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)
The divide and conquer approach for multiplying two large matrices of size 16x16 involves breaking down the problem into smaller subproblems. To multiply two matrices, A (of size 16x16) and B (of size 16x16), we divide each matrix into four submatrices. Each matrix is split into 4 submatrices, resulting in submatrices A1, A2, A3, A4 for matrix A and B1, B2, B3, B4 for matrix B. We then perform recursive multiplications of these smaller submatrices. The multiplication of matrices is done in a way that reduces the overall computational complexity by solving subproblems recursively. For matrix multiplication, we use the Strassen algorithm. This algorithm divides each matrix into 4 blocks (submatrices) and recursively computes _________ __________ ____ _______ ___.
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Differentiate between Quadratic-time and Exponential-time algorithms. Give example of one problem each for these two running times.
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.
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)
Write and explain the Insertion sort algorithm. Discuss its best and worst-case time complexity.
Expalin the PARTITION algorithm of the Quick Sort taking the last element as pivot. Demonstrate the use of Quick sort algorithm for sorting the following data of size 8: [1, 5, 8, 11, 16, 19, 29, 38]. Compute the complexity of the worst case of Quick Sort algorithm.
Design and develop an efficient algorithm to find all the prime numbers in the range 2 to 10000.Access the complexity of this algorithm?
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