Programming Assignment 3 Greedy Algorithms Solution

Introduction




In this programming assignment, you will be practicing implementing greedy solutions. As usual, in some problems you just need to implement an algorithm covered in the lectures, while for some others your goal will be to first design an algorithm and then to implement it. Thus, you will be practicing designing an algorithm, proving that it is correct, and implementing it.




Recall that starting from this programming assignment, the grader will show you only the first few tests.




Learning Outcomes




Upon completing this programming assignment you will be able to:




Apply greedy strategy to solve various computational problems. This will usually require you to design an algorithm that repeatedly makes the most profitable move to construct a solution. You will then need to show that the moves of your algorithm are safe, meaning that they are consistent with at least one optimal solution.



Design and implement an efficient greedy algorithm for the following problems:



changing money with a minimum number of coins;



maximizing the total value of a loot;



minimizing the number of tank refills;



maximizing revenue in online ad placement;



minimizing work while collecting signatures;



maximizing the number of prize places in a competition;



finally, maximizing your salary!



Passing Criteria: 3 out of 7




Passing this programming assignment requires passing at least 3 out of 7 programming challenges from this assignment. In turn, passing a programming challenge requires implementing a solution that passes all the tests for this problem in the grader and does so under the time and memory limits specified in the problem statement.








































1
Contents




1
Money Change
3
2
Maximum Value of the Loot
4
3
Car Fueling
5
4
Maximum Advertisement Revenue
7
5
Collecting Signatures
8
6
Maximum Number of Prizes
10
7
Maximum Salary
11













































































































































2
Money Change



Problem Introduction










In this problem, you will design and implement an elementary greedy algorithm used by cashiers all over the world millions of times per day.










Problem Description




Task. The goal in this problem is to find the minimum number of coins needed to change the input value (an integer) into coins with denominations 1, 5, and 10.




Input Format. The input consists of a single integer .




Constraints. 1 ≤ ≤ 103.




Output Format. Output the minimum number of coins with denominations 1, 5, 10 that changes .




Sample 1.




Input:




2




Output:







2




2=1+1.




Sample 2.




Input:




28




Output:







6




28=10+10+5+1+1+1.




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3
Maximum Value of the Loot



Problem Introduction










A thief finds much more loot than his bag can fit. Help him to find the most valuable combination of items assuming that any fraction of a loot item can be put into his bag.










Problem Description




Task. The goal of this code problem is to implement an algorithm for the fractional knapsack problem.




Input Format. The first line of the input contains the number of items and the capacity of a knapsack. The next lines define the values and weights of the items. The -th line contains integers and —the value and the weight of -th item, respectively.




Constraints. 1 ≤ ≤ 103, 0 ≤ ≤ 2 · 106; 0 ≤ ≤ 2 · 106, 0 < ≤ 2 · 106 for all 1 ≤ ≤ . All the numbers are integers.




Output Format. Output the maximal value of fractions of items that fit into the knapsack. The absolute value of the difference between the answer of your program and the optimal value should be at most 10−3. To ensure this, output your answer with at least four digits after the decimal point (otherwise your answer, while being computed correctly, can turn out to be wrong because of rounding issues).




Sample 1.




Input:




50



60 20




100 50




120 30




Output:







180.0000




To achieve the value 180, we take the first item and the third item into the bag.




Sample 2.




Input:




10



500 30




Output:







166.6667




Here, we just take one third of the only available item.




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4
Car Fueling



Problem Introduction




You are going to travel to another city that is located miles away from your home city. Your can can travel at most miles on a full tank and you start with a full tank. Along your way, there are gas stations at distances stop1, stop2, . . . , stop from your home city. What is the minimum number of refills needed?




Problem Description




Input Format. The first line contains an integer . The second line contains an integer . The third line specifies an integer . Finally, the last line contains integers stop1, stop2, . . . , stop .




Input Format. Assuming that the distance between the cities is miles, a car can travel at most miles on a full tank, and there are gas stations at distances stop1, stop2, . . . , stop along the way, output the minimum number of refills needed. Assume that the car starts with a full tank. If it is not possible to reach the destination, output −1.




Constraints. 1 ≤ ≤ 105. 1 ≤ ≤ 400. 1 ≤ ≤ 300. 0 < stop1 < stop2 < · · · < stop < .




Sample 1.




Input:




950




400




4




200 375 550 750




Output:







2




The distance between the cities is 950, the car can travel at most 400 miles on a full tank. It suffices to make two refills: at points 375 and 750. This is the minimum number of refills as with a single refill one would only be able to travel at most 800 miles.




Sample 2.




Input:




10




3




4




1259




Output:







-1




One cannot reach the gas station at point 9 as the previous gas station is too far away.











































5
Sample 3.




Input:




200




250




2




100




150




Output:







0




There is no need to refill the tank as the car starts with a full tank and can travel for 250 miles whereas the distance to the destination point is 200 miles.




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6
Maximum Advertisement Revenue



Problem Introduction




You have ads to place on a popular Internet page. For each ad, you know how much is the advertiser willing to pay for one click on this ad. You have set up slots on your page and estimated the expected number of clicks per day for each slot. Now, your goal is to distribute the ads among the slots to maximize the total revenue.







Problem Description




Task. Given two sequences 1, 2, . . . , ( is the profit per click of the -th ad) and 1, 2, . . . , ( is the average number of clicks per day of the -th slot), we need to partition them into pairs ( , ) such that the sum of their products is maximized.




