0-1 Knapsack Problem | DP-10 - GeeksforGeeks

08 Mar, 2022
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. 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).
Method 1: Recursion by Brute-Force algorithm OR Exhaustive Search.Approach: A simple solution is to consider all subsets of items and calculate the total weight and value of all subsets. Consider the only subsets whose total weight is smaller than W. From all such subsets, pick the maximum value subset.Optimal Sub-structure: To consider all subsets of items, there can be two cases for every item. 
Case 1: The item is included in the optimal subset.Case 2: The item is not included in the optimal set.
Case 1: The item is included in the optimal subset.
Case 2: The item is not included in the optimal set.
Therefore, the maximum value that can be obtained from ‘n’ items is the max of the following two values. 
Maximum value obtained by n-1 items and W weight (excluding nth item).Value of nth item plus maximum value obtained by n-1 items and W minus the weight of the nth item (including nth item).
Maximum value obtained by n-1 items and W weight (excluding nth item).
Value of nth item plus maximum value obtained by n-1 items and W minus the weight of the nth item (including nth item).
If the weight of ‘nth’ item is greater than ‘W’, then the nth item cannot be included and Case 1 is the only possibility.
Below is the implementation of the above approach: 
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Java
Python
C#
PHP
Javascript
/* A Naive recursive implementation of 0-1 Knapsack problem */#include <bits/stdc++.h>using namespace std;  // A utility function that returns// maximum of two integersint max(int a, int b) { return (a > b) ? a : b; }  // Returns the maximum value that// can be put in a knapsack of capacity Wint knapSack(int W, int wt[], int val[], int n){      // Base Case    if (n == 0 || W == 0)        return 0;      // If weight of the nth item is more    // than Knapsack capacity W, then    // this item cannot be included    // in the optimal solution    if (wt[n - 1] > W)        return knapSack(W, wt, val, n - 1);      // Return the maximum of two cases:    // (1) nth item included    // (2) not included    else        return max(            val[n - 1]                + knapSack(W - wt[n - 1],                            wt, val, n - 1),            knapSack(W, wt, val, n - 1));}  // Driver codeint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);    cout << knapSack(W, wt, val, n);    return 0;}  // This code is contributed by rathbhupendra
/* A Naive recursive implementationof 0-1 Knapsack problem */#include <stdio.h>  // A utility function that returns// maximum of two integersint max(int a, int b) { return (a > b) ? a : b; }  // Returns the maximum value that can be// put in a knapsack of capacity Wint knapSack(int W, int wt[], int val[], int n){    // Base Case    if (n == 0 || W == 0)        return 0;      // If weight of the nth item is more than    // Knapsack capacity W, then this item cannot    // be included in the optimal solution    if (wt[n - 1] > W)        return knapSack(W, wt, val, n - 1);      // Return the maximum of two cases:    // (1) nth item included    // (2) not included    else        return max(            val[n - 1]                + knapSack(W - wt[n - 1],                            wt, val, n - 1),            knapSack(W, wt, val, n - 1));}  // Driver program to test above functionint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);    printf("%d", knapSack(W, wt, val, n));    return 0;}
/* A Naive recursive implementationof 0-1 Knapsack problem */class Knapsack {      // A utility function that returns    // maximum of two integers    static int max(int a, int b)     {       return (a > b) ? a : b;     }      // Returns the maximum value that    // can be put in a knapsack of    // capacity W    static int knapSack(int W, int wt[], int val[], int n)    {        // Base Case        if (n == 0 || W == 0)            return 0;          // If weight of the nth item is        // more than Knapsack capacity W,        // then this item cannot be included        // in the optimal solution        if (wt[n - 1] > W)            return knapSack(W, wt, val, n - 1);          // Return the maximum of two cases:        // (1) nth item included        // (2) not included        else            return max(val[n - 1]                       + knapSack(W - wt[n - 1], wt,                                  val, n - 1),                       knapSack(W, wt, val, n - 1));    }      // Driver code    public static void main(String args[])    {        int val[] = new int[] { 60, 100, 120 };        int wt[] = new int[] { 10, 20, 30 };        int W = 50;        int n = val.length;        System.out.println(knapSack(W, wt, val, n));    }}/*This code is contributed by Rajat Mishra */
# A naive recursive implementation# of 0-1 Knapsack Problem  # Returns the maximum value that# can be put in a knapsack of# capacity W    def knapSack(W, wt, val, n):      # Base Case    if n == 0 or W == 0:        return 0      # If weight of the nth item is    # more than Knapsack of capacity W,    # then this item cannot be included    # in the optimal solution    if (wt[n-1] > W):        return knapSack(W, wt, val, n-1)      # return the maximum of two cases:    # (1) nth item included    # (2) not included    else:        return max(            val[n-1] + knapSack(                W-wt[n-1], wt, val, n-1),            knapSack(W, wt, val, n-1))  # end of function knapSack    #Driver Codeval = [60, 100, 120]wt = [10, 20, 30]W = 50n = len(val)print knapSack(W, wt, val, n)  # This code is contributed by Nikhil Kumar Singh
/* A Naive recursive implementation of0-1 Knapsack problem */using System;  class GFG {      // A utility function that returns    // maximum of two integers    static int max(int