Differentiate between Cubic-time and Factorial-time algorithms. Give example of one algorithm each for these two running times.
Cubic-time algorithms have a time complexity of O(n³), meaning their running time increases proportionally to the cube of the input size. These algorithms are typically feasible for moderate input sizes. An example is __________ __________ ________ ________ ___ __________ ___ ____.
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Write and explain the Left to Right binary exponentiation algorithm. Demonstrate the use of this algorithm to compute the value of 32 (Show the 29 steps of computation). Explain the worst-case complexity of this algorithm.
Explain the term Decision problem with the help of an example. Define the following problems and identify if they are decision problem or optimisation problem? Give reasons in support of your answer.
(i) Travelling Salesman Problem
(ii) Graph Colouring Problem
(iii) 0-1 Knapsack Problem
Explain the Merge procedure of the Merge Sort algorithm. Demonstrate the use of recursive Merge sort algorithm for sorting the following data of size 8: [19, 18, 16, 12, 11, 10, 9, 8]. Compute the complexity of Merge Sort algorithm.
What is an Optimisation Problem? Explain with the help of an example. When would you use a Greedy Approach to solve optimisation problem? Formulate the Task Scheduling Problem as an optimisation problem and write a greedy algorithm to solve this problem. Also, solve the following fractional Knapsack problem using greedy approach. Show all the steps.
Suppose there is a knapsack of capacity 20 Kg and the following 6 items are to packed in it. The weight and profit of the items are as under:
Select a subset of the items that maximises the profit while keeping the total weight below or equal to the given capacity.
SAT Problem
Write and explain the Bubble sort algorithm. Discuss its best and worst-case time complexity.
What are P and NP class of Problems? Explain each class with the help of at least two examples.