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Unit 1
2 Marks
**1. **What is an Algorithm? What are the criteria for writing an algorithm?**** Ans: - An algorithm is a step-by-step set of instructions for solving a particular problem or performing a specific task. It's a precise and unambiguous sequence of operations that, when executed, produces a desired result. Criteria for writing an algorithm: - Input: It should specify what data is needed as input. - Output: It should define the expected result or output. - Finiteness: The algorithm must terminate after a finite number of steps. - Definiteness: Each step must be precisely defined and unambiguous. - Effectiveness: Every step should be executable and achieve a specific task. - Generality: It should be applicable to a range of input values or instances. 2. What are the methods of specifying an algorithm? Ans: - Algorithms can be specified using various methods, including: - Natural Language (e.g., English) - Pseudocode - Flowcharts - Programming languages - Decision tables - State diagrams 3. List the steps of Algorithm design and analysis process: Ans: - Problem definition - Design a solution (algorithm) - Analysis of the algorithm - Implementation - Testing and debugging - Documentation 4. What is an exact algorithm and approximation algorithm? Give an example: Ans: - An exact algorithm is one that guarantees to find the optimal solution to a problem. It provides the most accurate and precise result. - An approximation algorithm is a heuristic method that finds a solution that is not necessarily optimal but is close to the optimal solution. It is often used for complex problems when finding an exact solution is computationally infeasible. Example: - Exact algorithm: Solving the traveling salesman problem to find the shortest route that visits a set of cities exactly once. - Approximation algorithm: The nearest neighbor algorithm for the traveling salesman problem, which finds a good but not necessarily optimal solution.
5. List the important Problem Types: Ans: - Sorting problems - Searching problems - Optimization problems - Combinatorial problems - Graph problems - Geometric problems - Network flow problems - Decision problems 6. Define the different methods for measuring algorithm efficiency: Ans: - Time complexity: Measures the time an algorithm takes to execute as a function of the input size. - Space complexity: Measures the memory space used by an algorithm in relation to the input size. - Computational complexity: Evaluates an algorithm's performance under different resource constraints. - Asymptotic analysis: Analyzes algorithm efficiency as input size approaches infinity. 7. Write the Euclid algorithm to find the GCD of 2 numbers: Ans: Python code: def euclid_gcd(a, b): while b: a, b = b, a % b return a 8. What are combinatorial problems? Give an example: Ans: - Combinatorial problems involve counting, arranging, or selecting objects in a specific way. They often deal with discrete, finite sets of elements. Example: The traveling salesman problem, where you need to find the shortest route that visits a set of cities exactly once. 9. Define the following data structures: a) Single linked list : A data structure in which each element points to the next one, forming a linear sequence. b) Double linked list : Similar to a single linked list, but each element has pointers to both the next and previous elements. c) Stack: A linear data structure that follows the Last-In-First-Out (LIFO) principle. d) Queue : A linear data structure that follows the First-In-First-Out (FIFO) principle. e) Graph : A collection of nodes (vertices) and edges that connect pairs of nodes. f) Tree: A hierarchical data structure with a single root node and child nodes, which are organized in a branching structure.
- Worst case: It refers to the scenario in which an algorithm performs with the maximum possible time or resource usage. It represents the upper bound on the algorithm's performance. 15. Why is order growth necessary in algorithm analysis: Ans: - Order growth (Big O notation) is necessary for algorithm analysis because it provides a way to describe the behavior of an algorithm as the input size grows. It allows us to understand how the algorithm's performance scales with larger inputs and make informed decisions about choosing the right algorithm for a specific problem. Order growth analysis helps in comparing and selecting algorithms based on their efficiency. 16. What are asymptotic notations? Why are they required: Ans: - Asymptotic notations are mathematical notations used to describe the limiting behaviour of functions as their input values approach infinity. They are required for algorithm analysis to provide a concise and standardized way of expressing the upper and lower bounds of algorithm performance, helping us compare algorithms without getting lost in specific constant factors. 17. What is Big O notation? Give an example: Ans: - Big O notation, often denoted as O(f(n)), describes the upper bound on the growth rate of a function, indicating the worst-case time complexity of an algorithm. It provides an asymptotic upper limit. Example: O(n^2) represents a quadratic time complexity, which means an algorithm's running time grows no faster than the square of the input size. 18. What is Big Omega notation? Give an example: Ans: - Big Omega notation, denoted as Ω(f(n)), describes the lower bound on the growth rate of a function, indicating the best-case time complexity of an algorithm. It provides an asymptotic lower limit. Example: Ω(n) represents a linear time complexity, which means an algorithm's running time grows at least as fast as the input size. 19. Define Big Theta notation. Give an example: Ans - Big Theta notation, denoted as Θ(f(n)), describes both the upper and lower bounds on the growth rate of a function, indicating the tightest possible bound on an algorithm's time complexity. Example: Θ(n) represents linear time complexity, meaning an algorithm's running time grows exactly in proportion to the input size. 20. Define Little Oh notation. Give an example: Ans: - Little Oh notation, denoted as o(f(n)), describes a growth rate strictly less than a function, indicating that an algorithm's performance is better than the specified function. Example: If an algorithm has a time complexity of o(n), it means the algorithm is faster-growing than linear (i.e., better than linear).
