");
// Record the start time

clock_t start = clock();

// Perform selection sort

selectionSort(arr, n);
// Record the end time
clock_t end = clock();
// Calculate the time taken for sorting
double time_taken = ((double)(end - start)) / CLOCKS_PER_SEC;
// Output the time taken to sort for the current value of n
printf("\nTime taken to sort for n = %d: %lf seconds\n\n", n, time_taken);
// Display sorted numbers
printf("Sorted numbers for n = %d:\n", n);
for (int i = 0; i < n; i++) {
}
printf("\n\n");
// Free the dynamically allocated memory
free(arr);
return 0;
}
Output:

Quick Sort method and compute its time complexity. Run the program for varied values of
n> 5000 and record the time taken to sort. Plot a graph of the time taken versus n. The
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
// Function to swap two elements

void swap(int* a, int* b) {
int temp = *a;
*a = *b;
*b = temp;
}
// Function to partition the array and return the pivot index

int partition(int arr[], int low, int high) {
int pivot = arr[high];
int i = (low - 1);
for (int j = low; j <= high - 1; j++) {

if (arr[j] < pivot) {
i++;
swap(&arr[i], &arr[j]);
}
}
swap(&arr[i + 1], &arr[high]);
return (i + 1);
}
// Function to perform Quick Sort

void quickSort(int arr[], int low, int high) {
if (low < high) {
int pi = partition(arr, low, high);
quickSort(arr, low, pi - 1);

quickSort(arr, pi + 1, high);
}
}

}

int main() {
// Set n value
int n = 6000;


srand(time(NULL));
for (int i = 0; i < n; i++) {
}
printf("\n");

// Perform quick sort

quickSort(arr, 0, n - 1);




for (int i = 0; i < n; i++) {
}
printf("\n\n");

free(arr);
return 0;
}

Output:

Merge Sort method and compute its time complexity. Run the program for varied values of
n> 5000, and record the time taken to sort. Plot a graph of the time taken versus n. The
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
// Merge two subarrays of arr[]
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r) {
int i, j, k;
int n1 = m - l + 1;
int n2 = r - m;
// Create temporary arrays

int L[n1], R[n2];
// Copy data to temporary arrays L[] and R[]

for (i = 0; i < n1; i++)
L[i] = arr[l + i];
for (j = 0; j < n2; j++)
R[j] = arr[m + 1 + j];
// Merge the temporary arrays back into arr[l..r]

i = 0; // Initial index of first subarray
j = 0; // Initial index of second subarray
k = l; // Initial index of merged subarray
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
} else {
arr[k] = R[j];
j++;
}
k++;
}
// Copy the remaining elements of L[], if there are any

while (i < n1) {
arr[k] = L[i];
i++;
k++;
}

// Copy the remaining elements of R[], if there are any

while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
// Merge sort function

void mergeSort(int arr[], int l, int r) {
if (l < r) {
// Same as (l+r)/2, but avoids overflow for large l and r
int m = l + (r - l) / 2;
// Sort first and second halves

mergeSort(arr, l, m);
mergeSort(arr, m + 1, r);
// Merge the sorted halves

merge(arr, l, m, r);
}
}

}
int main() {
// Set n value
int n = 6000;


srand(time(NULL));
for (int i = 0; i < n; i++) {
}
printf("\n");


// Perform merge sort
mergeSort(arr, 0, n - 1);

for (int i = 0; i < n; i++) {
}
printf("\n\n");
free(arr);
return 0;
}
Output:

