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.
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Explain the algorithm to find the optimal Binary Search Tree. Demonstrate this algorithm to find the Optimal Binary Search Tree for the following probability data (where pi represents the probability that the search will be for the key node ki, whereas qi represents that the search is for dummy node di. Make suitable assumptions, if any)
| i | 0 | 1 | 2 | 3 | 4 |
| pi | 0.10 | 0.15 | 0.20 | 0.10 | |
| qi | 0.05 | 0.10 | 0.10 | 0.10 | 0.10 |
What are asymptotic bounds for analysis of efficiency of algorithms? Why are asymptotic bounds used? What are their shortcomings? Explain the Big O and Big notation with the help of a diagram. Find the Big O-notation and -notation for the function:
Consider the following Graph:
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.
Define the NP-Hard and NP-Complete problem. How are they different from each other. Explain the use of polynomial time reduction with the help of an example
Write an algorithm to multiply two square matrices of order n*n. Also explain the time complexity of this algorithm.
Explain the divide and conquer approach of multiplying two large integers. Compute the time complexity of this approach. Also, explain the binary search algorithm and find its time complexity