Consider the following functions as time complexities of some algorithms. (a) First write the worst-case runtime of each algorithm in Big-O notation. (b) Then arrange functions from low to high (Consider their Big-O notations as you arrange them; as n grows, the function that grows slower should come sooner than the one that grows faster). (c) Arbitrary, select TWO of arrangements in part (b) and for each one of them justify why you have chosen such an ordering for the corresponding function. Provide formal proof. For example if fl<=f2<= f3<= f4<=f5 you may choose to formally justify why (1) fl<=f2 and why (2) f4<=f5. fl(n)=n°,

Consider a sequence of integers defined by the following recurrence: f(0) = 0, f(1) = 1, and f(i) = f(i – 1) + f([i/2]) for i> 2. We would like to compute f(n) using DP. If we use bottom-up approach, what is the running time? Explain how you obtained your answer. (Assume every basic operation such as plus takes O(1) time as usual.) Give a pseudo-code for a top-down approach with memoization.

Please provide a correct answer with a full explanation by yourself. Don't copy. Thank you.

Find the running time for each of the following algorithms. Show work by finding a table of values for each while loop, writing the summations, then solving. Be sure to show work on both the upper bound and lower bound, justify the split, and check that the bounds differ by only a constant factor. Use asymptotic notation to write the answer. c) Func4(n) 1 2 3 4 567∞∞ s = 0; for i 1 to 5n do 8 j← 3i; while (j < i3) do s+ s + i - j; j+5 x j; end end 8 return (s);

Given two sequences X and Y, we want to find Z i.e., the longest common subsequence (LCS) of both X and Y. Because we know that subsequences of interest are of length >= a and <= b, we threshold the LCS definition to |Z| in [a, b] (where |Z| is the sequence length). In python state the recursive formula for this problem. Implement the algorithm with bottom-up approach dynamic programming and backtracking. Create an iterative function that prints the longest common subsequence(LCS).

4. Give an analysis of the running time (Big-Oh will do). sum = 0; for( i = 0; i < n; ++i ) for( j = 0; j < n * n; ++j ) ++sum;

PLEASE explain how to solve this STEP BY STEP

Find the running time for each of the following algorithms. Show work by finding a table of values for each while loop, writing the summations, then solving. Be sure to show work on both the upper bound and lower bound, justify the split, and check that the bounds differ by only a constant factor. Use asymptotic notation to write the answer. b) Func3(n) 1 2 3 4 5 6 7 8 9 10 11 S← 0; i+n; while (i 8) do s+ s + i - j; j← lj/9]; end i + 4 x i; end return (s);

Give the asymptotic complexity of each of the following functions in simplest terms and then order the functions by asymptotic dominance. That is, produce a permutation, f1(n), f2(n), such that fi(n) = O(fi+1(n)). Note if any two functions are asymptotically equivalent, i.e., if fi (n) = O(fi+1(n)). Note: 'Ig x' is a simpler notation for 'log2 x'. fe(n)= 4 lg (n2+ 2n) + 6no.7 f(n) = n5 + 1010 lg (n+100) fe(n)= 3n lg (n2 + 2) + n2

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