21. What is a recurrence relation? Give an example: Ans: - A recurrence relation is a mathematical equation or formula that expresses a function's value in terms of one or more of its previous values. It is often used to describe the time complexity of recursive algorithms. Example: The Fibonacci sequence is described by the recurrence relation F(n) = F(n-
- F(n-2) with initial values F(0) = 0 and F(1) = 1, where F(n) represents the nth Fibonacci number. 22. Prove the following statements: a) 100n + 5 = O(n^2) - True, because it is a linear function, and n^2 is an upper bound. b) n^2 + 5n + 7 = Θ(n^2) - True, as it matches the tightest bound for a quadratic function. c) n^2 + n = O(n^3) - True, because it is an upper bound and n^3 grows faster than n^2. d) ½n(n-1) = Θ(n^2) - True, as it is a quadratic function and matches both upper and lower bounds. e) 5n^2 + 3n + 20 = O(n^2) - True, as it is a quadratic function, and n^2 is an upper bound. f) ½n^2 + 3n = Θ(n^2) - True, as it is a quadratic function and matches both upper and lower bounds. g) n^3 + 4n^2 = Ω(n^ 2 ) - True, because it is a lower bound, and n^2 is a lower bound as well. 23. Algorithm Sum(n): S <- 0 For i<- 1 to n do S <- S + i Return S a) What does this algorithm compute? Ans: - This algorithm computes the sum of all integers from 1 to n. b) What is its basic operation? Ans: - The basic operation is the addition operation (S = S + i). c) How many times is the basic operation executed? Ans: - The basic operation is executed 'n' times since it's inside a loop that runs from 1 to 'n'. d) What is the efficiency class of this algorithm? Ans: - The efficiency class of this algorithm is O(n) because it has a linear time complexity, and the number of basic operations is directly proportional to the input size 'n'.
Step 3: Now, 12 is less than 18, so swap 12 and 18 to get a = 18 and b = 12. Step 4: Subtract 12 from 18 to get 6. Step 5: Now, 6 is less than 12, so swap 6 and 12 to get a = 12 and b = 6. Step 6: Subtract 6 from 12 to get 6. Step 7: Now, 6 equals 6, so a = 6 and b = 6. Step 8: Subtract 6 from 6 to get 0. So, the GCD of 48 and 18 is 6.
3. Explain Consecutive integer checking methods of find the GCD of two numbers. Ans: Consecutive Integer Checking Algorithm: The Consecutive Integer Checking Algorithm is a simple, but inefficient, method to find the GCD of two numbers. Here’s how it works:
- Start with the smaller of the two numbers.
- Check if it divides both numbers. If it does, it’s the GCD.
- If not, subtract one and try again.
- Repeat until you find a number that divides both numbers evenly. For example, let’s find the GCD of 48 and 18:
- Start with 18 (the smaller number).
- Check if 18 divides both 48 and 18. It doesn’t (48 / 18 leaves a remainder), so subtract 1 to get 17.
- 17 doesn’t divide both numbers, so subtract 1 to get 16.
- Continue this process until you reach 6.
- 6 divides both 48 and 18, so the GCD of 48 and 18 is 6. This method is straightforward, but it can be slow for large numbers because it potentially requires checking all integers down to 1. Euclid’s algorithm, on the other hand, is much faster and more efficient for large numbers. 4. Explain Algorithm design and analysis process with flow diagram. Ans: Algorithm Design and Analysis Process: Algorithm Design and Analysis Process:
- Problem Definition: Clearly define the problem that the algorithm aims to solve.
- Algorithm Construction: Develop the algorithm using suitable methods (pseudocode, flowcharts, etc.).
