Algorithm Complexity and Sorting Algorithms
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Define Algorithm, Time Complexity and Space Complexity.

An algorithm is a step-by-step procedure for solving a problem. Time complexity measures the time an algorithm takes to complete as a function of the length of the input, while space complexity measures the amount of memory space required by the algorithm as a function of the length of the input.

Explain Worst Case, Best Case and Average Case Complexity with suitable examples.

Worst case complexity refers to the maximum time taken by an algorithm for any input size; Best case refers to the minimum time taken for the most favorable input; Average case considers the expected time taken on random inputs.

What is the best case, worst case and average case for Quick Sort algorithm?

Best case: O(n log n), Worst case: O(n^2), Average case: O(n log n)

Write an algorithm for Selection Sort Method.

<ol> <li>Loop through the array to find the smallest element. 2. Swap it with the first element. 3. Repeat the process for the remaining elements.</li> </ol> Signup and view all the answers

Demonstrate Binary Search method to search for Key = 14 from the array A.

<ol> <li>Check the middle element of the array. 2. If it's 14, return the index. 3. If 14 is less, search the left half; if more, search the right half. Repeat until found.</li> </ol> Signup and view all the answers

What is the Master theorem?

<p>The Master theorem provides a way to analyze the time complexity of divide and conquer algorithms through recurrence relations.</p> Signup and view all the answers

Solve the recurrence T(n) = T(n/2) + 1 using the Master theorem.

<p>T(n) = O(log n)</p> Signup and view all the answers

Write the recurrence relation T(n) = 2T(n/2) + n log n and find its solution.

<p>The solution is T(n) = O(n log^2 n).</p> Signup and view all the answers

What is the Principle of Optimality?

<p>It states that an optimal solution to any instance of an optimization problem is composed of optimal solutions to its subproblems.</p> Signup and view all the answers

Find the Longest Common Subsequence (LCS) of A={K,A,N,D,L,A,P} and B={A,N,D,L}.

<p>The LCS is {A,N,D,L}.</p> Signup and view all the answers

Discuss the Assembly Line Scheduling problem using dynamic programming with an example.

<p>The assembly line scheduling problem involves optimizing the processing of tasks across multiple assembly lines, often using dynamic programming to find the minimum cost to complete them.</p> Signup and view all the answers

Give the characteristics of Greedy Algorithms.

<p>Greedy algorithms make local optimum choices at each step with the hope that these choices will lead to a global optimum.</p> Signup and view all the answers

What is the difference between greedy approach and dynamic programming?

<p>Greedy algorithms make choices based on local optimization, while dynamic programming solves problems by combining solutions to subproblems, usually leading to optimal solutions.</p> Signup and view all the answers

Consider Knapsack capacity W=15, weights w=(4, 5, 6, 3) and values v=(10, 15, 12, 8). Find the maximum profit using greedy method.

<p>The maximum profit is 22 using items with weights 3 and 5.</p> Signup and view all the answers

Explain Articulation Point, Graph, and Tree.

<p>An articulation point in a graph is a vertex that, when removed, increases the number of connected components. A graph is a set of vertices connected by edges, and a tree is a connected graph without cycles.</p> Signup and view all the answers

Find the Minimum Spanning Tree for the given graph using Prim’s Algorithm.

<p>Prim's algorithm starts from an initial vertex and adds the least weight edge connecting a vertex in the growing spanning tree to another vertex until all vertices are included.</p> Signup and view all the answers

Explain Breadth-First Traversal Method for Graph with an algorithm and example.

<p>Breadth-first traversal explores the neighbor nodes at the present depth before moving on to nodes at the next depth level, implemented using a queue.</p> Signup and view all the answers

Explain Huffman code with Example.

<p>Huffman coding is a compression method that uses variable-length codes for encoding characters based on their frequencies. For example, more frequent characters will have shorter codes.</p> Signup and view all the answers

Write the Kruskal’s Algorithm to find out Minimum Spanning Tree.

<ol> <li>Sort edges by weight. 2. Add edges one by one to the tree, ensuring no cycles are formed. 3. Stop when there are V-1 edges in the tree.</li> </ol> Signup and view all the answers

Explain the fractional knapsack problem with example.

<p>The fractional knapsack problem allows breaking items into smaller parts to maximize value within a given capacity, unlike the 0/1 knapsack which requires whole items.</p> Signup and view all the answers

What is the string-matching problem? Define valid shift and invalid shift.

<p>String-matching involves finding occurrences of a pattern within a text. A valid shift occurs when the pattern aligns with a potential match in the text; an invalid shift does not.</p> Signup and view all the answers

Define P, NP, NP-Hard and NP-Complete Problem.

<p>P problems can be solved in polynomial time, NP problems can be verified in polynomial time, NP-Hard problems are at least as hard as NP problems, and NP-Complete problems are NP problems that are as hard as any problem in NP.</p> Signup and view all the answers

Explain Backtracking Method. What is N-Queens Problem? Give the solution of 4-Queens Problem using Backtracking Method.

<p>Backtracking is a method for finding solutions to problems by incrementally building candidates and abandoning them if they fail to satisfy the constraints. The N-Queens Problem involves placing N queens on an N×N chessboard so no two queens threaten each other. The solution for 4-Queens can be represented with positions (2, 4, 1, 3).</p> Signup and view all the answers

Explain 'P = NP?' problem.

