Question

The Knapsack problem is an optimization problem that asks to fill a knapsack with maximum possible value. Using greedy paradigm, we can solve this problem easily. Your task is the following:

(a) Write the pseudo-code of a greedy solution for knapsack problem.

(b) Give is the time complexity of your solution in part (a).

(c) Implement part (a) in C programming language.

Answer 1

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.

In the 0-1 Knapsack problem, we are not allowed to break items. We either take the whole item or don’t take it.

Input:
  Items as (value, weight) pairs
  arr[] = {{60, 10}, {100, 20}, {120, 30}}
  Knapsack Capacity, W = 50;
Output:
  Maximum possible value = 220
  by taking items of weight 20 and 30 kg 

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

In Fractional Knapsack, we can break items for maximizing the total value of knapsack. This problem in which we can break an item is also called the fractional knapsack problem.

Input : 
   Same as above
Output :
   Maximum possible value = 240
   By taking full items of 10 kg, 20 kg and 
   2/3rd of last item of 30 kg

A brute-force solution would be to try all possible subset with all different fraction but that will be too much time taking.

An efficient solution is to use Greedy approach. The basic idea of the greedy approach is to calculate the ratio value/weight for each item and sort the item on basis of this ratio. Then take the item with the highest ratio and add them until we can’t add the next item as a whole and at the end add the next item as much as we can. Which will always be the optimal solution to this problem.

A simple code with our own comparison function can be written as follows, please see sort function more closely, the third argument to sort function is our comparison function which sorts the item according to value/weight ratio in non-decreasing order.
After sorting we need to loop over these items and add them in our knapsack satisfying above-mentioned criteria.

• C++
• Java
• Python3

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// C/C++ program to solve fractional Knapsack Problem #include <bits/stdc++.h>    using namespace std;    // Structure for an item which stores weight and corresponding // value of Item struct Item {     int value, weight;        // Constructor     Item(int value, int weight) : value(value), weight(weight)     {} };    // Comparison function to sort Item according to val/weight ratio bool cmp(struct Item a, struct Item b) {     double r1 = (double)a.value / a.weight;     double r2 = (double)b.value / b.weight;     return r1 > r2; }    // Main greedy function to solve problem double fractionalKnapsack(int W, struct Item arr[], int n) {     //    sorting Item on basis of ratio     sort(arr, arr + n, cmp);        //    Uncomment to see new order of Items with their ratio     /*     for (int i = 0; i < n; i++)     {         cout << arr[i].value << " " << arr[i].weight << " : "              << ((double)arr[i].value / arr[i].weight) << endl;     }     */        int curWeight = 0; // Current weight in knapsack     double finalvalue = 0.0; // Result (value in Knapsack)        // Looping through all Items     for (int i = 0; i < n; i++)     {         // If adding Item won't overflow, add it completely         if (curWeight + arr[i].weight <= W)         {             curWeight += arr[i].weight;             finalvalue += arr[i].value;         }            // If we can't add current Item, add fractional part of it         else         {             int remain = W - curWeight;             finalvalue += arr[i].value * ((double) remain / arr[i].weight);             break;         }     }        // Returning final value     return finalvalue; }    // driver program to test above function int main() {     int W = 50;   //    Weight of knapsack     Item arr[] = {{60, 10}, {100, 20}, {120, 30}};        int n = sizeof(arr) / sizeof(arr[0]);        cout << "Maximum value we can obtain = "          << fractionalKnapsack(W, arr, n);     return 0; }

Output :

Maximum value in Knapsack = 240

As main time taking step is sorting, the whole problem can be solved in O(n log n) only.