How can an optimisation problem be represented mathematically. Explain with the help of an example. Explain the steps of designing a Greedy solution for an optimisation problem. Also explain the concept of local and global optimal solutions. Solve the following fractional Knapsack problem using greedy approach. Show all the steps.
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Differentiate between Quadratic-time and Exponential-time algorithms. Give example of one problem each for these two running times.
Write and explain the Insertion sort algorithm. Discuss its best and worst-case time complexity.
Write the Right to Left binary exponentiation algorithm. Demonstrate the use of this algorithm to compute the value of 523. Show all steps of the computation. Explain the worst-case complexity of this algorithm.
Explain the divide and conquer approach of multiplying two large matrices of size 16 x 16. Compute the time complexity of this approach.
Consider the following Graph:
Figure 1: A sample weighted Grари
a) Write the Kruskal's algorithm to find the minimum cost spanning tree of a graph. Also, find the time complexity of Kruskal's algorithm. Demonstrate the use of Kruskal's algorithm and Prim's algorithm to find the minimum cost spanning tree for the Graph given in Figure 1. Show all the steps.
Bellman-Ford's shortest path algorithm for the graph given in Figure 1. Show all the steps of computation.
search (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)
What is a recurrence relations? When are they used? Solve the following recurrence relations using the Master's method
Here are the equations from the image written in LaTeX format, along with a plain text version.$
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Explain the Big Theta (O) and Big Omega (2) notations with the help of a diagram. What is the need of asymptotic bounds? Also, list the shortcomings of the asymptotic bounds. Find the Big O, O and 2 bounds for the following function:
f(n)=999n3+999999n+100
How can an optimisation problem be represented mathematically. Explain with the help of an example. Explain the steps of designing a Greedy solution for an optimisation problem. Also explain the concept of local and global optimal solutions. Solve the following fractional Knapsack problem using greedy approach. Show all the steps.