- Verification: Check the algorithm for correctness and accuracy.
- Analysis: Evaluate the algorithm's efficiency, time complexity, & space complexity.
- Optimization: Modify the algorithm to improve efficiency if necessary.
- Documentation: Provide clear documentation for future reference. Flow Diagram:
[Problem Definition] --> [Algorithm Construction] --> [Verification] --> [Analysis] --> [Optimization] --> [Documentation] **5. Explain any FIVE Problem types. Ans:** Five Problem Types: 1. Sorting Problems: Involving the arrangement of elements in a specific order. The sorting problem involves rearranging the items of a given list in a specific order (usually non-decreasing). This is a fundamental operation in many applications, including ranking search results, data analysis, and preparing data for other algorithms. For example, sorting algorithms include QuickSort, MergeSort, and BubbleSort. 2. Searching Problems: Finding a particular element in a set of data. Searching involves finding a specific item in a data structure. This is a key operation in many applications, such as looking up a contact in a phone book or finding a webpage in a search engine’s index. Examples of searching algorithms include Binary Search and Linear Search. 3. String Processing: String processing problems involve manipulation and analysis of strings. This includes operations like searching for patterns in text, comparing strings, and transforming one string into another. Examples of algorithms used in string processing include the Knuth-Morris-Pratt (KMP) algorithm for pattern searching and the Levenshtein distance algorithm for measuring the difference between two strings 4. Graph Problems: Dealing with graphs and networks. Graph problems involve the study of graphs, which are mathematical structures used to model pairwise relations between objects. Examples of graph problems include finding the shortest path between two nodes, determining whether a graph is connected, and finding a cycle in a graph. Algorithms used to solve these problems include Dijkstra’s algorithm for shortest paths and Kruskal’s algorithm for minimum spanning trees. Combinatorial Problems: Involving arrangements and combinations. 6. Geometric Problems: Related to geometry and spatial relationships. These problem types encompass a wide range of challenges that algorithms are designed to address, showcasing the diverse applications of algorithmic solutions in various domains. **6. Explain following a. Graph problem b. Combinatorial problems c. Geometrical problems. Ans:** a. Graph Problem: A graph problem involves the study and manipulation of graphs, which are mathematical structures representing relationships between pairs of objects. Graph problems can include finding paths, determining connectivity, and analyzing network structures. **8. Write a note on Graph Data Structure: Ans:** Graph:- A collection of nodes (vertices) and edges connecting pairs of nodes. A Graph is a non-linear data structure that consists of vertices (or nodes) and edges. The edges are lines or arcs that connect any two nodes in the graph. More formally, a Graph is composed of a set of vertices (V) and a set of edges (E), and is denoted by G (E, V). key components and concepts related to Graphs: - Vertices: Vertices are the fundamental units of the graph. Sometimes, vertices are also known as vertex or nodes. - Edges: Edges are drawn or used to connect two nodes of the graph. It can be an ordered pair of nodes in a directed graph. - Directed and Undirected Graphs: In a directed graph, edges form an ordered pair. Edges represent a specific path from some vertex A to another vertex B. In an undirected graph, edges are not associated with directions2. - Weighted Graph: In a weighted graph, each edge is assigned with some data such as length or weight. - Degree of the Node: A degree of a node is the number of edges that are connected with that node. - Graphs are used to solve many real-life problems. They are used to represent networks, which may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like LinkedIn, Facebook. **9. Write a note on following data structures. a. Tree b. Sets c. Dictionary Ans:** a. **Tree**: A tree is a non-linear data structure that consists of nodes connected by edges⁴. It has a hierarchical relationship between the nodes⁴. The tree data structure stems from a single node called a root node and has subtrees connected to the root⁸. Examples of non-linear data structures are trees and graphs². b. **Sets**: A set is a data structure that stores a collection of unique elements, with no duplicates allowed¹. Sets can be implemented using a variety of data structures, including arrays, linked lists, binary search trees, and hash tables¹. The various operations that are performed on a set are add or insert, replace or reassign, delete or remove, find or lookup¹. c. **Dictionary**: A dictionary is a general-purpose data structure for storing a group of objects⁹. A dictionary has a set of keys and each key has a single associated value⁹. When presented with a key, the dictionary will return the associated value⁹. The various operations that are performed on a dictionary are add or insert, replace or reassign, delete or remove, find or lookup⁹. **10. Explain Space complexity and Time complexity with example. Ans:** 1. **Time Complexity**: Time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the input¹. It is the time needed for the completion of an algorithm¹. For example, consider the problem of finding whether a pair (X, Y) exists in an array, A of N elements whose sum is Z. The simplest idea is to consider every pair and check if it satisfies the given condition or not¹. The pseudo-code is as follows: ```python int a [n] for (int i = 0; i < n; i++) cin >> a [i] for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) if (i != j && a [i] + a [j] == z) return true return false Assuming that each of the operations in the computer takes approximately