a, int b)     {          return (a > b) ? a : b;     }      // Returns the maximum value that can    // be put in a knapsack of capacity W    static int knapSack(int W, int[] wt,                         int[] val, int n)    {          // Base Case        if (n == 0 || W == 0)            return 0;          // If weight of the nth item is        // more than Knapsack capacity W,        // then this item cannot be        // included in the optimal solution        if (wt[n - 1] > W)            return knapSack(W, wt,                             val, n - 1);          // Return the maximum of two cases:        // (1) nth item included        // (2) not included        else            return max(val[n - 1]                       + knapSack(W - wt[n - 1], wt,                                  val, n - 1),                       knapSack(W, wt, val, n - 1));    }      // Driver code    public static void Main()    {        int[] val = new int[] { 60, 100, 120 };        int[] wt = new int[] { 10, 20, 30 };        int W = 50;        int n = val.Length;          Console.WriteLine(knapSack(W, wt, val, n));    }}  // This code is contributed by Sam007
<?php// A Naive recursive implementation// of 0-1 Knapsack problem   // Returns the maximum value that// can be put in a knapsack of // capacity Wfunction knapSack($W, $wt, $val, $n){    // Base Case    if ($n == 0 || $W == 0)        return 0;          // If weight of the nth item is     // more than Knapsack capacity     // W, then this item cannot be    // included in the optimal solution    if ($wt[$n - 1] > $W)        return knapSack($W, $wt, $val, $n - 1);          // Return the maximum of two cases:     // (1) nth item included     // (2) not included    else        return max($val[$n - 1] +                knapSack($W - $wt[$n - 1],                $wt, $val, $n - 1),                knapSack($W, $wt, $val, $n-1));}      // Driver Code    $val = array(60, 100, 120);    $wt = array(10, 20, 30);    $W = 50;    $n = count($val);    echo knapSack($W, $wt, $val, $n);  // This code is contributed by Sam007?>
<script>      /* A Naive recursive implementation of    0-1 Knapsack problem */          // A utility function that returns    // maximum of two integers    function max(a, b)    {         return (a > b) ? a : b;    }       // Returns the maximum value that can    // be put in a knapsack of capacity W    function knapSack(W, wt, val, n)    {           // Base Case        if (n == 0 || W == 0)            return 0;           // If weight of the nth item is        // more than Knapsack capacity W,        // then this item cannot be        // included in the optimal solution        if (wt[n - 1] > W)            return knapSack(W, wt, val, n - 1);           // Return the maximum of two cases:        // (1) nth item included        // (2) not included        else            return max(val[n - 1] +             knapSack(W - wt[n - 1], wt, val, n - 1),            knapSack(W, wt, val, n - 1));    }            let val = [ 60, 100, 120 ];    let wt = [ 10, 20, 30 ];       let W = 50;    let n = val.length;      document.write(knapSack(W, wt, val, n));      </script>
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It should be noted that the above function computes the same sub-problems again and again. See the following recursion tree, K(1, 1) is being evaluated twice. The time complexity of this naive recursive solution is exponential (2^n).  
In the following recursion tree, K() refers 
to knapSack(). The two parameters indicated in the
following recursion tree are n and W.
The recursion tree is for following sample inputs.
wt[] = {1, 1, 1}, W = 2, val[] = {10, 20, 30}
                       K(n, W)
                       K(3, 2)  
                   /            \ 
                 /                \               
            K(2, 2)                  K(2, 1)
          /       \                  /    \ 
        /           \              /        \
       K(1, 2)      K(1, 1)        K(1, 1)     K(1, 0)
       /  \         /   \              /        \
     /      \     /       \          /            \
K(0, 2)  K(0, 1)  K(0, 1)  K(0, 0)  K(0, 1)   K(0, 0)
Recursion tree for Knapsack capacity 2 
units and 3 items of 1 unit weight.
Complexity Analysis: 
Time Complexity: O(2n). As there are redundant subproblems.
Auxiliary Space :O(1). As no extra data structure has been used for storing values.
Since subproblems are evaluated again, this problem has Overlapping Sub-problems property. So the 0-1 Knapsack problem has both properties (see this and this) of a dynamic programming problem. 
Method 2: Like other typical Dynamic Programming(DP) problems, re-computation of same subproblems can be avoided by constructing a temporary array K[][] in bottom-up manner. Following is Dynamic Programming based implementation.
Approach: In the Dynamic programming we will work considering the same cases as mentioned in the recursive approach. In a DP[][] table let’s consider all the possible weights from ‘1’ to ‘W’ as the columns and weights that can be kept as the rows. The state DP[i][j] will denote maximum value of ‘j-weight’ considering all values from ‘1 to ith’. So if we consider ‘wi’ (weight in ‘ith’ row) we can fill it in all columns which have ‘weight values > wi’. Now two possibilities can take place: 
Fill ‘wi’ in the given column.
Do not fill ‘wi’ in the given column.
Now we have to take a maximum of these two possibilities, formally if we do not fill ‘ith’ weight in ‘jth’ column then DP[i][j] state will be same as DP[i-1][j] but if we fill the weight, DP[i][j] will be equal to the value of ‘wi’+ value of the column weighing ‘j-wi’ in the previous row. So we take the maximum of these two possibilities to fill the current state. This visualisation will make the concept clear:  
Let weight elements = {1, 2, 3}
Let weight values = {10, 15, 40}
Capacity=6

