ADA
Unit-1
2 marks
1. **What is an Algorithm? What are the criteria for writing an algorithm?**
- **Algorithm Definition:** An algorithm is a step-by-step procedure or a set
of rules for solving a specific problem or accomplishing a particular task. It is a
finite sequence of well-defined, unambiguous instructions that a computer can
execute to achieve a specific goal.
- **Criteria for Writing an Algorithm:**
- **Well-defined:** The steps of the algorithm must be clear and
unambiguous.
- **Input and Output:** An algorithm should take zero or more inputs and
produce at least one output.
- **Finiteness:** The algorithm should terminate after a finite number of
steps.
- **Effectiveness:** Each step of the algorithm should be executable and
should contribute to solving the problem.
2. **What are the methods of specifying an algorithm?**
- **Natural Language:** Describing the algorithm in plain, human-readable
language.
- **Pseudocode:** A mix of natural language and programming-like
constructs.
- **Flowcharts:** Graphical representation using symbols and arrows.
- **Programming Language:** Writing the algorithm in a specific
programming language.
3. **List the steps of the Algorithm design and analysis process**
- **Understand the Problem**
- **Devise a Plan**
- **Specify the Algorithm Precisely**
- **Analyze the Algorithm**
- **Implement the Algorithm**
- **Test and Debug**
4. **What is an exact algorithm and approximation algorithm? Give an
example.**
- **Exact Algorithm:** Provides an optimal solution to the problem. It
guarantees the best possible solution. For example, the exhaustive search
algorithm for the Traveling Salesman Problem.
- **Approximation Algorithm:** Provides a solution that is close to the
optimal, but not necessarily the best. It trades optimality for efficiency. An
example is the greedy algorithm for the Traveling Salesman Problem.
5. **List the important Problem Types.**
- **Sorting Problems**
- **Searching Problems**
- **Graph Problems**
- **String Problems**
- **Geometric Problems**
- **Numeric Problems**
6. **Define the different methods for measuring algorithm efficiency.**
- **Time Complexity:** Measures the amount of time an algorithm takes to
complete.
- **Space Complexity:** Measures the amount of memory space an
algorithm uses.
- **Big-O Notation:** Provides an upper bound on the growth rate of an
algorithm's time or space complexity.
7. **Write the Euclid algorithm to find the GCD of 2 numbers (using Java).**
```java
public class EuclidAlgorithm {
public static int findGCD(int a, int b) {
while (b != 0) {
int temp = b;
b = a % b;
a = temp;
}
return a;
}
public static void main(String[] args) {
int num1 = 48;
int num2 = 18;
int gcd = findGCD(num1, num2);
System.out.println("GCD of " + num1 + " and " + num2 + " is: " + gcd);
}
}
```
8. **What are combinatorial problems? Give an example.**
- **Combinatorial Problems:** Involve counting, arranging, and selecting
elements without considering the specific order. These problems often require
combinatorial analysis.
- **Example:** The Traveling Salesman Problem, where the task is to find the
shortest possible tour that visits a set of cities and returns to the starting city.
Certainly! Let's go through each question:
**9. What are the following data structures?**
a) **Single Linked List:** A data structure where each element (node) points
to the next element in the sequence.
b) **Double Linked List:** Similar to a single linked list, but each node also
points to the previous node in addition to the next one.
c) **Stack:** A data structure that follows the Last In, First Out (LIFO)
principle. Elements are added and removed from the same end, often called
the "top."
d) **Queue:** A data structure that follows the First In, First Out (FIFO)
principle. Elements are added at the rear and removed from the front.
e) **Graph:** A collection of nodes (vertices) and edges that connect pairs of
nodes. Graphs can be directed or undirected.
f) **Tree:** A hierarchical data structure composed of nodes connected by
edges. Trees have a root node, and each node has zero or more child nodes.