constant time, let it be c. The number of lines of code executed actually depends on the value of Z¹. 2. **Space Complexity**: Space complexity is a parallel concept to time complexity. It is the amount of memory space that an algorithm needs to run to completion⁴. The space needed by an algorithm increases with the size of the input⁴. For example, if we need to create an array of size n, this will require O(n) space. If we create a two-dimensional array of size n*n, this will require O(n^2) space⁴. In recursive calls stack space also counts⁴. These complexities are fundamental to the study of algorithms, as they provide a measure of the efficiency of an algorithm. An algorithm that has good time and space complexity is considered efficient¹²³⁴. **11. Write an algorithm find sum of two matrixes also calculate its time complexity. Ans:** ```python Algorithm AddMatrices(A, B): Input: Matrices A and B of size m x n Output: Resultant matrix C of size m x n 1. Initialize an empty matrix C of size m x n. 2. For each row i from 1 to m: a. For each column j from 1 to n: i. Set C[i][j] = A[i][j] + B[i][j]. 3. Return matrix C.
For example, for a linear search algorithm, the worst case occurs when the element being searched for is at the end of the list, resulting in a time complexity of O(n).
- Best case: This refers to the simplest scenario for an algorithm. The best case analysis considers the scenario in which the algorithm takes the shortest possible time to complete its task. This is typically the scenario in which the input data is in a specific order or configuration that causes the algorithm to perform the minimum number of operations. For a linear search algorithm, the best case occurs when the element being searched for is at the beginning of the list, resulting in a time complexity of O(1)¹⁴.
- Average case: This refers to the average scenario for an algorithm. The average case analysis considers the average time it takes the algorithm to complete its task over all possible input data. This is typically calculated by assuming that all input data is equally likely and then averaging the time it takes the algorithm to complete its task for each input. For a linear search algorithm, the average case would also result in a time complexity of O(n), assuming that the element being searched for is equally likely to be at any position in the list¹⁴. 15. Write an algorithm to perform sequential search and also calculate its Worst case, Best case and average case complexity. Ans: Sequential search is a simple algorithm for finding a specific element in a list of elements. It works by iterating through the list and comparing each element to the element to be searched for until the element is found or the end of the list is reached. `algorithm in Python to perform a sequential search: def sequential_search(lst, item): for i in range(len(lst)): if lst[i] == item: return i return - 1 In this algorithm:
- The worst-case scenario is when the item is not in the list or is the last element in the list. The function would have to iterate through the entire list, resulting in a time complexity of O(n)¹⁵.
- The best-case scenario is when the item is the first element in the list. The function would find the item immediately, resulting in a time complexity of O(1)¹⁵.
- The average-case scenario, assuming that the item is equally likely to be at any position in the list, would result in a time complexity of O(n)¹⁵.
- The space complexity of this algorithm is O(1) because it uses a constant amount of space.
16. Explain Big O notation with example. Ans: In computer science, Big O Notation is a fundamental tool used to find out the time complexity of algorithms. Big O Notation allows programmers to classify algorithms depending on how their run time or space requirements vary as the input size varies. Examples: Runtime Complexity for Linear Search – O(n) Runtime Complexity for Binary Search – O(log n) Runtime Complexity for Bubble Sort, Selection Sort, Insertion Sort, Bucket Sort - O(n^c). Runtime Complexity for Exponential algorithms like Tower of Hanoi - O(c^n). Runtime Complexity for Heap Sort, Merge Sort - O(n log n). Big O Notation gives the upper-bound runtime or worst-case complexity of an algorithm. It analyzes and classifies algorithms depending on their run time or space requirements. 17. Explain Big Omega notation with example. Ans: Big Omega notation is used to describe the lower bound of an algorithm's time complexity, representing the best-case scenario¹. It provides a guarantee that the algorithm will not complete in less time than this bound¹. For example, in the best-case scenario for a simple search algorithm (where the target value is the first element in the list), the algorithm runs in Ω(1) time, meaning it will take at least constant time¹. 18. Explain Big Theta notation with example. Ans: Big Theta notation is used when the algorithm's upper and lower bounds are the same¹¹. It provides a tight bound on the time complexity of an algorithm¹¹. For example, an algorithm that always takes a constant amount of time, regardless of the size of the input, is said to run in Θ(1) time¹¹. This means that the algorithm's time complexity is exactly constant, neither growing with the size of the input nor depending on the specific values of the input¹¹. 19. Explain asymptotic notations Big O, Big Ω and Big θ that are used to compare the order of growth of an algorithm with example. Ans: 1. Big O Notation (O-notation): Big O notation is used to describe the asymptotic upper bound, which provides an upper limit on the time complexity of an algorithm¹. It represents the worst-case scenario in terms of execution time¹. For example, if we have a simple search algorithm that checks each element of a list to find a target value, the worst-case scenario is that the target is at the end of the list or not in the list at all. In this case, the algorithm runs in O(n) time, where n is the number of elements in the list¹.