   0   1   2   3   4   5   6

0  0   0   0   0   0   0   0

1  0  10  10  10  10  10  10

2  0  10  15  25  25  25  25

3  0
 
Explanation:
For filling 'weight = 2' we come 
across 'j = 3' in which 
we take maximum of 
(10, 15 + DP[1][3-2]) = 25   
  |        |
'2'       '2 filled'
not filled  

   0   1   2   3   4   5   6

0  0   0   0   0   0   0   0

1  0  10  10  10  10  10  10

2  0  10  15  25  25  25  25

3  0  10  15  40  50  55  65

Explanation:
For filling 'weight=3', 
we come across 'j=4' in which 
we take maximum of (25, 40 + DP[2][4-3]) 
= 50

For filling 'weight=3' 
we come across 'j=5' in which 
we take maximum of (25, 40 + DP[2][5-3])
= 55

For filling 'weight=3' 
we come across 'j=6' in which 
we take maximum of (25, 40 + DP[2][6-3])
= 65
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Java
Python
C#
PHP
Javascript
// A dynamic programming based// solution for 0-1 Knapsack problem#include <bits/stdc++.h>using namespace std;  // A utility function that returns// maximum of two integersint max(int a, int b){    return (a > b) ? a : b;}  // Returns the maximum value that// can be put in a knapsack of capacity Wint knapSack(int W, int wt[], int val[], int n){    int i, w;      vector<vector<int>> K(n + 1, vector<int>(W + 1));      // Build table K[][] in bottom up manner    for(i = 0; i <= n; i++)    {        for(w = 0; w <= W; w++)        {            if (i == 0 || w == 0)                K[i][w] = 0;            else if (wt[i - 1] <= w)                K[i][w] = max(val[i - 1] +                                 K[i - 1][w - wt[i - 1]],                                K[i - 1][w]);            else                K[i][w] = K[i - 1][w];        }    }    return K[n][W];}  // Driver Codeint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);          cout << knapSack(W, wt, val, n);          return 0;}  // This code is contributed by Debojyoti Mandal
// A Dynamic Programming based// solution for 0-1 Knapsack problem#include <stdio.h>  // A utility function that returns// maximum of two integersint max(int a, int b){     return (a > b) ? a : b; }  // Returns the maximum value that// can be put in a knapsack of capacity Wint knapSack(int W, int wt[], int val[], int n){    int i, w;    int K[n + 1][W + 1];      // Build table K[][] in bottom up manner    for (i = 0; i <= n; i++)     {        for (w = 0; w <= W; w++)         {            if (i == 0 || w == 0)                K[i][w] = 0;            else if (wt[i - 1] <= w)                K[i][w] = max(val[i - 1]                          + K[i - 1][w - wt[i - 1]],                          K[i - 1][w]);            else                K[i][w] = K[i - 1][w];        }    }      return K[n][W];}  // Driver Codeint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);    printf("%d", knapSack(W, wt, val, n));    return 0;}
// A Dynamic Programming based solution// for 0-1 Knapsack problemclass Knapsack {      // A utility function that returns    // maximum of two integers    static int max(int a, int b)     {           return (a > b) ? a : b;     }      // Returns the maximum value that can    // be put in a knapsack of capacity W    static int knapSack(int W, int wt[],                         int val[], int n)    {        int i, w;        int K[][] = new int[n + 1][W + 1];          // Build table K[][] in bottom up manner        for (i = 0; i <= n; i++)         {            for (w = 0; w <= W; w++)             {                if (i == 0 || w == 0)                    K[i][w] = 0;                else if (wt[i - 1] <= w)                    K[i][w]                        = max(val[i - 1]                         + K[i - 1][w - wt[i - 1]],                         K[i - 1][w]);                else                    K[i][w] = K[i - 1][w];            }        }          return K[n][W];    }      // Driver code    public static void main(String args[])    {        int val[] = new int[] { 60, 100, 120 };        int wt[] = new int[] { 10, 20, 30 };        int W = 50;        int n = val.length;        System.out.println(knapSack(W, wt, val, n));    }}/*This code is contributed by Rajat Mishra */