12. Design and implement C/C++ Program for N Queen's problem using Backtracking.
#include <iostream>
#include <vector>
using namespace std;
// Function to print the solution
void printSolution(const vector<vector<char>>& board) {
for (const auto& row : board) {
for (char cell : row)
cout << " " << cell << " ";
cout << endl;
}
}
// Function to check if a queen can be placed on board[row][col]
bool isSafe(const vector<vector<char>>& board, int row, int col) {
int i, j;
int n = board.size();
// Check the row on the left side

for (i = 0; i < col; i++)
if (board[row][i] == 'Q')
return false;
// Check upper diagonal on the left side
for (i = row, j = col; i >= 0 && j >= 0; i--, j--)
if (board[i][j] == 'Q')
return false;
// Check lower diagonal on the left side
for (i = row, j = col; j >= 0 && i < n; i++, j--)
if (board[i][j] == 'Q')
return false;
return true;
}
// Recursive function to solve N Queens problem
bool solveNQUtil(vector<vector<char>>& board, int col) {
int n = board.size();
// If all queens are placed, return true
if (col >= n)
return true;
// Consider this column and try placing this queen in all rows one by one
for (int i = 0; i < n; i++) {
// Check if the queen can be placed on board[i][col]
if (isSafe(board, i, col)) {
// Place this queen in board[i][col]
board[i][col] = 'Q';
// Recur to place rest of the queens
if (solveNQUtil(board, col + 1))

return true;
// If placing queen in board[i][col] doesn't lead to a solution, then remove queen from
board[i][col]
board[i][col] = '-';
}
}
// If the queen cannot be placed in any row in this column, then return false
return false;
}
// Function to solve N Queens problem for 4 queens
void solve4Queens() {
int n = 4;
vector<vector<char>> board(n, vector<char>(n, '-'));
if (solveNQUtil(board, 0) == false) {
cout << "Solution does not exist" << endl;
return;
}
printSolution(board);
}
// Driver function
int main() {
cout << "Solution for 4 Queens problem:" << endl;
solve4Queens();
return 0;
}
Output:

VIVA QUESTIONS
1. What is an Algorithm?
Algorithm is a Step by step procedure to Solve a given problem for a finite number of input
producing finitenumber of output with desired output.
2. What is a Flow Chart?
Flow chart is a Graphical Representation of a solution to the Problem.
3. What is the difference between Algorithm, Flow Chart, Program?
 Algorithm specifies the different things to be followed for solving a Problem.
 Flow Chart is a Graphical Representation of a Solution to the Problem. Both
Algorithm and Flow Chartare Machine Independent.
 Program is a Set of Instructions which is used as a tool to communicate to the
machine to get our workdone,Program is Machine Dependent for particular Machine.
4. What is the Aim of DAA lab or why we need to study DAA Lab
DAA is a discipline, where we are dealing with designing or writing the algorithm
keeping in Consideration of Space and Time Complexity, Such that Our Algorithm
should execute in a very minimum amount of time by Minimum Space or RAM.
5. Define Space and Time Complexity? Among this which one is more prioritized?
Space Complexeity is a measure of Amount of Space taken by a Program to finish its
Execution.
Time Complexeity is a measure of amount of time taken by a program to complte its
Execution.Depending Upon Application it is considered,EX:For Mobile or Handheld
Devices,We give Prefernce for both Spaceand time.
For a Huge and Inter active Systems like Web Applications we give more Preferences to
time Complexeity.
6. What is Design and what is Analysis of a Program?
Design is a Process of Writing an algorithm to a given Problem so that it should accept
finite number of input and finite number of output with a definite output and Should Exit
appropriately.
Analysis:Analysis is a next Phase of Writing an Algorithm ,in this phase we calculate
the Efficiency of anAlgorithm i.e time and space needed by an algorithm.

