Design and Analysis of Algorithm MCQ

A. A step-by-step procedure for solving a problem. An algorithm is a step-by-step procedure for solving a problem.

D. Ambiguity. An algorithm must be free from ambiguity.

C. To clarify the purpose of the algorithm. Well-defined inputs and outputs clarify the purpose and behavior of the algorithm.

A. Have a defined end. An algorithm must have a defined end to ensure it does not run indefinitely.

C. It provides a logical structure for problem-solving. Algorithmic thinking provides a logical structure for approaching and solving problems.

B. Encourages creative approaches to problems. Algorithmic thinking encourages systematic and creative approaches to solving problems.

A. By reducing the problem into smaller, manageable steps. Algorithms break down a problem into smaller steps that can be more easily solved.

B. Problem definition. Problem definition focuses on clearly defining what the algorithm is meant to solve.

B. The worst-case time complexity of an algorithm. Big O notation describes the worst-case time complexity of an algorithm.

A. It measures the best-case performance of an algorithm. Ω (Omega) notation describes the best-case performance of an algorithm.

B. Dividing the problem into smaller subproblems, solving them, and combining the results. Divide and conquer involves dividing the problem into smaller subproblems, solving them independently, and combining the results.

A. Into two equal parts. Binary search divides the search space into two equal parts at each step.

A. Quick Sort. Quick Sort is an example of a divide and conquer algorithm.

B. O(n log n). The time complexity of merge sort is O(n log n).

A. Choosing a pivot. Choosing a pivot is a key step in the Quick Sort algorithm.

A. The subproblems are solved independently. In divide and conquer, subproblems are solved independently before combining the results.

A. Solve the problem by making a series of local optimal choices. Greedy algorithms work by making a series of local optimal choices to build the solution.

A. Huffman Coding. Huffman Coding is solved using the greedy algorithm approach.

A. Selecting the vertex with the smallest edge cost. Prim’s algorithm makes the greedy choice by selecting the vertex with the smallest edge cost.

A. Optimal substructure. Greedy algorithms rely on the problem having an optimal substructure.

C. Dynamic Programming. The 0/1 knapsack problem is solved using dynamic programming, not the greedy approach.

A. Selecting the smallest edge that does not form a cycle. Kruskal’s algorithm selects the smallest edge that does not form a cycle.

A. Dynamic programming solves each subproblem once and stores the solution. Dynamic programming solves each subproblem once, stores the solution, and reuses it.

A. Matrix Chain Multiplication. Matrix Chain Multiplication is a problem that can be solved using dynamic programming.

A. Storing solutions to subproblems to avoid redundant calculations. Memoization is a technique used to store the results of subproblems to avoid recalculating them.

A. Longest Common Subsequence. The Longest Common Subsequence algorithm is based on dynamic programming.

C. O(nW), where W is the capacity of the knapsack. The dynamic programming solution to the 0/1 knapsack problem has a time complexity of O(nW), where W is the capacity of the knapsack.

A. Try all possible solutions and backtrack when a solution fails. Backtracking involves trying possible solutions and backtracking when a solution is not feasible.

A. N-Queens Problem. The N-Queens Problem is often solved using the backtracking technique.

A. It explores all possible solutions. Backtracking explores all possible solutions by recursively trying each possibility.

A. To find an optimal solution by systematically exploring solution space. Branch and bound is used to find the optimal solution by systematically exploring the solution space.

A. Travelling Salesman Problem. The Travelling Salesman Problem can be solved using the branch and bound technique.

A. By eliminating branches that do not provide better solutions. Branch and bound prunes the solution space by eliminating branches that do not provide better solutions.

A. It makes decisions randomly to solve a problem. A randomized algorithm makes some decisions randomly to solve a problem.

A. Randomized Quick Sort. Randomized Quick Sort is an example of a randomized algorithm.

A. The amount of time an algorithm takes relative to its input size. Time complexity measures the amount of time an algorithm takes as a function of the input size.

A. O(1). In the best-case scenario, the time complexity of linear search is O(1), when the target element is the first in the list.

A. Time complexity. Recurrence relations help analyze the time complexity of recursive algorithms.

A. The amount of memory an algorithm uses as a function of input size. Space complexity refers to the amount of memory an algorithm uses as its input size grows.

A. The extra space or temporary storage used by an algorithm. Auxiliary space refers to the extra space or temporary storage used by an algorithm during its execution.

A. Non-deterministic Polynomial time. NP stands for Non-deterministic Polynomial time.

A. Travelling Salesman Problem. The Travelling Salesman Problem is a classic example of an NP-Complete problem.

A. The first proof that SAT is NP-Complete. The Cook-Levin theorem was the first proof that SAT is NP-Complete.

A. P. P is the class of problems that are both in NP and can be solved in polynomial time.

A. NP-Hard Problems. Approximation algorithms are typically used for NP-Hard problems where finding an exact solution is computationally infeasible.

B. O(n^2). The time complexity of selection sort is O(n^2) as it involves two nested loops.

A. Bubble Sort. Bubble Sort repeatedly swaps adjacent elements if they are in the wrong order.

B. O(n). The best-case time complexity of insertion sort is O(n), which occurs when the input array is already sorted.

A. Quick Sort. Quick Sort is a divide and conquer algorithm with an average time complexity of O(n log n).

A. O(n log n). Merge sort has a time complexity of O(n log n) because it divides the array and then merges the sorted subarrays.

A. Counting Sort. Counting Sort is a non-comparative sorting algorithm, as it sorts elements by counting occurrences rather than comparing elements.

A. O(n). In the worst case, the time complexity of linear search is O(n), as every element may need to be checked.

A. The search space is halved with each comparison. Binary search has O(log n) complexity because the search space is halved with each comparison.

