# 0/1 Knapsack Problem to print all possible solutions

Given weights and profits of N items, put these items in a knapsack of capacity W. The task is to print all possible solutions to the problem in such a way that there are no remaining items left whose weight is less than the remaining capacity of the knapsack. Also, compute the maximum profit.
Examples:

Input: Profits[] = {60, 100, 120, 50}
Weights[] = {10, 20, 30, 40}, W = 40
Output:
10: 60, 20: 100,
10: 60, 30: 120,
Maximum Profit = 180
Explanation:
Maximum profit from all the possible solutions is 180

Input: Profits[] = {60, 100, 120, 50}
Weights[] = {10, 20, 30, 40}, W = 50
Output:
10: 60, 20: 100,
10: 60, 30: 120,
20: 100, 30: 120,
Maximum Profit = 220
Explanation:
Maximum profit from all the possible solutions is 220

Approach: The idea is to make pairs for the weight and the profits of the items and then try out all permutations of the array and including the weights until their is no such item whose weight is less than the remaining capacity of the knapsack. Meanwhile after including an item increment the profit for that solution by the profit of that item.

Below is the implementation of the above approach:

## C++

 `// C++ implementation to print all` `// the possible solutions of the` `// 0/1 Knapsack problem`   `#include `   `using` `namespace` `std;`   `// Utility function to find the` `// maximum of the two elements` `int` `max(``int` `a, ``int` `b) { ` `    ``return` `(a > b) ? a : b; ` `}`   `// Function to find the all the` `// possible solutions of the ` `// 0/1 knapSack problem` `int` `knapSack(``int` `W, vector<``int``> wt, ` `            ``vector<``int``> val, ``int` `n)` `{` `    ``// Mapping weights with Profits` `    ``map<``int``, ``int``> umap;` `    `  `    ``set>> set_sol;` `    ``// Making Pairs and inserting` `    ``// into the map` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``umap.insert({ wt[i], val[i] });` `    ``}`   `    ``int` `result = INT_MIN;` `    ``int` `remaining_weight;` `    ``int` `sum = 0;` `    `  `    ``// Loop to iterate over all the ` `    ``// possible permutations of array` `    ``do` `{` `        ``sum = 0;` `        `  `        ``// Initially bag will be empty` `        ``remaining_weight = W;` `        ``vector> possible;` `        `  `        ``// Loop to fill up the bag ` `        ``// until there is no weight` `        ``// such which is less than` `        ``// remaining weight of the` `        ``// 0-1 knapSack` `        ``for` `(``int` `i = 0; i < n; i++) {` `            ``if` `(wt[i] <= remaining_weight) {`   `                ``remaining_weight -= wt[i];` `                ``auto` `itr = umap.find(wt[i]);` `                ``sum += (itr->second);` `                ``possible.push_back({itr->first,` `                     ``itr->second` `                ``});` `            ``}` `        ``}` `        ``sort(possible.begin(), possible.end());` `        ``if` `(sum > result) {` `            ``result = sum;` `        ``}` `        ``if` `(set_sol.find(possible) == ` `                        ``set_sol.end()){` `            ``for` `(``auto` `sol: possible){` `                ``cout << sol.first << ``": "` `                     ``<< sol.second << ``", "``;` `            ``}` `            ``cout << endl;` `            ``set_sol.insert(possible);` `        ``}` `        `  `    ``} ``while` `(` `        ``next_permutation(wt.begin(), ` `                           ``wt.end()));` `    ``return` `result;` `}`   `// Driver Code` `int` `main()` `{` `    ``vector<``int``> val{ 60, 100, 120 };` `    ``vector<``int``> wt{ 10, 20, 30 };` `    ``int` `W = 50;` `    ``int` `n = val.size();` `    ``int` `maximum = knapSack(W, wt, val, n);` `    ``cout << ``"Maximum Profit = "``;` `    ``cout << maximum;` `    ``return` `0;` `}`

