In Duolingo, it is the case that whenever the streak - number of days it is used in a row - reaches a number n divisible by 10, one obtains n/10 so called lingots. There is a possiblility to buy streak freezes, each costing 10 lingots. How many days in a row, starting from day zero, one needs to keep up the streak without using a streak freeze in order to gain so many lingots that one can only use streak freezes from that day on?

In Duolingo, it is the case that whenever the streak - number of days it is used in a row - reaches a number n divisible by 10, one obtains n/10 so called lingots. There is a possiblility to buy streak freezes, each costing 10 lingots. How many days in a row, starting from day zero, one needs to keep up the streak without using a streak freeze in order to gain so many lingots that one can only use streak freezes from that day on?

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 18SA

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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.
Imagine there are N teams competing in a tournament, and that each team plays each of the other teams once. If a tournament were to take place, it should be demonstrated (using an example) that every team would lose to at least one other team in the tournament.
There is an upcoming football tournament, and the n participating teams are labelled from 1 to n. Each pair of teams will play against each other exactly once. Thus, a total of [n(n-1)/2] matches will be held, and each team will compete in n − 1 of these matches. There are only two possible outcomes of a match: 1. The match ends in a draw, in which case both teams will get 1 point. 2. One team wins the match, in which case the winning team gets 3 points and the losing team gets 0 points. Design an algorithm which runs in O(n2 ) time and provides a list of results in all [n(n-1)/2] matches which: (a) ensures that all n teams finish with the same points total, and (b) includes the fewest drawn matches among all lists satisfying (a). Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English list of results mean Any combination of wins, losses and draws. You may wish to view this as a mapping from the set of distinct…
A country has coins with k denominations 1 = d1 < d2 < ... < dk, and you want to make change for n cents using the smallest number of coins. For example, in the United States we have d1 = 1, d2 = 5, d3 = 10, d4 = 25, and the change for 37 cents with the smallest number of coins is 1 quarter, 1 dime, and 2 pennies, which are a total of 4 coins. To solve for the general case (change for n cents with k denominations d1 ... dk), we refer to dynamic programming to design an algorithm. 1. We will come up with sub-problems and recursive relationship for you. Let be the minimum number of coins needed to make change for n cents, then we have: Explain why the above recursive relationship is correct. [Formal proof is not required] 2. Use the relationship above to design a dynamic programming algorithm, where the inputs include the k denominations d1 ... dk and the number of cents n to make changes for, and the output is the minimum number of coins needed to make change for n. Provide…
There is a legend about a magical park with N × N trees. The trees are positioned in a square grid with Nrows (numbered from 1 to N from north to south) and N columns (numbered from 1 to N from west to east).The height (in metres) of each tree is an integer between 1 and N × N, inclusive. Magically, the height ofall trees is unique.Bunga is standing on the northmost point of the park and wants to count the number of visible trees for eachcolumn. Similarly, Lestari is standing on the westmost point of the park and wants to count the number ofvisible trees for each row. A tree X is visible if all other trees in front of the tree X are shorter than the treeX.For example, let N = 3 and the height (in metres) of the trees are as follows.6 1 87 5 32 9 4• On the first column, Bunga can see two trees, as the tree on the third row is obstructed by the othertrees.• On the second column, Bunga can see all three trees.• On the third column, Bunga can see only the tree on the first row, as the…
There is an upcoming football tournament, and the n participating teams are labelled from 1 to n. Each pair of teams will play against each other exactly once. Thus, a total of matches will be held, and each team will compete in n − 1 of these matches. There are only two possible outcomes of a match: 1. The match ends in a draw, in which case both teams will get 1 point. 2. One team wins the match, in which case the winning team gets 3 points and the losing team gets 0 points. Design an algorithm which runs in O(n2 ) time and provides a list of results in all matches which: (a) ensures that all n teams finish with the same points total, and (b) includes the fewest drawn matches among all lists satisfying (a). Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English.Please give time complexity. list of results mean Any combination of wins, losses and draws. You may wish to view this as a mapping from the set of…
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). Provide two different algorithms for calculating f(n): one using Recursion, and one using Dynamic Programming. Explain why both algorithms are guaranteed to return the correct value of f(n).
The Josephus problem is the following game: N people, numbered 1 to N, are sitting in a circle. Starting at person 1, a hot potato is passed. After M passes, the person holding the hot potato is eliminated, the circle closes ranks, and the game continues with the person who was sitting after the eliminated person picking up the hot potato. The last remaining person wins. Thus, if M = 0 and N = 5, players are eliminated in order, and player 5 wins. If M = 1 and N = 5, the order of elimination is 2, 4, 1, 5. Write a C program to solve the Josephus problem for general values of M and N. Try to make your program as efficient as possible. Make sure you dispose of cells. What is the running time of your program? If M = 1, what is the running time of your program? How is the actual speed affected by the delete routine for large values of N (N > 100,000)? ps. provide a screenshot of output, thankss
There 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 0
ProblemGiven 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) ++.