**10. Explain the terms (w.r.t graph):**
a) **Directed Graph:** A graph where edges have a direction, i.e., they go
from one vertex to another.
b) **Undirected Graph:** A graph where edges have no direction; they
simply connect vertices without a specified starting or ending point.
c) **Adjacency Matrix:** A matrix representation of a graph where each cell
indicates whether an edge exists between the corresponding vertices.
d) **Adjacency Lists:** A list representation of a graph where each vertex has
a list of its neighboring vertices.
e) **Weighted Graph:** A graph in which each edge is assigned a weight,
representing some quantity like distance, cost, etc.
f) **Path:** A sequence of vertices where each adjacent pair is connected by
an edge.
g) **Cycle:** A path that starts and ends at the same vertex, forming a closed
loop.
**11. Explain the terms (w.r.t trees):**
a) **Free Tree:** A tree where each edge is undirected, and there are no
restrictions on the arrangement of nodes.
b) **Forest:** A collection of disjoint trees.
c) **Rooted Tree:** A tree in which one node is designated as the "root," and
every edge is directed away from the root.
d) **Ordered Tree:** A tree in which the children of each node are ordered.
e) **Binary Search Tree:** A binary tree where the left subtree of a node
contains only nodes with values less than the node's value, and the right
subtree contains only nodes with values greater than the node's value. This
ordering property facilitates efficient search operations.
I'll provide concise answers to each question based on the general concepts
covered in algorithm textbooks, including "Introduction to the Design and
Analysis of Algorithms" by Anany Levitin:
**12. Define Sets and Dictionaries:**
- **Sets:** A collection of distinct elements. Sets do not allow duplicate
elements.
- **Dictionaries:** Also known as maps or associative arrays, dictionaries
store key-value pairs, allowing efficient retrieval of values based on their
associated keys.
**13. Define the two types of efficiencies used in algorithm:**
- **Time Efficiency:** Measures the amount of time an algorithm takes to
complete.
- **Space Efficiency:** Measures the amount of memory space an algorithm
uses.
**14. What are Best case and Worst case in algorithm:**
- **Best Case:** The minimum amount of time or space required by an
algorithm for any input.
- **Worst Case:** The maximum amount of time or space required by an
algorithm for any input.
**15. Why is order growth necessary in algorithm analysis:**
- Order growth helps in understanding how the performance of an algorithm
scales with the size of the input. It provides a high-level view of efficiency and
allows for comparison between algorithms.
**16. What are asymptotic notation? Why is it required:**
- **Asymptotic Notation:** Describes the limiting behavior of a function as
the input approaches infinity.
- **Requirement:** It provides a concise way to express the efficiency of an
algorithm without getting into the details of constant factors and lower-order
terms.
**17. What is Big O notation? Give an example:**
- **Big O Notation:** Represents an upper bound on the growth rate of a
function.
- **Example:** O(n^2) for an algorithm with a quadratic time complexity.
**18. What is Big Omega notation? Give an example:**
- **Big Omega Notation:** Represents a lower bound on the growth rate of a
function.
- **Example:** Ω(n) for an algorithm with linear time complexity.
**19. Define Big Theta notation. Give an example:**
- **Big Theta Notation:** Represents both upper and lower bounds,
indicating a tight bound on the growth rate of a function.
- **Example:** Θ(n log n) for an algorithm with linearithmic time complexity.
**20. Define Little Oh notation. Give an example:**
- **Little Oh Notation:** Represents an upper bound that is not
asymptotically tight.
- **Example:** o(n^2) for an algorithm that grows slower than quadratic
time.
**21. What is recurrence relation? Give an example:**
- **Recurrence Relation:** Describes a function in terms of its values for
smaller inputs.
- **Example:** Fibonacci sequence: F(n) = F(n-1) + F(n-2) with initial
conditions F(0) = 0, F(1) = 1.
**22. Prove the following statements:**
a) 100n + 5 =`O(n2)
b) n 2 + 5n + 7= Θ(n2 )
c) n 2 + n =O(n3 )
d) ½ n(n-1)=Θ(n2 )
e) 5n2 + 3n + 20 =O(n2 )
f) ½n2+3n=Θ(n2 )
g) n 3 + 4n2 =Ω(n2 )
**23. Algorithm Sum(n):**
S <- 0
For i<- 1 to n do
S <-S + i
Return S
// a) What does this algorithm compute?
// It computes the sum of integers from 1 to n.
