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COMPLEXITY NOTATIONS. If function f(n) belongs to O, Θ, Ω notations, indicate by labeling along the side as correct; otherwise, label it as wrong.
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- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.
- Show that for f(n) = 2n2 and g(n) = 20n + 3n2 , f(n) is θ(g(n)). How many ways can it be shown? Also Show that for g(n) = 10n2and f(n) = n! + 3 , f(n) is Ω(g(n)). How many ways can it be shown? Discuss with the instructor.2. Prove from the definition that 2n2 + 100n log n + 1000 = O(n2) 3. For what values of n is 50 n lg n greater than 0.5 n2? Why do we say that 0.5 n2 is asymptotically larger, if 50 n lg n is larger for many values? (Hint: you may need to graph the functions or play around with your calculator.)Let n be the product of two large primes. Alice wants to send a message m to Bob, where gcd(m, n) = 1. Alice and Bob choose integers a and b that are relatively prime to Φ(n). Alice computes c ≡ ma (mod n) and sends c to Bob. Bob computes d ≡ cb (mod n) and sends d back to Alice. Since Alice knows a, she finds a1 such that aa1 ≡ 1 (mod Φ(n)). Then she computes e ≡ da1 (mod n) and sends e to Bob. Explain what Bob must do to obtain m and show that this works.
- Subject : calculas Show that: ¬q 1) p→¬q 2) (p∧r)∨s 3) s→(t∨u) 4) ¬t∧¬u where ¬ is denied.Give big-O estimates for the factorial function and the logarithm of the factorial function, wherethe factorial function f(n) = n! is defined byn! = 1 · 2·3· ... ·nwhenever n is a positive integer, and O! = 1 . For example,1 ! = 1, 2! = 1 · 2=2, 3 ! =1· 2·3 = 6, 4! = 1 · 2·3· 4=24.Note that the function n! grows rapidly. For instance,20! = 2,432,902,008,176,640,000.not handwritten 1. Consider f(n) = 2n2 - 4n. Is it O(n2)? 2. Consider f(n) = 4n2 + 4n. Is it O(n2)?
- PLEASE HELP ME. kindly show all your work 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}. 3. Let n ∈ N \ {0}. Describe the largest set of values n for which you think 2n < n!. Use induction toprove that your description is correct.Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! ∈ O(n log2 n). Thank you. But please show all work and all stepsPlease help me with these question. SHow all you work. Thank you 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}).Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}Suppose that f(x) is O(g(x)). Does it follow that 2f(x) is O(2g(x))? Prove your answer.