Given weights and values of N items, we need to put these items in a knapsack of capacity W to get the maximum total value in the knapsack.Note: Unlike 0/1 knapsack, you are allowed to break the item here.
Question
Given weights and values of N items, we need to put these items in a knapsack of capacity W to get the maximum total value in the knapsack.Note: Unlike 0/1 knapsack, you are allowed to break the item here.
Solution
This problem is a classic example of the fractional knapsack problem, which is a variation of the knapsack problem. Here's a step-by-step solution:
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Create a list of objects to represent the items. Each object should contain the weight and value of an item, as well as the ratio of value to weight.
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Sort the list of objects in descending order based on their value to weight ratio.
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Initialize a variable to keep track of the total value of items in the knapsack. This will be set to 0 at the start.
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Iterate over the sorted list of objects. For each object, do the following:
- If the weight of the object is less than or equal to the remaining capacity of the knapsack, add the object to the knapsack and subtract its weight from the remaining capacity. Add its value to the total value.
- If the weight of the object is greater than the remaining capacity, calculate the fraction of the object that can be added to the knapsack (remaining capacity divided by object's weight). Add this fraction of the object's value to the total value and set the remaining capacity to 0, since the knapsack is now full.
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Once the iteration is complete, the total value variable will hold the maximum total value that can be achieved.
This solution works because it always selects the item with the highest value to weight ratio that fits in the remaining capacity of the knapsack, ensuring that the total value is maximized.
Similar Questions
Question1Max. score: 100.000/1 Knapsack ProblemYou are given weights and values of N items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack. Note that we have only one quantity of each item.In other words, given two integer arrays val[0..N-1] and wt[0..N-1] which represent values and weights associated with N items respectively. Also given an integer W which represents knapsack capacity, find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to W. You cannot break an item, either pick the complete item, or don’t pick it (0-1 property).Input:First Line Containf the size of the itemsSecond line contains Profit valuesThird Line Contains Weights Fourth line contains Capacity Output: Output=Max ProfitExample 1:Input:31 2 34 5 1 3Output: 3
Given items as {value,weight} pairs {{40,20},{30,10},{20,5}}. The capacity of knapsack=20. Find the maximum value output assuming items to be divisible. Options 40 80 100 None of the mentioned
ou are given a knapsack that can carry a maximum weight of 60. There are 4 items with weights {20, 30, 40, 70} and values {70, 80, 90, 200}. What is the maximum value of the items you can carry using the knapsack?
Which of the following problems is equivalent to the 0-1 Knapsack problem?a.You are given a bag that can carry a maximum weight of W. You are given N items which have a weight of {w1, w2, w3,…., wn} and a value of {v1, v2, v3,…., vn}. You can break the items into smaller pieces. Choose the items in such a way that you get the maximum valueb.You are studying for an exam and you have to study N questions. The questions take {t1, t2, t3,…., tn} time(in hours) and carry {m1, m2, m3,…., mn} marks. You can study for a maximum of T hours. You can either study a question or leave it. Choose the questions in such a way that your score is maximizedc.You are given a suitcase that can carry a maximum weight of 15kg. You are given 4 items which have a weight of {10, 20, 15,40} and a value of {1, 2, 3,4}. You can break the items into smaller pieces. Choose the items in such a way that you get the maximum valued.You are given infinite coins of denominations {v1, v2, v3,….., vn} and a sum S. You have to find the minimum number of coins required to get the sum S
Solve the following instance of greedy knapsack problem where n=4, m=10, p = (40, 42, 25,12) and w = (4, 7, 5, 3)
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