// b) What is its basic operation?
// The basic operation is the addition operation (S = S + i).
// c) How many times is the basic operation executed?
// The basic operation is executed n times (for each value of i from 1 to n).
// d) What is the efficiency class of this algorithm?
// The time complexity of this algorithm is O(n).
Long Answers:
1. What is an Algorithm? Explain the various criteria for writing an algorithm
with an example.
**Algorithm Definition:**
An algorithm is a step-by-step procedure or set of rules for solving a specific
problem or accomplishing a particular task. It is a finite sequence of well-
defined, unambiguous instructions that a computer can execute to achieve a
specific goal.
**Criteria for Writing an Algorithm:**
1. **Well-defined:** The steps of the algorithm must be clear and
unambiguous.
2. **Input and Output:** An algorithm should take zero or more inputs and
produce at least one output.
3. **Finiteness:** The algorithm should terminate after a finite number of
steps.
4. **Effectiveness:** Each step of the algorithm should be executable and
should contribute to solving the problem.
**Example: Finding the Maximum Element in an Array**
```java
Algorithm FindMax(arr):
Input: An array arr of n elements
Output: The maximum element in arr
1. Set max to the first element of arr (max = arr[0]).
2. For each element elem in arr from index 1 to n-1:
a. If elem > max, set max to elem.
3. Return max.
```
2. Explain Euclid Algorithm with an example to find the GCD of two numbers.
The Euclidean Algorithm is a method for finding the greatest common divisor
(GCD) of two numbers.
**Euclid Algorithm:**
Given two integers, a and b, where \(a \geq b\), the algorithm involves
repeatedly replacing a with b and b with the remainder of the division a by b
until b becomes 0. At this point, the GCD is the non-zero remainder obtained in
the previous step.
**Example: GCD of 48 and 18**
```
a = 48, b = 18
Step 1: 48 = 18 * 2 + 12
Step 2: 18 = 12 * 1 + 6
Step 3: 12 = 6 * 2 + 0
The GCD is the non-zero remainder from the last step, which is 6.
```
3. Explain Consecutive Integer Checking methods to find the GCD of two
numbers.
**Consecutive Integer Checking:**
This method involves checking consecutive integers starting from 1 to find the
largest number that divides both given integers without leaving a remainder.
**Example: GCD of 48 and 18**
```
1 does not divide both 48 and 18.
2 divides 48 but not 18.
3 divides both 48 and 18.
4 does not divide both 48 and 18.
5 does not divide both 48 and 18.
6 divides both 48 and 18.
The largest number dividing both 48 and 18 is 6.
```
4. Explain the Algorithm design and analysis process with a flow diagram.
**Algorithm Design and Analysis Process:**
1. **Understand the Problem:** Clearly define the problem and its
requirements.
2. **Devise a Plan:** Develop a strategy or plan to solve the problem.
3. **Specify the Algorithm Precisely:** Express the algorithm using a suitable
representation like pseudocode or a programming language.
4. **Analyze the Algorithm:** Evaluate the algorithm's time and space
complexity.
5. **Implement the Algorithm:** Code the algorithm in a programming
language.
6. **Test and Debug:** Verify the correctness of the implementation through
testing and debugging.
**Flow Diagram for Algorithm Design and Analysis:**
This iterative process ensures a systematic approach to problem-solving and
algorithm development, leading to effective and efficient solutions.
### 5. Explain any FIVE Problem Types.
1. **Sorting Problems:**
- **Definition:** Involves arranging elements in a specific order, such as
ascending or descending.
- **Example:** Sorting a list of names alphabetically.
2. **Searching Problems:**
- **Definition:** Involves finding the location or presence of a particular item
in a collection.
- **Example:** Searching for a specific book in a library.
3. **Graph Problems:**
- **Definition:** Involves operations on graphs, such as finding the shortest
path, detecting cycles, or determining connectivity.
- **Example:** Finding the shortest route between two cities on a map.
4. **String Problems:**
- **Definition:** Involves operations on strings, such as pattern matching,
substring search, or string manipulation.
- **Example:** Searching for a specific word in a document.