## Java

 `// Java implementation to print all` `// the possible solutions of the` `// 0/1 Knapsack problem`   `import` `java.util.*;`   `public` `class` `Main {`   `    ``// Utility function to find the maximum of the two` `    ``// elements` `    ``static` `int` `max(``int` `a, ``int` `b) { ``return` `(a > b) ? a : b; }`   `    ``// Function to find the all the possible solutions of` `    ``// the 0/1 knapSack problem` `    ``static` `int` `knapSack(``int` `W, List wt,` `                        ``List val, ``int` `n)` `    ``{`   `        ``// Mapping weights with Profits` `        ``Map umap = ``new` `HashMap<>();`   `        ``Set > > setSol` `            ``= ``new` `HashSet<>();`   `        ``// Making Pairs and inserting into the map` `        ``for` `(``int` `i = ``0``; i < n; i++) {` `            ``umap.put(wt.get(i), val.get(i));` `        ``}`   `        ``int` `result = Integer.MIN_VALUE;` `        ``int` `remaining_weight;` `        ``int` `sum = ``0``;`   `        ``// Loop to iterate over all the possible` `        ``// permutations of array` `        ``do` `{` `            ``sum = ``0``;`   `            ``// Initially bag will be empty` `            ``remaining_weight = W;` `            ``List > possible` `                ``= ``new` `ArrayList<>();`   `            ``// Loop to fill up the bag until there is no` `            ``// weight such which is less than remaining` `            ``// weight of the 0-1 knapSack` `            ``for` `(``int` `i = ``0``; i < n; i++) {` `                ``if` `(wt.get(i) <= remaining_weight) {`   `                    ``remaining_weight -= wt.get(i);` `                    ``Integer valAtWtI = umap.get(wt.get(i));` `                    ``sum += valAtWtI;` `                    ``possible.add(` `                        ``new` `AbstractMap.SimpleEntry<>(` `                            ``wt.get(i), valAtWtI));` `                ``}` `            ``}` `            ``Collections.sort(` `                ``possible,` `                ``Comparator.comparingInt(Map.Entry::getKey));` `            ``if` `(sum > result) {` `                ``result = sum;` `            ``}` `            ``if` `(!setSol.contains(possible)) {` `                ``for` `(Map.Entry sol :` `                     ``possible) {` `                    ``System.out.print(sol.getKey() + ``": "` `                                     ``+ sol.getValue()` `                                     ``+ ``", "``);` `                ``}` `                ``System.out.println();` `                ``setSol.add(possible);` `            ``}`   `        ``} ``while` `(nextPermutation(wt));`   `        ``return` `result;` `    ``}`   `    ``// Utility function to generate the next permutation` `    ``static` `boolean` `nextPermutation(List arr)` `    ``{` `        ``int` `i = arr.size() - ``2``;` `        ``while` `(i >= ``0` `&& arr.get(i) >= arr.get(i + ``1``)) {` `            ``i--;` `        ``}` `        ``if` `(i < ``0``) {` `            ``return` `false``;` `        ``}` `        ``int` `j = arr.size() - ``1``;` `        ``while` `(arr.get(j) <= arr.get(i)) {` `            ``j--;` `        ``}` `        ``int` `temp = arr.get(i);` `        ``arr.set(i, arr.get(j));` `        ``arr.set(j, temp);`   `        ``Collections.reverse(arr.subList(i + ``1``, arr.size()));` `        ``return` `true``;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``List val` `            ``= ``new` `ArrayList<>(Arrays.asList(``60``, ``100``, ``120``));` `        ``List wt` `            ``= ``new` `ArrayList<>(Arrays.asList(``10``, ``20``, ``30``));` `        ``int` `W = ``50``;` `        ``int` `n = val.size();` `        ``int` `maximum = knapSack(W, wt, val, n);` `        ``System.out.println(``"Maximum Profit = "` `+ maximum);` `    ``}` `}` `// This code was contributed by rutikbhosale`