- Big Omega Notation (Ω-notation): Big Omega notation provides an asymptotic lower bound, which provides a lower limit on the time complexity of an algorithm¹. It represents the best-case scenario¹. For example, in the best-case scenario for a simple
16. Define Big Omega notation and prove a) n^3 ∈ Ω (n^2) b) Prove that 2n + 3 = Ω (n) c)1/2n(n-1) ∈ Ω (n^2) d) Prove that n^3 + 4 n^2 = Ω (n^2) Ans: Big Omega (Ω) notation is used to describe the lower bound of an algorithm's time complexity. It provides an asymptotic lower bound, guaranteeing that the algorithm will not perform faster beyond a certain input size. a) n^3 ∈ Ω(n^2) This is true because n^3 grows at least as fast as n^2 for all n > 1. So we can choose c = 1 and n0 = 1, and we have 0 ≤ c*g(n) ≤ f(n) for all n ≥ n0. b) 2n + 3 = Ω(n) This is true because for n > 3, 2n + 3 is always greater than n. So we can choose c = 1 and n0 = 3, and we have 0 ≤ c*g(n) ≤ f(n) for all n ≥ n0. c) 1/2*n(n-1) ∈ Ω(n^2) This is true because 1/2*n(n-1) simplifies to 1/2*n^2 - 1/2*n, and for n > 2, 1/2*n^2 - 1/2*n is always greater than n^2. So we can choose c = 1 and n0 = 2, and we have 0 ≤ c*g(n) ≤ f(n) for all n ≥ n0. d) n^3 + 4n^2 = Ω(n^2) This is true because n^3 + 4n^2 grows at least as fast as n^2 for all n > 0. So we can choose c = 1 and n0 = 0, and we have 0 ≤ c*g(n) ≤ f(n) for all n ≥ n0. In each case, the function on the left grows at least as fast as the function on the right for n greater than n0, so the statement is true according to the definition of Big Omega notation. 17. Define Big Theta notation and prove a) n^2+5n+7 = Θ (n^2) b) 1/2n^2 +3n =Θ (n^2) c)1/2n(n-1) ∈ Θ (n^2) Ans: Big Theta (Θ) notation is used to describe the average-case performance of an algorithm. It provides an asymptotic tight bound, meaning it gives both an upper and a lower bound on the growth rate of runtime of an algorithm⁶⁷⁸⁹[^10^]. For a given function g(n), we denote f(n) ∈ Θ(g(n)) if there are positive constants c1, c2 and n0 such that 0 ≤ c1*g(n) ≤ f(n) ≤ c2*g(n) for all n ≥ n0. This means f(n) grows at the same rate as g(n). a) n^2+5n+7 = Θ(n^2): This is true because for n ≥ 1, n^2 ≤ n^2 + 5n + 7 ≤ 13n^2¹¹. So we can choose c1 = 1, c2 = 13, and n0 = 1. b) 1/2n^2 +3n =Θ(n^2): This is true because for n ≥ 6, 1/2n^2 ≤ 1/2n^2 + 3n ≤ 2n^2¹. So we can choose c1 = 1/2, c2 = 2, and n0 = 6.
c) 1/2n(n-1) = Θ(n^2): This is true because 1/2n(n-1) simplifies to 1/2n^2 - 1/2n, and for n ≥ 2, 1/2n^2 - 1/2n grows at the same rate as n^2. So we can choose c1 = 1/2, c2 = 1, and n0 = 2. In each case, the function on the left grows at the same rate as the function on the right for n greater than n0, so the statement is true according to the definition of Big Theta notation.