# A Dynamic Programming based Python# Program for 0-1 Knapsack problem# Returns the maximum value that can# be put in a knapsack of capacity W    def knapSack(W, wt, val, n):    K = [[0 for x in range(W + 1)] for x in range(n + 1)]      # Build table K[][] in bottom up manner    for i in range(n + 1):        for w in range(W + 1):            if i == 0 or w == 0:                K[i][w] = 0            elif wt[i-1] <= w:                K[i][w] = max(val[i-1]                          + K[i-1][w-wt[i-1]],                                K[i-1][w])            else:                K[i][w] = K[i-1][w]      return K[n][W]    # Driver codeval = [60, 100, 120]wt = [10, 20, 30]W = 50n = len(val)print(knapSack(W, wt, val, n))  # This code is contributed by Bhavya Jain
// A Dynamic Programming based solution for// 0-1 Knapsack problemusing System;  class GFG {      // A utility function that returns    // maximum of two integers    static int max(int a, int b)     {          return (a > b) ? a : b;     }      // Returns the maximum value that    // can be put in a knapsack of    // capacity W    static int knapSack(int W, int[] wt,                         int[] val, int n)    {        int i, w;        int[, ] K = new int[n + 1, W + 1];          // Build table K[][] in bottom        // up manner        for (i = 0; i <= n; i++)         {            for (w = 0; w <= W; w++)             {                if (i == 0 || w == 0)                    K[i, w] = 0;                                  else if (wt[i - 1] <= w)                    K[i, w] = Math.Max(                        val[i - 1]                        + K[i - 1, w - wt[i - 1]],                        K[i - 1, w]);                else                    K[i, w] = K[i - 1, w];            }        }          return K[n, W];    }      // Driver code    static void Main()    {        int[] val = new int[] { 60, 100, 120 };        int[] wt = new int[] { 10, 20, 30 };        int W = 50;        int n = val.Length;          Console.WriteLine(knapSack(W, wt, val, n));    }}  // This code is contributed by Sam007
<?php// A Dynamic Programming based solution// for 0-1 Knapsack problem  // Returns the maximum value that// can be put in a knapsack of // capacity Wfunction knapSack($W, $wt, $val, $n){          $K = array(array());          // Build table K[][] in    // bottom up manner    for ($i = 0; $i <= $n; $i++)    {        for ($w = 0; $w <= $W; $w++)        {            if ($i == 0 || $w == 0)                $K[$i][$w] = 0;            else if ($wt[$i - 1] <= $w)                    $K[$i][$w] = max($val[$i - 1] +                                      $K[$i - 1][$w -                                      $wt[$i - 1]],                                      $K[$i - 1][$w]);            else                    $K[$i][$w] = $K[$i - 1][$w];        }    }          return $K[$n][$W];}      // Driver Code    $val = array(60, 100, 120);    $wt = array(10, 20, 30);    $W = 50;    $n = count($val);    echo knapSack($W, $wt, $val, $n);      // This code is contributed by Sam007.?>
<script>    // A Dynamic Programming based solution    // for 0-1 Knapsack problem          // A utility function that returns    // maximum of two integers    function max(a, b)    {          return (a > b) ? a : b;    }       // Returns the maximum value that can    // be put in a knapsack of capacity W    function knapSack(W, wt, val, n)    {        let i, w;        let K = new Array(n + 1);           // Build table K[][] in bottom up manner        for (i = 0; i <= n; i++)        {            K[i] = new Array(W + 1);            for (w = 0; w <= W; w++)            {                if (i == 0 || w == 0)                    K[i][w] = 0;                else if (wt[i - 1] <= w)                    K[i][w]                        = max(val[i - 1]                         + K[i - 1][w - wt[i - 1]],                         K[i - 1][w]);                else                    K[i][w] = K[i - 1][w];            }        }           return K[n][W];    }          let val = [ 60, 100, 120 ];    let wt = [ 10, 20, 30 ];    let W = 50;    let n = val.length;    document.write(knapSack(W, wt, val, n));</script>