5. **Geometric Problems:**
- **Definition:** Involves operations on geometric shapes, such as finding
areas, distances, or intersections.
- **Example:** Calculating the area of a polygon.
### 6. Explain the following:
a. **Graph Problem:**
- **Definition:** Involves analyzing and solving problems related to graphs.
Common graph problems include finding the shortest path, detecting cycles,
and determining connectivity.
- **Example:** Finding the optimal route for a delivery truck to visit multiple
locations.
b. **Combinatorial Problems:**
- **Definition:** Involves counting, arranging, and selecting elements
without considering the specific order. Combinatorial problems often require
combinatorial analysis.
- **Example:** Counting the number of ways to choose a committee from a
group of people.
c. **Geometrical Problems:**
- **Definition:** Involves operations on geometric shapes and structures,
such as finding areas, distances, or intersections.
- **Example:** Determining the intersection point of two lines in a plane.
### 7. Explain the Fundamentals of Data Structure.
- **Data Structure Definition:** A data structure is a way of organizing and
storing data to perform operations efficiently. It defines the relationship
between data elements and allows the design of efficient algorithms.
- **Fundamentals:**
- **Organization:** How the data is arranged in memory.
- **Access:** Methods for retrieving and manipulating data.
- **Operations:** Functions or methods that can be performed on the data.
- **Representation:** The format used to store data.
### 8. Write a Note on Graph Data Structure.
- **Graph Definition:** A graph is a collection of nodes (vertices) and edges
that connect pairs of nodes. Graphs can be directed or undirected and may
have weighted edges.
- **Components of a Graph:**
- **Vertices (Nodes):** Represent entities.
- **Edges:** Represent relationships between entities.
- **Types of Graphs:**
- **Directed Graph (Digraph):** Edges have a direction.
- **Undirected Graph:** Edges have no direction.
- **Weighted Graph:** Edges have weights.
### 9. Write a Note on the Following Data Structures:
a. **Tree:**
- **Definition:** A hierarchical data structure consisting of nodes connected
by edges. It has a root node, and each node has zero or more child nodes.
- **Types:** Binary Tree, Binary Search Tree, AVL Tree, etc.
b. **Sets:**
- **Definition:** A collection of distinct elements. Sets do not allow duplicate
elements.
- **Operations:** Union, Intersection, Difference, etc.
c. **Dictionary:**
- **Definition:** Also known as maps or associative arrays, dictionaries store
key-value pairs, allowing efficient retrieval of values based on their associated
keys.
- **Operations:** Insertion, Deletion, Lookup, etc.
### 10. Explain Space Complexity and Time Complexity with Examples.
- **Space Complexity:**
- **Definition:** The amount of memory space required by an algorithm or
program during its execution.
- **Example:** In an algorithm that sorts an array of n elements, the space
complexity might be O(1) for an in-place sorting algorithm or O(n) for an
algorithm that requires additional memory.
- **Time Complexity:**
- **Definition:** The amount of time an algorithm takes to complete as a
function of the size of the input.
- **Example:** In a sorting algorithm, the time complexity might be O(n log
n) for efficient algorithms like merge sort or O(n^2) for less efficient algorithms
like bubble sort.
### 13. Explain the following w.r.t algorithm efficiency:
a. **Measuring Input Size:**
- **Definition:** The size of the input to an algorithm is a critical factor in
determining its efficiency. It is often denoted by the symbol \(n\), representing
the number of elements in the input.
- **Example:** In the context of sorting algorithms, the input size (\(n\))
could be the number of elements in an array to be sorted. For a graph
algorithm, \(n\) might represent the number of vertices.
b. **Unit for Measuring Run Time:**
- **Definition:** The run time of an algorithm is the time it takes to complete
its execution. It is measured in some unit of time, often denoted as seconds,
milliseconds, or other relevant time units.
- **Example:** If an algorithm takes 2.5 seconds to process a set of data, the
unit for measuring the run time is seconds.
c. **Order Growth:**
- **Definition:** Order growth, also known as asymptotic growth, refers to
how the time or space complexity of an algorithm scales with the size of the
input (\(n\)) in the worst-case scenario.