## Python3

 `# Python3 implementation to print all` `# the possible solutions of the` `# 0/1 Knapsack problem`     `INT_MIN``=``-``2147483648` `def` `nextPermutation(nums: ``list``) ``-``> ``None``:` `        ``"""` `        ``Do not return anything, modify nums in-place instead.` `        ``"""` `        ``if` `sorted``(nums,reverse``=``True``)``=``=``nums:` `            ``return` `None` `        ``n``=``len``(nums)` `        ``brk_point``=``-``1` `        ``for` `pos ``in` `range``(n``-``1``,``0``,``-``1``):` `            ``if` `nums[pos]>nums[pos``-``1``]:` `                ``brk_point``=``pos` `                ``break` `        ``else``:` `            ``nums.sort()` `            ``return` `        ``replace_with``=``-``1` `        ``for` `j ``in` `range``(brk_point,n):` `            ``if` `nums[j]>nums[brk_point``-``1``]:` `                ``replace_with``=``j` `            ``else``:` `                ``break` `        ``nums[replace_with],nums[brk_point``-``1``]``=``nums[brk_point``-``1``],nums[replace_with]` `        ``nums[brk_point:]``=``sorted``(nums[brk_point:])` `        ``return` `nums`   `# Function to find the all the` `# possible solutions of the ` `# 0/1 knapSack problem` `def` `knapSack(W, wt, val, n):` `    ``# Mapping weights with Profits` `    ``umap``=``dict``()` `    `  `    ``set_sol``=``set``()` `    ``# Making Pairs and inserting` `    ``# o the map` `    ``for` `i ``in` `range``(n) :` `        ``umap[wt[i]]``=``val[i]` `    `    `    ``result ``=` `INT_MIN` `    ``remaining_weight``=``0` `    ``sum` `=` `0` `    `  `    ``# Loop to iterate over all the ` `    ``# possible permutations of array` `    ``while` `True``:` `        ``sum` `=` `0` `        `  `        ``# Initially bag will be empty` `        ``remaining_weight ``=` `W` `        ``possible``=``[]` `        `  `        ``# Loop to fill up the bag ` `        ``# until there is no weight` `        ``# such which is less than` `        ``# remaining weight of the` `        ``# 0-1 knapSack` `        ``for` `i ``in` `range``(n) :` `            ``if` `(wt[i] <``=` `remaining_weight) :`   `                ``remaining_weight ``-``=` `wt[i]` `                ``sum` `+``=` `(umap[wt[i]])` `                ``possible.append((wt[i],` `                     ``umap[wt[i]])` `                ``)` `            `  `        `  `        ``possible.sort()` `        ``if` `(``sum` `> result) :` `            ``result ``=` `sum` `        `  `        ``if` `(``tuple``(possible) ``not` `in` `set_sol):` `            ``for` `sol ``in` `possible:` `                ``print``(sol[``0``], ``": "``, sol[``1``], ``", "``,end``=``'')` `            `  `            ``print``()` `            ``set_sol.add(``tuple``(possible))` `        `  `        `  `        ``if` `not` `nextPermutation(wt):` `            ``break` `    ``return` `result`     `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    ``val``=``[``60``, ``100``, ``120``] ` `    ``wt``=``[``10``, ``20``, ``30``] ` `    ``W ``=` `50` `    ``n ``=` `len``(val)` `    ``maximum ``=` `knapSack(W, wt, val, n)` `    ``print``(``"Maximum Profit ="``,maximum)`   `#This code was contributed by Amartya Ghosh`

## Javascript

 `// Utility function to find the maximum of the two elements` `function` `max(a, b) { ` `    ``return` `(a > b) ? a : b; ` `}`   `// Function to find the all the possible solutions of the 0/1 knapSack problem` `function` `knapSack(W, wt, val, n) {` `    ``// Mapping weights with Profits` `    ``let umap = ``new` `Map();` `    ``let set_sol = ``new` `Set();`   `    ``// Making Pairs and inserting into the map` `    ``for` `(let i = 0; i < n; i++) {` `        ``umap.set(wt[i], val[i]);` `    ``}`   `    ``let result = Number.MIN_SAFE_INTEGER;` `    ``let remaining_weight, sum;` `    `  `    ``// Loop to iterate over all the possible permutations of array` `    ``do` `{` `        ``sum = 0;` `        `  `        ``// Initially bag will be empty` `        ``remaining_weight = W;` `        ``let possible = [];` `        `  `        ``// Loop to fill up the bag until there is no weight such which is less than remaining weight of the 0-1 knapSack` `        ``for` `(let i = 0; i < n; i++) {` `            ``if` `(wt[i] <= remaining_weight) {` `                ``remaining_weight -= wt[i];` `                ``let val = umap.get(wt[i]);` `                ``sum += val;` `                ``possible.push([wt[i], val]);` `            ``}` `        ``}` `        `  `        ``possible.sort((a, b) => a[0] - b[0]);` `        `  `        ``if` `(sum > result) {` `            ``result = sum;` `        ``}` `        `  `        ``if` `(!set_sol.has(JSON.stringify(possible))) {` `            ``for` `(let i = 0; i < possible.length; i++) {` `                ``console.log(possible[i][0] + ``": "` `+ possible[i][1] + ``", "``);` `            ``}` `            `  `            ``console.log();` `            ``set_sol.add(JSON.stringify(possible));` `        ``}` `    ``} ``while` `(nextPermutation(wt));`   `    ``return` `result;` `}`   `// Function to generate the next permutation of array` `function` `nextPermutation(a) {` `    ``let i = a.length - 2;` `    ``while` `(i >= 0 && a[i] >= a[i + 1]) {` `        ``i--;` `    ``}`   `    ``if` `(i < 0) {` `        ``return` `false``;` `    ``}`   `    ``let j = a.length - 1;` `    ``while` `(a[j] <= a[i]) {` `        ``j--;` `    ``}`   `    ``let temp = a[i];` `    ``a[i] = a[j];` `    ``a[j] = temp;`   `    ``for` `(let l = i + 1, r = a.length - 1; l < r; l++, r--) {` `        ``temp = a[l];` `        ``a[l] = a[r];` `        ``a[r] = temp;` `    ``}`   `    ``return` `true``;` `}`   `// Driver Code` `function` `main() {` `    ``let val = [60, 100, 120];` `    ``let wt = [10, 20, 30];` `    ``let W = 50;` `    ``let n = val.length;` `    ``let maximum = knapSack(W, wt, val, n);` `    ``console.log(``"Maximum Profit = "` `+ maximum);` `}`   `main();`