7. Write the general plan for analyzing the recursive algorithms.

 Identify the inputs.
 Identify the output is depended only on number of inputs.
 Identify the Basic Operation in Algorithm.
 Form or write the Recursive Relation to the Algorithm.
8. What are the various notations used to write an algorithm?
(i) Pseudocode (ii)Natural Language and etc..
9. What is a Pseudocode?
It’s a notation which is having the combination of Programming Constructs and English like
Statements.
10. What is the Time Complexeity of Bubble Sort,Selection Sort ,Merge Sort,Quick
Sort?(L3)
Bubble Sort-n2, Selection Sort- n2 Merge Sort-nlog.n
Quick Sort -nLogn, Worst case for Quick Sort- n2
11. Which sorting agorithm is more Efficient and why?
Quick Sorting is More Efficient ,because this algorithm is
instable algorithm andinplace.
12. What do you mean by the term Instable Algorithms?
The Instable Algorithms are one, which divides the array as certainly depending upon
pivot or key elementand hence i index precedes index j
13. Which algorithms are faster?
Instable Algorithms are much Faster compared to Stable Algorithms.
14. For what type of instance Merge sort do better than Quick Sort?
For a Larger input and a sorted input values.
15. For what type of instance Quick sort do better than Merge Sort?
For Smaller Set of input numbers.
16. What are Inplace Algorithms?
Inplace Algorithms are the one which doesn't occupies Extra Space.
17. Write the order of growth terms as per the time Execution in Ascending Order.
logn,n,nlogn,n2,n3,. .... nn,2n,n!

18. What is Brute Force Technique? When We Should Use?

Brute Force is a straight Forward Technique to solve a problem, We used to solve a
Problem throughthis approach when we don't have sufficient data to solve a problem in
Efficient Way.
19. What is the difference between Divide and Conquer, Decrease and Conquer?
Divide and Conquer can be solved to solve a problem with a larger data set and
when there is no dependency between any of the data sets.
 Divide and Solve as Small as Small sets.
 Conquer or Merge it get one final resultant data set.
Decrease and Conquer is almost similar to Divide and Conquer but we are finding a
solutions to the problem in a different variations,EX:Decrease by Constant (Usually
by One),Decrease by Constant factor which is almost similar to Divide and Conquer
Technique(Usually by two),Decrease by Variable(The Dividing Criteria changes for
each iteration depends upon the data set.
20. Define Greedy Technique.
Greedy Technique is always applied for the problem of the type optimization type,
which reduces loss and increases profit.
21. Define Optimal and Feasible Solution.
Optimal Solution is a solution which is best among N Feasible Solution. Feasible Solution is
a solution which Satisfies a Problem Constraints/conditions.
22. Can A Problem solved by all the algorithmic Techniques.
Yes,but some problems will give better results with some Algorithmic Technique
and it may give worstresult when it is applied with other technique.
23. State and Explain Knapsack Problem.
Filling the Maximum number of items to the Knapsack (Container) Which
Increases the profit anddecreases the Loss.
24. Which one is Most Admired algorithmic Technique?
Dynamic Programming.
25. What is Spanning tree and Minimum Spanning tree?
A tree Without Cycles are called as Spanning tree . A Minimum Spanning Tree is a
spanning tree which yeilds the very less Cost when all the edges costsummed up.

26. How Many Spanning Tree can a Tree can have?

A tree can have 1 to many number of Possible ways of Spanning Tree.
27. Differentiate between Prims and Kruskals Algorithm for finding MST.
In Prims We consider any one vertex in the graph as Source and We compute the
distance from that source to other vertices ,after computing the vertices which has
minimum value among (n-1) vertices is added to tree vertices and that respective edges
added to tree Edges Set.The above mentioned Process continues till we reach (n-1)
vertices.
In Kruskals we first arrange the edges in Ascending Order and then we start to form the
tree which wont form cycles,if adding that edges forms cycles then that edges is dropped
from adding to tree edges.The above saidprocess is continues till we reach the count of
(n-1) Vertices.
28. What is the Application of Prims and Kruskals Algorithm?

