Compute the computational complexity of the following code snippets (in “Big-O” notation): (a) for (i = 1; i < n; i *= 2) { for (j = n; j > 0; j /= 2) { for (k = j; k < n; k += 2) { sum + (i+j k ); } (b) Assume that n = 2m. } for (i = n; i > 0; i--) { for(j= 1; j
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- Question 2 Consider the following algorithm: g1 = 7 g2 = 6 for k in range(3,8): gk = (k-1)·gk-1 + gk-2 What is the last term, g8, of the recursive sequence generated as a result of executing this algorithm? Your Answer: Question 2 options: AnswerPlease provide at least one example of an algorithm for each of the following complexity classes: log2n, n, n log2n, n2, n3, 2n, n!, Undecidable..Big-O notation. (a) Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n2, f6(n) = 2n, f7(n) = n!, f8(n) = nn. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n2 < 2n < n! < nn. Prove this is true. [Hint: You are expected to prove the following asymptotics by using the definition of big-O notation: 1 = O(log n), log n = O(n), n = O(n log n), n log n = O(n2), n2 = O(2n), 2n = O(n!), n! = O(nn). (b) Let f, g : N → R+, prove that O(f(n) + g(n)) = O(max{f(n), g(n)}). [Hint: The key is max{f(n), g(n)} ≤ f(n) + g(n) ≤ 2 · max{f(n), g(n)}. Note: Proving this will help you to understand why we can leave out the insignificant parts in big-O notation and only keep the dominate part, e.g., O(n2+n log n+n) = O(n2).] (c) Let f, g : N → R+, prove that Ω(f(n) + g(n)) = Ω(max{f(n), g(n)}). [Note: Proving this will help you understand…
- Assume that you were given N cents (N is an integer) and you were asked to break up the N cents into coins consisting of 1 cent, 2 cents and 5 cents. Write a dynamicprogramming based recursive algorithm, which returns the smallest (optimal) number of coins needed to solve this problem. For example, if your algorithm is called A, and N = 13, then A(N) = A(13) returns 4, since 5+5+2+1 = 13 used the smallest (optimal) number of coins. In contrast, 5+5+1+1+1 is not an optimal answer.Big-O Complexity Remember, Big-O time complexity gives us an idea of the growth rate of a function. In other words, for a large input size N, as N increases, in what order of magnitude is the volume of statements executed expected to increase? A) Rearrange the following functions in increasing order of their Big-O Complexity B) Also Find the time if n=5: 4n2 log3 n 20n n2.5 n0.00000001 log n! nn 2n 2n+1 22n 3n n log n 100 n2/3 log [(log n)2 ] N! (n−1)!Question No 2: Compare the complexities of the following two pieces of code using the cyclomatic complexity metrics. What conclusions can you draw about relative complexity of the code? Code A Code B int i = 1; int j = 1; while (i <= 5) { int i = 2; Playacard(i); if (j =i); if (playhaswon(i)) j++; else i++; i++; else System.out.println(j); i--; i++; }
- What is the leading term for the following expressions and specify the lowest Big O Complexity for each algorithm?Complete the following analysis of an algorithm such that the time complexity, T(n), of the given algorithm is determined. The analysis uses instruction count (i.e. number of times the statement is executed) as the measure for the analysis. The instruction count of each instruction in the algorithm is indicated at the right of the statement after the //. You should replace each of the question marks(?) by the corresponding instruction count in order to complete the analysis. The total instruction count is obtained by adding all the instruction counts. Note that in this exercise, the linear search algorithm is analyzed in a case where the item to be searched is assumed not found in the array (i.e.Worst Case Analysis) int linearSearch(item, array) { // worst case instruction countint index = 0; // ?while (index < array.length) { // ?if (array[index] == item) { // ?return index; // ?}index++; // ?}return -1; // ?} // ------// T(n) = ? where n is the array sizeComplete the following analysis of an algorithm such that the time complexity, T(n), of the given algorithm is determined. The analysis uses instruction count (i.e. number of times the statement is executed) as the measure for the analysis. The instruction count of each instruction in the algorithm is indicated at the right of the statement after the //. You should replace each of the question marks(?) by the corresponding instruction count in order to complete the analysis. The total instruction count is obtained by adding all the instruction counts. Note that in this exercise, the binary search algorithm is analyzed in a case where the item to be searched is assumed to be NOT found in the sorted array (i.e. Worst Case Analysis). // Worst Case Analysis (e.g., item not in array)int binarySearch(item, sortedArray) { // Instruction Countint low = 0; // 1int high = sortedArray.length - 1; // 1int mid = 0; // 1while (low <= high) { // ≈ log(n) + 1mid = (low + high) / 2; // ≈ log(n)if…
- Modeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…Due to its poor design, the standard Turing architecture loses a considerable amount of energy. Explain what it is about the Turing computer that creates this challenge, and then present an example of a non-Turing computer, such as Steve Furber's SpiNNaker, and explain how it circumvents the issue described in the preceding statement using that example.I need help understanding the concept of complexity theory in algorithms. Can you explain the different complexity classes, such as P, NP, and NP-complete, and their significance?