- **Example:** If an algorithm has a time complexity of \(O(n^2)\), it means
that as the input size (\(n\)) increases, the running time of the algorithm grows
quadratically.
In summary, measuring the input size (\(n\)) helps us understand how the
algorithm performs as the problem size increases. The unit for measuring run
time provides a quantitative measure of the actual time taken by the algorithm.
Order growth, expressed through Big O notation, provides a high-level
understanding of how the algorithm's efficiency scales with larger inputs.
### 14. Explain Worst Case, Best Case, and Average Case with Examples:
1. **Worst Case:**
- **Definition:** The worst-case time complexity represents the maximum
amount of time an algorithm takes for any input of size \(n\). It considers the
scenario where the algorithm performs most poorly.
- **Example:** Consider the linear search algorithm. In the worst case, the
element being searched for is at the end of the array, requiring \(n\)
comparisons.
2. **Best Case:**
- **Definition:** The best-case time complexity represents the minimum
amount of time an algorithm takes for any input of size \(n\). It considers the
scenario where the algorithm performs most efficiently.
- **Example:** In the context of the linear search algorithm, the best case
occurs when the element being searched for is at the beginning of the array,
requiring only one comparison.
3. **Average Case:**
- **Definition:** The average-case time complexity represents the expected
time an algorithm takes for a random input of size \(n\), considering all possible
inputs and their probabilities.
- **Example:** For a sorting algorithm like quicksort, the average case occurs
when the input array is divided into approximately equal parts in each
recursive call. The expected time complexity in this case is often better than the
worst case.
In summary, the worst-case scenario is used to provide an upper bound on the
algorithm's performance, the best-case scenario represents an idealized lower
bound, and the average case provides a more realistic expectation based on
the probability distribution of inputs. These analyses help in understanding
how an algorithm behaves across different input scenarios.
### 16. Explain Big O Notation with Example:
**Big O Notation:**
- **Definition:** Big O notation represents the upper bound or worst-case
scenario for the growth rate of an algorithm's time or space complexity.
- **Example:** Consider a simple algorithm to find the maximum element in
an array.
```java
// Algorithm to find the maximum element in an array
int findMax(int[] arr) {
int max = arr[0]; // Assume the first element is the maximum
for (int i = 1; i < arr.length; i++) {
if (arr[i] > max) {
max = arr[i]; // Update max if a larger element is found
}
}
return max;
}
```
**Big O Notation Example:**
- In this case, the time complexity of the algorithm is \(O(n)\), where \(n\) is
the size of the array. This is because the algorithm iterates through the entire
array once to find the maximum element.
### 17. Explain Big Omega Notation with Example:
**Big Omega Notation:**
- **Definition:** Big Omega notation represents the lower bound or best-case
scenario for the growth rate of an algorithm's time or space complexity.
- **Example:** Consider a simple algorithm to find the minimum element in an
array.
```java
// Algorithm to find the minimum element in an array
int findMin(int[] arr) {
int min = arr[0]; // Assume the first element is the minimum
for (int i = 1; i < arr.length; i++) {
if (arr[i] < min) {
min = arr[i]; // Update min if a smaller element is found
}
}
return min;
}
```
**Big Omega Notation Example:**
- In this case, the time complexity of the algorithm is \(\Omega(n)\), where
\(n\) is the size of the array. This is because the algorithm needs to check each
element in the array to find the minimum, and in the best-case scenario, it has
to go through all elements.
### 18. Explain Big Theta Notation with Example:
**Big Theta Notation:**
- **Definition:** Big Theta notation represents both the upper and lower
bounds, providing a tight bound on the growth rate of an algorithm's time or
space complexity.
- **Example:** Consider a simple algorithm to calculate the sum of elements
in an array.
```java
// Algorithm to calculate the sum of elements in an array
int calculateSum(int[] arr) {
int sum = 0;
for (int i = 0; i < arr.length; i++) {
sum += arr[i];
}
return sum;
}
```
**Big Theta Notation Example:**
- In this case, the time complexity of the algorithm is \(\Theta(n)\), where \(n\)
is the size of the array. The algorithm's performance is directly proportional to
the size of the input array, both in the best and worst-case scenarios.