Suppose three algorithms A, B, and C, can be implemented to perform a task. The
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- Suppose three algorithms A, B, and C, can be implemented to perform a task. The algorithms have the following time complexities: A~0(n), B~0(n²), C~ 0(log2(n)). If the task runs an average input of size 100, rank the algorithms from least time- complex to most-time complex. Explain your choice.2. For a problem we have come up with three algorithms: A, B, and C. Running time of Algorithm A is O(n¹000), Algorithm B runs in 0(2¹) and Algorithm C runs in O(n!). How do these algorithms compare in terms of speed, for large input? Explain why.A graph is a collection of vertices and edges G(V, E). A weighted graph has weights (numbers, etc.) on every edge. A multigraph can have more than one edges between any vertices. Explain why a person should use a weighted graph instead of a multigraph. Give examples. An adjacency matrix might be a better choice for speeding up a program, however, it consumes huge memory for large graphs. How this situation can be improved? What programming constructs better suit graph representation? Explain with example
- Let us assume that we have two algorithms A and B. Algorithm A has run time given by (10n)2 and algorithm B has run time given by (Sin2n + Cos2n +1)n where n is the input size. Also assume that both algorithms are run on the same computer. Find the smallest value of n for which the runtime of algorithm A is less than the run time of algorithm B.Given two sorted arrays A and B, design a linear (O(IA|+|B|)) time algorithm for computing the set C containing elements that are in A or B, but not in both. That is, C = (AU B) \ (AN B). You can assume that elements in A have different values and elements in B also have different values. Please state the steps of your algorithm clearly, prove that it is correct, and analyze its running time. Pls give the code in C++, or very clear steps of the algorithmGiven is a strictly increasing function, f(x). Strictly increasing meaning: f(x)< f(x+1). (Refer to the example graph of functions for a visualization.) Now, define an algorithm that finds the smallest positive integer, n, at which the function, f(n), becomes positive. The things left to do is to: Describe the algorithm you came up with and make it O(log n).
- Consider a function f: N → N that represents the amount of work done by some algorithm as follow: f(n) = {(1 if n is oddn if n is even)┤ A. Prove or disprove. f(n) is O(n).Suppose that the running time of an algorithm A is T(n) = n2. If the time it takes for algorithm A to finish on an input of size n 10 is 100ms, what will be the time that it will take for A to finish on an input of size 20? Assume that the algorithm runs on the same machine.Consider two algorithms for the same problem: • Algorithm A, which runs in O(n) and produces a correct answer with probability 0.7, and a wrong answer with probability 0.3. • Algorithm B, which runs in O(n log n) and produces a correct answer with probability 0.99, and a wrong answer with probability 0.01. Which of the two algorithms should you use to build a more asymptotically (in n) efficient algorithm with probability 0.99 of producing a correct answer? Justify. Assume that you can always check if an answer is correct in constant time.
- Assume that each of the expressions below gives the processing time T(n) spent by an algorithm for solving a problem of size n. Select the dominant term(s) having the steepest increase in n and specify the lowest Big-Oh complexity of each algorithm. For example, the dominant term in 0.1n + 10n4 is 10n4 and it is O(n4). Expression Dominant term(s) O(. . .) 5 + 0.001n3 + 0.025n 500n + 100n1.5 + 50n log10 n 0.3n + 5n1.5 + 2.5 · n1.75 n2 log2 n + n(log2 n)2 n log3 n + n log2 n 100n + 0.01n2 0.01n + 100n2 2n + n0.5 + 0.5n1.25 0.01n log2 n + n(log2 n)2 100n log3 n + n3 + 100nConsider a function f: N → N that represents the amount of work done by some algorithm as follow: f(n) = {(1 if n is oddn if n is even)┤ Prove or disprove. f(n) is O(n). Please show proof or disproofFor a given problem with inputs of size n, algorithms running time,one of the algorithm is o(n), one o(nlogn) and one o(n2). Some measured running times of these algorithm are given below: Identify which algorithm is which and explain the observed running Times which algorithm would you select for different value of n?