<p>The 'P = NP?' problem questions whether every problem for which a solution can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time).</p> Signup and view all the answers

Explain Minimax Principle.

<p>The Minimax Principle is a decision rule used for minimizing the possible loss for a worst-case scenario, particularly in game theory and decision-making.</p> Signup and view all the answers

What is Finite Automata? Explain the use of finite automata for string matching with suitable example.

<p>Finite automata are theoretical machines used to recognize patterns within input strings. They can be used for string matching algorithms like the Knuth-Morris-Pratt algorithm for efficiently finding substrings.</p> Signup and view all the answers

Study Notes

Algorithm Complexity

  • An algorithm is a set of well-defined instructions for solving a problem
  • Time Complexity refers to the number of operations an algorithm performs as a function of the input size
  • Space Complexity refers to the amount of memory an algorithm uses as a function of the input size
  • Worst Case Complexity is the maximum time or space required for any input of a given size
  • Best Case Complexity is the minimum time or space required for any input of a given size
  • Average Case Complexity is the average time or space required for all inputs of a given size

Sorting Algorithms

  • Quick Sort is a divide and conquer algorithm that partitions the input array into two subarrays around a pivot element
  • The pivot element is placed at its correct position in the sorted array, and the subarrays are recursively sorted
  • Worst Case of Quick Sort occurs when the pivot is always the smallest or largest element in the input array
  • Best Case of Quick Sort occurs when the pivot is always the median element in the input array
  • Average Case of Quick Sort occurs when the pivot is randomly selected for each partition

Searching Algorithms

  • Binary Search is an efficient searching algorithm that works on a sorted input array
  • It repeatedly divides the search interval in half until the target element is found or the search interval becomes empty
  • Selection Sort is an in-place sorting algorithm that iterates through the input array and selects the minimum element and places it at its correct position in the sorted array

Recurrence Relations

  • A recurrence relation is an equation that defines a sequence of numbers in terms of previous terms in the sequence
  • Master Theorem is a theorem that can be used to solve recurrence relations of the form T(n) = aT(n/b) + f(n)
  • Iterative Method is a method for solving recurrence relations by repeatedly substituting the recurrence relation into itself until a closed form solution is obtained

Dynamic Programming

  • Dynamic Programming is a technique for solving optimization problems by breaking them down into smaller subproblems
  • The solutions to the subproblems are stored in a table to avoid recomputing them
  • Principle of Optimality states that the optimal solution to a problem can be obtained by combining the optimal solutions to its subproblems
  • Longest Common Subsequence (LCS) problem is a dynamic programming problem that finds the longest subsequence common to two input sequences
  • Assembly Line Scheduling Problem is a dynamic programming problem that optimizes task allocation in an assembly line

Greedy Algorithms

  • Greedy Algorithm is a technique that makes the locally optimal choice at each step in the hope of finding the globally optimal solution
  • Knapsack Problem is a classic optimization problem that involves maximizing the value of items that can be placed into a knapsack with a limited capacity

Graph Algorithms

  • Articulation Point is a vertex in a graph whose removal would increase the number of connected components in the graph
  • Minimum Spanning Tree (MST) is a tree that spans all vertices of a connected graph with the minimum total edge weight
  • Prim's Algorithm is a greedy algorithm that finds an MST by repeatedly adding the minimum-weight edge that connects a vertex in the MST to a vertex outside the MST
  • Breath First Traversal (BFT) is a graph traversal algorithm that visits the vertices of a graph in a level-by-level manner

Huffman Coding

  • Huffman Coding is a data compression algorithm that uses variable-length codes representing frequently occurring symbols with shorter codes and less frequent symbols with longer codes
  • Kruskal's Algorithm is a greedy algorithm that finds an MST by repeatedly adding the minimum-weight edge that does not create a cycle

String Matching

  • String Matching Problem is a problem that involves finding all occurrences of a pattern string within a text string
  • Valid Shift is a shift of the pattern that aligns at least one character of the pattern with a matching character in the text
  • Invalid Shift is a shift of the pattern that does not align any character of the pattern with a matching character in the text

Computational Complexity

  • P is the class of decision problems that can be solved in polynomial time
  • NP is the class of decision problems that can be verified in polynomial time
  • NP-Hard is the class of problems that are at least as hard as any NP problem
  • NP-Complete is the class of problems that are both NP and NP-Hard
  • P = NP? is a major unsolved problem in computer science that asks whether every problem that can be verified in polynomial time can also be solved in polynomial time

Game Theory

  • Minimax Principle is a strategy for playing games against an adversarial opponent
  • Finite Automata is a mathematical model of computation used for recognizing patterns in strings
  • Backtracking is a general algorithm design technique that systematically explores all possible solutions to a problem, often using a tree structure
  • N-Queens Problem is a classic backtracking problem that asks to place N chess queens on an N×N chessboard such that no two queens threaten each other

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Analysis and Design of Algorithms Past Paper PDF - GUJARAT Technical University - Summer 2022

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Explore key concepts of algorithm complexity including time and space complexity, as well as worst, best, and average case scenarios. Delve into sorting algorithms, specifically Quick Sort, and understand how they operate and their efficiency under different conditions. Test your knowledge with this insightful quiz!

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