Input Format. The first
line
contains
an integer ,
the second one contains a
sequence of integers
1, 2, . . . , , the third one contains a sequence of integers 1, 2, . . . , .


Constraints. 1 ≤ ≤ 103; −105 ≤ ,
≤ 105 for all 1 ≤ ≤ .


1, 2,..., .










the


∑
is a permutation of
Output Format. Output
maximum value of
, where 1, 2, . . . ,
=1







Sample 1.




Input:




1




23




39




Output:







897




897 = 23 · 39.




Sample 2.




Input:




3




13-5




-241




Output:







23




23=3·4+1·1+(−5)·(−2).




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7
Collecting Signatures



Problem Introduction




You are responsible for collecting signatures from all tenants of a certain build-ing. For each tenant, you know a period of time when he or she is at home. You would like to collect all signatures by visiting the building as few times as possible.




The mathematical model for this problem is the following. You are given a set of segments on a line and your goal is to mark as few points on a line as possible so that each segment contains at least one marked point.




Problem Description




Task. Given a set of segments {[ 0, 0], [ 1, 1], . . . , [ −1, −1]} with integer coordinates on a line, find the minimum number of points such that each segment contains at least one point. That is, find a set of integers of the minimum size such that for any segment [ , ] there is a point ∈ such that ≤ ≤ .




Input Format. The first line of the input contains the number of segments. Each of the following lines contains two integers and (separated by a space) defining the coordinates of endpoints of the -th segment.




Constraints. 1 ≤ ≤ 100; 0 ≤ ≤ ≤ 109 for all 0 ≤ < .




Output Format. Output the minimum number of points on the first line and the integer coordinates of points (separated by spaces) on the second line. You can output the points in any order. If there are many such sets of points, you can output any set. (It is not difficult to see that there always exist a set of points of the minimum size such that all the coordinates of the points are integers.)




Sample 1.




Input:




3




3



2 5




3 6




Output:







1




3




In this sample, we have three segments: [1, 3], [2, 5], [3, 6] (of length 2, 3, 3 respectively). All of them




contain the point with coordinate 3: 1 ≤ 3 ≤ 3, 2 ≤ 3 ≤ 5, 3 ≤ 3 ≤ 6.

















































8
Sample 2.




Input:




4




7



1 3




2 5




5 6




Output:







2




3 6




The second and the third segments contain the point with coordinate 3 while the first and the fourth segments contain the point with coordinate 6. All the four segments cannot be covered by a single point, since the segments [1, 3] and [5, 6] are disjoint.




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9
Maximum Number of Prizes



Problem Introduction




You are organizing a funny competition for children. As a prize fund you have candies. You would like to use these candies for top places in a competition with a natural restriction that a higher place gets a larger number of candies. To make as many children happy as possible, you are going to find the largest value of for which it is possible.







Problem Description




Task. The goal of this problem is to represent a given positive integer as a sum of as many pairwise distinct positive integers as possible. That is, to find the maximum such that can be written as 1 + 2 + · · · + where 1, . . . , are positive integers and ̸= for all 1 ≤ < ≤ .




Input Format. The input consists of a single integer .




Constraints. 1 ≤ ≤ 109.




Output Format. In the first line, output the maximum number such that can be represented as a sum of pairwise distinct positive integers. In the second line, output pairwise distinct positive integers that sum up to (if there are many such representations, output any of them).




Sample 1.




Input:




6




Output:







3




1 2 3




Sample 2.




Input:




8




Output:







3




1 2 5




Sample 3.




Input:




2




Output:







1




2







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10
7 Maximum Salary




Problem Introduction







As the last question of a successful interview, your boss gives you a few pieces of paper with numbers on it and asks you to compose a largest number from these numbers. The resulting number is going to be your salary, so you are very much interested in maximizing this number. How can you do this?




In the lectures, we considered the following algorithm for composing the largest number out of the given single-digit numbers.







LargestNumber(Digits):




answer ← empty string




while Digits is not empty:

maxDigit ← −∞




for digit in Digits:

if digit ≥ maxDigit:




maxDigit ← digit

append maxDigit to answer




remove maxDigit from Digits




return answer




Unfortunately, this algorithm works only in case the input consists of single-digit numbers. For example, for an input consisting of two integers 23 and 3 (23 is not a single-digit number!) it returns 233, while the largest number is in fact 323. In other words, using the largest number from the input as the first number is not a safe move.




Your goal in this problem is to tweak the above algorithm so that it works not only for single-digit numbers, but for arbitrary positive integers.




Problem Description




Task. Compose the largest number out of a set of integers.




Input Format. The first line of the input contains an integer . The second line contains integers

1, 2,..., .




Constraints. 1 ≤ ≤ 100; 1 ≤ ≤ 103 for all 1 ≤ ≤ .




Output Format. Output the largest number that can be composed out of 1, 2, . . . , .




Sample 1.




Input:




2




21 2




Output:







221




Note that in this case the above algorithm also returns an incorrect answer 212.




























11
Sample 2.




Input:




5




94619




Output:







99641




In this case, the input consists of single-digit numbers only, so the algorithm above computes the right answer.




Sample 3.




Input:




3




23 39 92




Output:







923923




As a coincidence, for this input the above algorithm produces the right result, though the input does not have any single-digit numbers.




What To Do




Interestingly, for solving this problem, all you need to do is to replace the check digit ≥ maxDigit with a call




IsGreaterOrEqual(digit, maxDigit) for an appropriately implemented function IsGreaterOrEqual.

For example, IsGreaterOrEqual(2, 21) should return True.




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