A. A value smaller than the parent node. In a BST, the left child contains a value smaller than the parent node.

A. Self-balancing to ensure logarithmic search time. AVL trees are self-balancing, which ensures that operations like search, insert, and delete take logarithmic time.

A. An adjacency matrix uses a 2D array, while an adjacency list uses linked lists. An adjacency matrix uses a 2D array, while an adjacency list uses linked lists to represent the graph.

A. Breadth-First Search (BFS). Breadth-First Search (BFS) explores all adjacent vertices before moving to the next level.

A. By exploring as far as possible along each branch before backtracking. In DFS, the graph is traversed by exploring as far as possible along each branch before backtracking.

A. The minimum spanning tree of a graph. Prim’s algorithm is used to find the minimum spanning tree of a graph.

A. By selecting the smallest edge that does not form a cycle. Kruskal’s algorithm selects the smallest edge that does not form a cycle.

A. O(E log V). The time complexity of Prim’s algorithm when implemented using a binary heap is O(E log V), where E is the number of edges and V is the number of vertices.

A. The shortest path from a single source to all other vertices. Dijkstra’s algorithm computes the shortest path from a single source to all other vertices in a graph.

A. Negative weight edges. The Bellman-Ford algorithm can handle graphs with negative weight edges.

A. O(V^3). The time complexity of the Floyd-Warshall algorithm is O(V^3), where V is the number of vertices.

A. Computing the maximum flow in a flow network. The Ford-Fulkerson algorithm is used to compute the maximum flow in a flow network.

A. Edmonds-Karp Algorithm. The Edmonds-Karp algorithm improves the Ford-Fulkerson method by using breadth-first search to find augmenting paths.

A. Pair vertices of two sets such that the number of edges is maximized. The goal of maximum bipartite matching is to pair vertices from two sets such that the number of edges is maximized.

A. Optimization problems in logistics and traffic flow. Network flow algorithms are often used in optimization problems such as logistics and traffic flow.

B. O(n * m). The time complexity of the naive string matching algorithm in the worst case is O(n * m), where n is the length of the text and m is the length of the pattern.

A. It uses hashing to find multiple pattern occurrences efficiently. The Rabin-Karp algorithm uses hashing to efficiently find multiple occurrences of a pattern in a text.

A. It avoids rechecking characters after a mismatch. The KMP algorithm avoids rechecking characters after a mismatch, which improves its efficiency over the naive method.

A. It skips sections of the text based on mismatches. The Boyer-Moore algorithm skips sections of the text based on mismatches, which makes it more efficient.

A. O(log n). The time complexity of inserting an element into a binary heap is O(log n).

A. The root. In a max heap, the largest element is always at the root.

A. Fibonacci heaps allow faster decrease-key operations. Fibonacci heaps allow faster decrease-key operations compared to binary heaps.

A. To flatten the structure, making future operations faster. Path compression flattens the structure, making future operations faster by keeping the tree shallower.

A. Reduce the height of the tree in union-find operations. Union by rank helps to reduce the height of the tree in union-find operations.

A. To map data of arbitrary size to fixed-size values. A hash function maps data of arbitrary size to fixed-size values for efficient lookups.

A. Linear probing. Linear probing is a collision resolution technique that uses open addressing.

A. Efficiently storing and searching strings. A trie is primarily used for efficiently storing and searching strings.

A. A character of a string. In a trie, each node represents a character of a string.

A. Efficient range queries and updates. Segment trees are used for efficient range queries and updates on arrays.

A. Performing range sum queries. A Binary Indexed Tree (BIT) is used for performing efficient range sum queries.

A. NP-Hard problems. Approximation algorithms are commonly used for NP-Hard problems where finding an exact solution is impractical.

A. Approximation algorithms. The Traveling Salesman Problem is often solved using approximation algorithms due to its NP-Hard nature.

A. The ratio of the approximate solution to the optimal solution. The performance ratio is the ratio of the approximate solution’s value to the optimal solution’s value.

A. It produces a correct result with high probability but not always. Monte Carlo algorithms produce a correct result with high probability but not always.

A. Las Vegas algorithms always produce a correct result. Las Vegas algorithms always produce a correct result but may take a variable amount of time.

A. The future state depending only on the current state. A Markov chain is characterized by the fact that the future state depends only on the current state, not on the sequence of events that preceded it.

A. P. A problem that can be solved in polynomial time is classified under the complexity class P.

A. They can be verified in polynomial time. NP problems can be verified in polynomial time, even though they may not be solvable in polynomial time.

A. Guess solutions and verify correctness. Non-deterministic algorithms guess solutions and verify their correctness, as opposed to following a fixed sequence of steps.

A. Shor’s Algorithm. Shor’s Algorithm is a quantum algorithm used for factoring large integers.

A. To search an unsorted database in $O(\sqrt{n})$ time. Grover’s algorithm is used to search an unsorted database in $O(\sqrt{n})$ time.

A. Problems involving large-scale data search. Quantum algorithms, such as Grover’s, offer significant speedups in problems involving large-scale data search.

A. Divide tasks among multiple processors to reduce computation time. Parallel algorithms are designed to divide tasks among multiple processors to reduce computation time.

A. Performed across multiple networked computers. Distributed algorithms involve computation that is spread across multiple networked computers.

A. Large data sets that cannot fit into memory. Streaming algorithms are designed to handle large data sets that cannot fit into memory by processing data in small chunks.

A. They mimic the process of natural selection to solve optimization problems. Genetic algorithms mimic the process of natural selection to find approximate solutions to optimization problems.

A. Selection, crossover, and mutation. In evolutionary algorithms, a population of solutions evolves over time through processes like selection, crossover, and mutation.