In Networks to remove the Cyclicity of the Network.
29. Explain Job Sequencing With Deadlines?
Placing or scheduling the maximum number of Jobs to a machine without violating the
deadlines constraintof any of the Jobs in Sequence.
30. Why the Name Bubble Sort named?
Because in first Pass the first highest data will bubbles up,so since the largest
element bubbles up in the first and second largest element bubbles up in the Second pass
and so on, so hence the name bubble sort.
31. Why the Name Selection Sort?(L3)
The Selection sort is named because we initially first select an arrays first element as
minimum and will compare with other elements ,so in pass one first least element
goes to the first position and so on so forth for 2nd,3rd and so on. Selecting
32. What is the difference between Brute force strings matching to Horspool String
Matching Method? (L2)In brute Force we compare each and every element of the
text to the pattern by shiftingthe text positionby one and in Horspool method we
shift it by number of shift positions recorded in theshift table.

33. Explain Merge Sort?
In Merge Sort will divide the entire input set by 2 until we reach low<high and later
will find a solution to each item by comparing half of the array data set to the other
half array data set and finally we mergeit to form a sinle array(conquer)
34. What is the Basic Operations in Merge sort and Quick sort?
In Merge Sort the Basic Operations is Comparisions and in Quick sort basic Operations
is Partitioning andhence also known as partitioning sort.
35. Why the Insertion Sort?
We are Inserting an element to its suitable place by comparing n elements for each pass.
36. What is the Use of DFS and BFS?
DFS and BFS both used to check the Connectivity of a graph,Cyclicity in a
graph,Spanning tree of agraph.
37. Differentiate between DFS and BFS.
DFS and BFS are both the Graph Traversing Technique,in which DFS Traverse the
Graph in a depthwise(Vertical) and BFS Traverse the Graph from left to
right(Horizontal)
38. Which Data structures used in BFS and DFS.
BFS USes Queue as its data structure and DFS uses as stack its Data structure.
39. What are back edges in DFS and Cross Edges in BFS.
Back Edges and Cross edges are the Edges which already visited by a ancestor node.
40. What is Topological Sorting?
Topological Sorting is a Sorting Technique used for sorting Vertices in the Graph.
41. What is the Conditions necessary for a Topological Sorting?
For a Topological Sorting the Graph Should be DAG(Directed Acyclic Graph)
42. What are the Different methods used to solve a topological Sorting ?
1. Source Removal Method
2. Using DFS based Scheme.
43. What is the Use of Topological Sorting?
Use of Topological Ordering is in the field of Operating System for Scheduling and in
Networks,Automation and Robotics.

44. What is Dijikstra's Algorithm?

Dijikstra's Algorithm is Used to find the Single shortest Path from source to the other
vertex.
45. What is a graph?
Graph is a component which is having a set of Edges and vertices G={V,E}
46. What are the different ways that can be represents a graph?
Adjaceny Matrix and Adjacency List.
47. What is Adjacency Matrix?
Is a Matrix which illustrates the Graph in the form of Matrix,if it is a weights Graph
then we initialize the value of the cost in that position(i,j) or else simply we write 1 to
mention ther exist an edge between (i,j) OR else we use 0 or 9999 to mention non
connectivity of a graph.
48. What is the limitations of Algorithms?
Algorithm can't find the better the soltions when we come across the tight lower
bound,So we can find thebetter solutins which is not possible with Algorithmic way.To
find the Tight lower bound we use DecisionTrees.
49. What is Tight lower Bound?
It is a Lower bound which is a best lower bound for an problem,beyond that no
algorithm will produce better results.
50. What are Decision Trees?
Decision trees are also known as Comparision Trees used to find the tight lower
bound for a particular Problem EX:Tight Lower Bound For Sorting is n.logn
and tight lower bound for Searching is logn which is not possible to get the better result.
51. What is a polynomial problem (P-type)
P-type problem are decision problems in which we can find the solutions in a
polynomial time and is oftype deterministic.
52. What is NP-problem?
NP-Problem belongs to decision problem and these problems are Non Deterministic
Polynomial i.e forwhich the problem doesn't have deterministic solutions and Can be
solved in Polynomial time
There are 2 phases in solving a problem.
(i) Guessing(Non-Deterministic stage) Producing N number of Candidate Outputs.