18. Explain with example mathematical analysis of non-recursive algorithm. Ans: Mathematical analysis of non-recursive algorithms often involves determining the time complexity of the algorithm, which is a measure of the amount of time an algorithm takes to run as a function of the size of the input to the program. For example, consider a simple non-recursive algorithm that sums all the elements in an array: Python' def sum_array(arr): total = 0 for num in arr: total += num return total The time complexity of this algorithm is O(n), where n is the number of elements in the array. This is because each operation (adding a number to the total) is performed n times. 19. Write an algorithm to Find the largest element in an array and also perform mathematical analysis. Ans: Python code: def find_largest(arr): largest = arr[0] for num in arr: if num > largest: largest = num return largest The time complexity of this algorithm is also O(n), where n is the number of elements in the array. This is because we’re comparing each element in the array to the current largest element once.
This algorithm uses bitwise operations to count the number of bits in n. The time complexity of this algorithm is O(log n), where n is the number being examined. This is because the number of bits in n is proportional to the logarithm (base 2) of n. The time complexity depends on the size of the input and the specific operations performed in the algorithm.
23. List the steps for analyzing the time efficiency of recursive algorithm. Ans: Steps for Analyzing the Time Efficiency of Recursive Algorithm:
- Define the Recurrence Relation:Identify the recursive relationship that describes how the problem is decomposed into smaller subproblems. This recurrence relation is often a key to understanding the algorithm's time complexity.
- Write Down the Base Case(s):Identify the base case(s) of the recursion, which represent the smallest subproblems that can be directly solved. Base cases are crucial for determining when the recursion stops.
- Determine the Size of Subproblems: Analyze how the size of the problem decreases with each recursive call. Define the size of the subproblems in terms of the input size.
- Express the Recurrence Relation: Write down the recurrence relation explicitly, expressing the time complexity of the algorithm in terms of the size of the input and the time complexity of smaller subproblems.
- Solve or Simplify the Recurrence: Solve the recurrence relation or simplify it to obtain a closed-form expression. This step involves expressing the time complexity of the algorithm as a function of the input size without recursion.
- Determine the Dominant Term: Identify the dominant term in the closed-form expression. The dominant term indicates the growth rate of the time complexity and is often the most significant factor in determining the overall efficiency.
- Analyze the Time Complexity: Express the time complexity of the recursive algorithm using Big O notation. The result should provide a clear understanding of how the algorithm's efficiency scales with the input size.
- Verify with Examples: Validate the derived time complexity by running the algorithm on different inputs and comparing the observed behavior with the theoretical analysis.
- Consider Best, Worst, and Average Cases: Analyze the best-case, worst-case, and average-case time complexities separately, if applicable. Different input scenarios may lead to distinct time complexities.
- Optimize if Necessary: If the derived time complexity is not acceptable, consider optimizing the algorithm or exploring alternative algorithmic approaches to achieve the desired efficiency. By following these steps, you can systematically analyze the time efficiency of a recursive algorithm, gaining insights into its performance characteristics and enabling informed decisions about its suitability for specific use cases.
23. Explain with example mathematical analysis of recursive algorithm. Ans: Let’s consider the recursive algorithm for factorial calculation: Python def factorial(n): if n == 0: return 1 else: return n * factorial(n-1) The base case is n == 0, where the function returns 1. The recurrence relation can be defined as T(n) = T(n-1) + c, where c is a constant representing the time to perform the multiplication2. This relation says that the time to compute factorial(n) is the time to compute factorial(n-1) plus the time for the multiplication operation. Solving this recurrence relation gives T(n) = cn, which means the time complexity of the algorithm is O(n). 24. Write an algorithm to find the factorial of a number using recursion and also perform mathematical analysis. Ans: Recursive Algorithm for Factorial Calculation: Python’ def factorial(n): if n == 0: return 1 else: return n * factorial(n - 1) Mathematical Analysis: The factorial of a non-negative integer n is the product of all positive integers from 1 to n. Mathematically, it is represented as: n! = n * (n - 1) * (n - 2) * ... * 1 Using the recursive definition of factorial, we can derive the following recurrence relation: n! = n * (n - 1)! This recurrence relation forms the basis of the recursive algorithm for calculating factorial. The base case of the recursion is n = 0, for which factorial(0) = 1.
