Give the pseudo-code of the following algorithm: 1 ALGORITHM GE(A[0...n-1, 0...n]) 2 // Input: An n (n+1) matrix A[0...n-1, 0...n] of real numbers 3 for i <- 0 to n-2 do 4 for j <- i+1 to n-1 do 5 for k <- i to n do 6. A[j, k] <- A[j, k] - A[i, k] A[j, i] / A[i, i] a. Assume that we only focus on the assignment operation in line 6. Write the function following n to show the time complexity for the above algorithm and then determine the corresponding Big-O growth function of the above algorithm. b. Determine which is not efficient in the above algorithm and propose the way to take it more efficient.
a. The time complexity of the algorithm can be analyzed as follows:
The dominant operation is in line 6, which gets executed for each iteration of all three loops.
So, the total number of times the key operation is executed: T(n) = n n n = n^3
Therefore, the time complexity of the algorithm, in Big-O notation, is O(n^3).
b. The efficiency of this algorithm depends on the specific use case. However, one general improvement could involve working with sparse matrices if applicable. Since the algorithm iterates through every entry in the matrix, if the majority of the entries in a matrix are zero (i.e., it's a sparse matrix), then time could be saved by skipping these zero entries.
Furthermore, you could improve performance by parallelizing the operations. Multiple computations from line 6 can occur simultaneously, which would lower the time complexity, assuming you have the necessary computational resources.
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