State and explain the theorems for computing the bounds O, and
Apply these theorem to find the O-notation, Ω-notation and Θ-notation for the function:
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Using Horner's rule, evaluate the polynomialAnalyse the computation time required for polynomial evaluation using Horner’s rule against the Brute force method.
Expalin the Partition procedure of the Quick Sort algorithm. Use this procedure and quick sort algorithm to sort the following array of size 8: [12, 9, 17, 15, 23, 19, 16, 24]. Compute the worst case and best case complexity of Quick sort algorithm.
Graph Colouring Problem
What is the use of Topological sorting? Write and explain the Topological sorting algorithm. Also, compute the time complexity for the topological sorting algorithm.
What is a Greedy Approach to Problem-solving? Formulate the fractional Knapsack Problem as an optimisation problem and write a greedy algorithm to solve this problem. Solve the following fractional Knapsack problem using this algorithm. Show all the steps. Suppose there is a knapsack of capacity 15 Kg and 6 items are to packed in it. The weight and profit of the items are as under:
(p1, p2,…, p6) = (3, 2, 4, 5, 1, 6)
(w1, w2,…, w6) = (2, 1, 2, 1, 5, 1)
Clique problem
What is the purpose of using Huffman Codes? Explain the steps of building a huffman tree. Design the Huffman codes for the following set of characters and their frequencies: a:15, e:19, s:5, d:6, f:4, g:7, h:8, t:10.