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Complexity Analysis: 
Time Complexity: O(N*W). where ‘N’ is the number of weight element and ‘W’ is capacity. As for every weight element we traverse through all weight capacities 1<=w<=W.
Auxiliary Space: O(N*W). The use of 2-D array of size ‘N*W’.
Scope for Improvement :-  We used the same approach but with optimized space complexity
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Java
Python3
C#
Javascript
#include <bits/stdc++.h>using namespace std;  // we can further improve the above Knapsack function's space// complexityint knapSack(int W, int wt[], int val[], int n){    int i, w;    int K[2][W + 1];    // We know we are always using the the current row or    // the previous row of the array/vector . Thereby we can    // improve it further by using a 2D array but with only    // 2 rows i%2 will be giving the index inside the bounds    // of 2d array K      for (i = 0; i <= n; i++) {        for (w = 0; w <= W; w++) {            if (i == 0 || w == 0)                K[i % 2][w] = 0;            else if (wt[i - 1] <= w)                K[i % 2][w] = max(                    val[i - 1]                        + K[(i - 1) % 2][w - wt[i - 1]],                    K[(i - 1) % 2][w]);            else                K[i % 2][w] = K[(i - 1) % 2][w];        }    }    return K[n % 2][W];}  // Driver Codeint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);      cout << knapSack(W, wt, val, n);      return 0;}  // This code was improved by Udit Singla
import java.util.*;class GFG {    // we can further improve the above Knapsack function's space  // complexity  static int knapSack(int W, int wt[], int val[], int n)  {    int i, w;    int [][]K = new int[2][W + 1];          // We know we are always using the the current row or    // the previous row of the array/vector . Thereby we can    // improve it further by using a 2D array but with only    // 2 rows i%2 will be giving the index inside the bounds    // of 2d array K    for (i = 0; i <= n; i++) {      for (w = 0; w <= W; w++) {        if (i == 0 || w == 0)          K[i % 2][w] = 0;        else if (wt[i - 1] <= w)          K[i % 2][w] = Math.max(          val[i - 1]          + K[(i - 1) % 2][w - wt[i - 1]],          K[(i - 1) % 2][w]);        else          K[i % 2][w] = K[(i - 1) % 2][w];      }    }    return K[n % 2][W];  }    // Driver Code  public static void main(String[] args)  {    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = val.length;      System.out.print(knapSack(W, wt, val, n));    }}  // This code is contributed by gauravrajput1
# we can further improve the above Knapsack function's space# complexitydef knapSack(W, wt, val, n):      K = [[0 for x in range(W+1)] for y in range(2)]          # We know we are always using the the current row or    # the previous row of the array/vector . Thereby we can    # improve it further by using a 2D array but with only    # 2 rows i%2 will be giving the index inside the bounds    # of 2d array K    for i in range(n + 1):        for w in range(W + 1):            if (i == 0 or w == 0):                K[i % 2][w] = 0            elif (wt[i - 1] <= w):                K[i % 2][w] = max(                    val[i - 1]                    + K[(i - 1) % 2][w - wt[i - 1]],                    K[(i - 1) % 2][w])              else:                K[i % 2][w] = K[(i - 1) % 2][w]      return K[n % 2][W]  # Driver Codeif __name__ == "__main__":      val = [60, 100, 120]    wt = [10, 20, 30]    W = 50    n = len(val)      print(knapSack(W, wt, val, n))      # This code is contributed by ukasp.