## C#

 `using` `System;` `using` `System.Collections.Generic;` `using` `System.Linq;`   `class` `MainClass {` `    ``// Utility function to find the maximum of the two elements` `    ``static` `int` `max(``int` `a, ``int` `b) { ``return` `(a > b) ? a : b; }`   `    ``// Function to find the all the possible solutions of` `    ``// the 0/1 knapSack problem` `    ``static` `int` `knapSack(``int` `W, List<``int``> wt,` `                        ``List<``int``> val, ``int` `n)` `    ``{`   `        ``// Mapping weights with Profits` `        ``Dictionary<``int``, ``int``> umap = ``new` `Dictionary<``int``, ``int``>();`   `        ``HashSet>> setSol` `            ``= ``new` `HashSet>>();`   `        ``// Making Pairs and inserting into the map` `        ``for` `(``int` `i = 0; i < n; i++) {` `            ``umap.Add(wt[i], val[i]);` `        ``}`   `        ``int` `result = ``int``.MinValue;` `        ``int` `remaining_weight;` `        ``int` `sum = 0;`   `        ``// Loop to iterate over all the possible permutations of array` `        ``do` `{` `            ``sum = 0;`   `            ``// Initially bag will be empty` `            ``remaining_weight = W;` `            ``List> possible` `                ``= ``new` `List>();`   `            ``// Loop to fill up the bag until there is no` `            ``// weight such which is less than remaining` `            ``// weight of the 0-1 knapSack` `            ``for` `(``int` `i = 0; i < n; i++) {` `                ``if` `(wt[i] <= remaining_weight) {`   `                    ``remaining_weight -= wt[i];` `                    ``int` `valAtWtI = umap[wt[i]];` `                    ``sum += valAtWtI;` `                    ``possible.Add(` `                        ``new` `KeyValuePair<``int``, ``int``>(` `                            ``wt[i], valAtWtI));` `                ``}` `            ``}` `            ``possible.Sort(` `                ``(x, y) => x.Key.CompareTo(y.Key));` `            ``if` `(sum > result) {` `                ``result = sum;` `            ``}` `            ``if` `(!setSol.Contains(possible)) {` `                ``foreach` `(KeyValuePair<``int``, ``int``> sol ``in` `possible) {` `                    ``Console.Write(sol.Key + ``": "` `+ sol.Value` `                                     ``+ ``", "``);` `                ``}` `                ``Console.WriteLine();` `                ``setSol.Add(possible);` `            ``}`   `        ``} ``while` `(nextPermutation(wt));`   `        ``return` `result;` `    ``}`   `    ``// Utility function to generate the next permutation` `    ``static` `bool` `nextPermutation(List<``int``> arr)` `    ``{` `        ``int` `i = arr.Count - 2;` `        ``while` `(i >= 0 && arr[i] >= arr[i + 1]) {` `            ``i--;` `        ``}` `        ``if` `(i < 0) {` `            ``return` `false``;` `        ``}` `        ``int` `j = arr.Count - 1;` `        ``while` `(arr[j] <= arr[i]) {` `            ``j--;` `        ``}` `        ``int` `temp = arr[i];` `        ``arr[i] = arr[j];` `        ``arr[j] = temp;`   `        ``arr.Reverse(i + 1, arr.Count - i - 1);` `        ``return` `true``;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `Main(``string``[] args)` `    ``{` `        ``List<``int``> val = ``new` `List<``int``>{60, 100, 120};` `        ``List<``int``> wt = ``new` `List<``int``>{10, 20, 30};` `        ``int` `W = 50;` `        ``int` `n = val.Count;` `        ``int` `maximum = knapSack(W, wt, val, n);` `        ``Console.WriteLine(``"Maximum Profit = "` `+ maximum);` `    ``}` `}`

Output

```10: 60, 20: 100,
10: 60, 30: 120,
20: 100, 30: 120,
Maximum Profit = 220```

Time complexity : O(N! * N), where N is the number of items. The code uses permutation to generate all possible combinations of items and then performs a search operation to find the optimal solution.

Space complexity : O(N), as the code uses a map and set to store the solution, which has a maximum size of N.

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