numerous bouquet combinations, including two 5-rose bouquets (total profit of $70), and a 4-rose bouquet with three 2-rose bouquets (total profit of $75). Provide two different algorithms for calculatin
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|
Roses |
1 |
2 |
3 |
4 |
5 |
|
Profit |
$5 |
$15 |
$24 |
$30 |
$35 |
For each positive integer n, let f(n) be the maximum profit that Flora can make with n roses.
For example, if n = 10, Flora can make numerous bouquet combinations, including two 5-rose bouquets (total profit of $70), and a 4-rose bouquet with three 2-rose bouquets (total profit of $75).
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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 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.we consider a problem where we are given a set of coins andour task is to form a sum of money n using the coins. The values of the coins arecoins = {c1, c2,..., ck}, and each coin can be used as many times we want. Whatis the minimum number of coins needed?For example, if the coins are the euro coins (in cents){1,2,5,10,20,50,100,200}and n = 520, we need at least four coins. The optimal solution is to select coins200+200+100+20 whose sum is 520.
- Let n be a natural number. Let's take the following 2 player game. We have .n matches. Player 1 takes one or two matches from the pile, then as long as there are still matches, player 2 takes one or two matches, and so on, alternating until there are no more matches left. The player who takes the last match loses. The losing player gets a benefit of -1 and the other player gets a benefit of 1. Represent this game for n=2 ,n=3 and n=4 and say how many strategies each player has.You are given an array A[1..n] of positive numbers where Ai] is the stock price on day i. You are allowed to buy the stock once and sell it at some point later. For each day you own the stock you pay S1 fee. Design a divide-and-conquer algorithm that will return a pair (i,j) such that buying the stock on day i and selling it on day j will maximize your gain The complexity of the algorithm has to be OSuppose there are a set of n precincts P1 . . . Pn with m voters in each precinct. Each voter supportsone of two possible political parties. (In this country, only 2 political parties exist).We are required to carve the precincts into two electoral districts which each have exactly n/2 precincts.We know exactly how many voters in each precinct support each party. The goal is to strategicallyassign the precincts to the two districts.Describe an algorithm which determines whether it is possible to define the two districts insuch a way that the same political party holds the majority in both districts. Assume n is even.Example: Consider n = 4, with the voters distributed as in the table.In this case. By grouping precincts 1 and 3 into a district and precincts 2 and 4 into a district, theRepulsive party will have a majority in both districts. Though, by total count, the Repulsive party onlyhas a tiny advantage over the Demonic party (205 to 195).
- We examine a problem in which we are handed a collection of coins and are tasked with forming a sum of money n out of the coins. The currency numbers are coins = c1, c2,..., ck, and each coin can be used as many times as we want. What is the bare amount of money required?If the coins are the euro coins (in euros) 1,2,5,10,20,50,100,200 and n = 520, we need at least four coins. The best option is to choose coins with sums of 200+200+100+20.The algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a We want to estimate its worst case running time using the big-Oh notation. Answer the following questions: a. Let x be a integer stored on n bits. How many bits will you need to store x/2? b. We note that if a ≥ b, then a mod b < a/2. Assume the values of the input integers a and b are encoded on n bits. How many bits will be used to store the values of a and b at the next iteration of the While loop? c. Deduce from this observation, the maximal number iterations of the While loop the algorithm will do.For any even integer n, it is always possible to find a pair ofintegers m and k such that n = m × 2k, where m is the smallest integer.1. Write an algorithm that finds a factorization of any even integer n as stated above.For instance, we have the following two factorizations:48 = 3 × 24 instead of 48 = 12 × 2252 = 13 × 22 instead of 52 = 26 × 2.2. Analyze the time of your algorithm by computing the number of its multiplications. Show
- You are going to purchase items from a store that can carry a maximal weight of 'w' into your knapsack. There are 5 items in store available and each items weight are Wi and the worth of these items are Pi dollars. What items should you take and how using Knapsack algorithm?The weight of knapsack depend upon the sum of your two digits of age for example suppose your age is 25 then sum of age becomes 7. The list of items and their respective weight withprice are given in table. Items Weight Price A Total count of your first name Total count of your first namedivide by 2 B 3 Your roll no mod 5 C Second digit of your roll no 5 D Total count of your last name Total count of your first namedivide by 2 E Your roll no mod 3 S Sum of roll no mod 13There are n people who want to carpool during m days. On day i, some subset ???? of people want to carpool, and the driver di must be selected from si . Each person j has a limited number of days fj they are willing to drive. Give an algorithm to find a driver assignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use network flow. For example, for the following input with n = 3 and m = 3, the algorithm could assign Tom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Driving Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0Assume 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 dynamic programming-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. Draw the recursion tree for the algorithm where N = 7. Derive the complexity bound of the algorithm. Do not need to prove the complexity bound formally, just derive it by analyzing each component in your algorithm.