(ii) Verification(Deterministic Stage) Verifying The correctness of N Number of

Candidate Outputs.
53. What is NP-Complete Problems?
Np_Complete Problems belongs to Decision problems and NP type Problems .
These problems can be find the solutions by converting or reducing to the problem
which we know theSolutions.
54. What is a trivial lower bound?

Trivial bound can be derived by formulating the number of inputs that has to be given
and number ofoutputs that has to be generated.
55. Explain Bactracking W.r.t
(I)Subset Problem (ii)N-Queens Problem
56. Explain Subset Problem.

In a given Set S ,find the Subset,in which the sum of all subset elements is equal to
the sum d which is predefined in a problem.
57. Explain N-Queens Problem.(
N-Queens Problem is of Placing a N-Queens in a N*N Chess board such that No 2-
Queens Should be placed in the same Row,Column and same diagnol(N=Should
consider both principal diagonal elements)
58. What is Hamiltonian Circuit?(L2)
Hamiltonian circuit is a problem in which that circuit starts from a source vertex and
has other vertex in any order without repeating and Should end with the Source vertex
only i.e source and Destination vertexshould be same.
59. Explain the Problem of TSP.(L2)
Travelling Sales Person Problem is a problem in which he should Visit N number of
cities with a minimum number of Cost by visiting every city Exactly one and the city
what he is started should end with same city.
60. What is the Concept of Dynamic Programming?(L3)
Deriving a Solution to the basic Condition and Extracting the solutions for the rest of
the other data sets by Previously drawnd Solution.Computer to other algorithmic
Technique Dynamic Programming because it avoids lot of reworking on the same
Solution what we have solved in the earlier phases of deriving the solution to the

problem.
61. What is the goal of Warshalls Algorithm?(L3)
Warshall’s algorithm is to find the shortest distance between a node to all the other
nodes in the graph. It Uses the Property of Transitive Closure i.e if there exist a path
between (i,k) and (k,j) then there surely exist a path between (i,j)
(i,k) & (k,j)---(I,J)
62. What is the use of Floyds algorithm?(L3)
It is use to find the All pairs shortest Path of an Graph.

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DEPARTMENT OF INFORMATION SCIENCE & ENGINEERING

Program Educational Objectives (PEO's):

Analysis & Design of Algorithms Lab Semester 4

Course outcomes (Course Skill Set):

Continuous Internal Evaluation (CIE):

Semester End Evaluation (SEE):

Suggested Learning Resources:

Year of Study: 2023-24

COURSE OUTCOME: On completion of this subject, students will be expected to:

Evaluation Criteria: Assessment Details (both CIE and SEE):

Faculty PAC HoD, ISE

Year of Study: 2023-24

Subject Code / Course code: BCSL404 Faculty Name: Mr. Anand M

Faculty PAC HOD-ISE

Dept. of ISE, GSSSIETW, Mysuru Page 1

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printf("\n Enter the number of nodes:");

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void floydWarshall(int graph[V][V]) {

for (int i = 0; i < V; i++)

for (int k = 0; k < V; k++)

printf("Shortest distances between every pair of vertices:\n");

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printf("Enter no. of objects: ");

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printf("Enter the no. of elements: ");

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// Allocate memory for the array

// Generate random elements for the array

// Record the start time

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// Perform selection sort

Dept. of ISE, GSSSIETW, Mysuru Page 20

// Function to swap two elements

// Function to partition the array and return the pivot index

for (int j = low; j <= high - 1; j++) {

// Function to perform Quick Sort

quickSort(arr, low, pi - 1);

// Function to generate random numbers between 0 and 999

Dept. of ISE, GSSSIETW, Mysuru Page 21

// Allocate memory for the array

// Generate random elements for the array

// Record the start time

// Perform quick sort

// Record the end time

// Calculate the time taken for sorting

// Output the time taken to sort for the current value of n