if n< d T(n)= - laT(n/b)+ f(n) if n2 d 1. if f(n) is O(n" 2. if f(n) is O(n™*“ log* n), then T(n) is O(n*.“ log**" n)| 3. if f(n) is 2(n*. *“), then T(n) is O(f (n)), provided af(n/b) S F (n) for some & <1. log, a-e ), then T(n) is O(no") log, a+e Which of the three cases above applies to an algorithm, the running time of which is described using the following recurrence relation: T(n) = 3 T(n/3) + n^/a Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2" Type N/A if the Master theorem is not applicable to the recurrence relation Answer:

if n< d T(n)= - laT(n/b)+ f(n) if n2 d 1. if f(n) is O(n" 2. if f(n) is O(n™“ log n), then T(n) is O(n*.“ log**" n)| 3. if f(n) is 2(n. “), then T(n) is O(f (n)), provided af(n/b) S F (n) for some & <1. log, a-e ), then T(n) is O(no") log, a+e Which of the three cases above applies to an algorithm, the running time of which is described using the following recurrence relation: T(n) = 3 T(n/3) + n^/a Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2" Type N/A if the Master theorem is not applicable to the recurrence relation Answer:

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter5: Control Structures Ii (repetition)

Section: Chapter Questions

Problem 7PE

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Heuristics Prove or disprove: If h1(n), ..., hk(n) are admissible, so is h(n) = h1(n) + ... + hk(n)
Find f(1), f(2), f(3), f(4), and f(5) if f(n) is defined re-cursively by f(0) = 3 and for n = 0, 1, 2, ... a) f(n + 1) = −2f(n).b) f(n + 1) = 3f(n) + 7.
Correct answer will be upvoted else downvoted. Computer science. Allow us to signify by d(n) the amount of all divisors of the number n, for example d(n)=∑k|nk. For instance, d(1)=1, d(4)=1+2+4=7, d(6)=1+2+3+6=12. For a given number c, track down the base n to such an extent that d(n)=c. Input The principal line contains one integer t (1≤t≤104). Then, at that point, t experiments follow. Each experiment is characterized by one integer c (1≤c≤107). Output For each experiment, output: "- 1" in case there is no such n that d(n)=c; n, in any case.
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.
Proof the following statement by definition of big O and big theta: If f(n) = O(g(n)), then f(n) + g(n) = Θ(g(n)).
Describe the worst case running time of the following pseudocode function in Big-Oh notation in terms of the variable n. Show your work to get full credit! void func(int n, int x, int y) { for (int i = 0; i < n; ++i) { if (x > y) { for (int j = 0; j < n; ++j) System.out.println(”j = ” + j); for (int k = 0; k < n * n; ++k) System.out.println(”k = ” + k); } else System.out.println(”i = ” + i); } }
Consider the functionf :: Int -> Intf n = if n==0then 0else 1 + (f(n-1))Use induction to show that the function f returns the value of n for all possible inputs n ≥0.Here are the steps:1. Verify that f 0 returns 0 to show the base case.2. Show that if f(n-1) returns n −1 then f n returns n.3. Since you have shown the base case and the induction step, you can confidentlystate that the function works for all possible nonnegative input values.Hint: To show that ”if f(n-1) returns n −1 then f n returns n” is valid you need toassume that f(n-1) returns n −1 and then argue that it must follow that f n returnsn. Use the definition of the function and just a little bit of algebra.Criteria for Success: You have clearly written down all three steps of the inductive proof. Your proof contains complete sentences which explain all the steps andthe algebra. I don’t want to see just a bunch of symbols on a page!
The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written using dynamic programming.
Problem1Given a value `value`, if we want to make change for `value` cents, and we have infinitesupply of each of coins = {S1, S2, .. , Sm} valued `coins`, how many ways can we make the change?The order of `coins` doesn't matter.For example, for `value` = 4 and `coins` = [1, 2, 3], there are four solutions:[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3].So output should be 4. For `value` = 10 and `coins` = [2, 5, 3, 6], there are five solutions: [2, 2, 2, 2, 2], [2, 2, 3, 3], [2, 2, 6], [2, 3, 5] and [5, 5].So the output should be 5. Time complexity: O(n * m) where n is the `value` and m is the number of `coins`Space complexity: O(n)""" def count(coins, value): """ Find number of combination of `coins` that adds upp to `value` Keyword arguments: coins -- int[] value -- int """ # initialize dp array and set base case as 1 dp_array = [1] + [0] * value.. (+.
Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω( g(n) ) if and only if there exists n0, such that:forall n ≥ n0, there exists c > 0 such that,f(n) ≥ c g(n)