using System;  public class GFG {      // we can further improve the above Knapsack function's space    // complexity    static int knapSack(int W, int []wt, int []val, int n) {        int i, w;        int[,] K = new int[2,W + 1];          // We know we are always using the the current row or        // the previous row of the array/vector . Thereby we can        // improve it further by using a 2D array but with only        // 2 rows i%2 will be giving the index inside the bounds        // of 2d array K        for (i = 0; i <= n; i++) {            for (w = 0; w <= W; w++) {                if (i == 0 || w == 0)                    K[i % 2, w] = 0;                else if (wt[i - 1] <= w)                    K[i % 2,w] = Math.Max(val[i - 1] + K[(i - 1) % 2,w - wt[i - 1]], K[(i - 1) % 2,w]);                else                    K[i % 2,w] = K[(i - 1) % 2,w];            }        }        return K[n % 2,W];    }      // Driver Code    public static void Main(String[] args) {        int []val = { 60, 100, 120 };        int []wt = { 10, 20, 30 };        int W = 50;        int n = val.Length;          Console.Write(knapSack(W, wt, val, n));      }}  // This code is contributed by umadevi9616
<script>    // we can further improve the above Knapsack function's space    // complexity    function knapSack(W , wt , val , n) {        var i, w;        var K = Array(2).fill().map(()=>Array(W + 1).fill(0));          // We know we are always using the the current row or        // the previous row of the array/vector . Thereby we can        // improve it further by using a 2D array but with only        // 2 rows i%2 will be giving the index inside the bounds        // of 2d array K        for (i = 0; i <= n; i++) {            for (w = 0; w <= W; w++) {                if (i == 0 || w == 0)                    K[i % 2][w] = 0;                else if (wt[i - 1] <= w)                    K[i % 2][w] = Math.max(val[i - 1] +                    K[(i - 1) % 2][w - wt[i - 1]],                    K[(i - 1) % 2][w]);                else                    K[i % 2][w] = K[(i - 1) % 2][w];            }        }        return K[n % 2][W];    }      // Driver Code        var val = [ 60, 100, 120 ];        var wt = [ 10, 20, 30 ];        var W = 50;        var n = val.length;          document.write(knapSack(W, wt, val, n));  // This code is contributed by Rajput-Ji</script>
Complexity Analysis:
Time Complexity: O(N*W).
Auxiliary Space: O(2*W) As we are using a 2-D array but with only 2 rows.
Method 3: This method uses Memoization Technique (an extension of recursive approach).This method is basically an extension to the recursive approach so that we can overcome the problem of calculating redundant cases and thus increased complexity. We can solve this problem by simply creating a 2-D array that can store a particular state (n, w) if we get it the first time. Now if we come across the same state (n, w) again instead of calculating it in exponential complexity we can directly return its result stored in the table in constant time. This method gives an edge over the recursive approach in this aspect.
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Python3
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Javascript
// Here is the top-down approach of// dynamic programming#include <bits/stdc++.h>using namespace std;  // Returns the value of maximum profitint knapSackRec(int W, int wt[],                 int val[], int i,                int** dp){    // base condition    if (i < 0)        return 0;    if (dp[i][W] != -1)        return dp[i][W];      if (wt[i] > W) {          // Store the value of function call        // stack in table before return        dp[i][W] = knapSackRec(W, wt,                                val, i - 1,                                dp);        return dp[i][W];    }    else {        // Store value in a table before return        dp[i][W] = max(val[i]                      + knapSackRec(W - wt[i],                                    wt, val,                                   i - 1, dp),                       knapSackRec(W, wt, val,                                    i - 1, dp));          // Return value of table after storing        return dp[i][W];    }}  int knapSack(int W, int wt[], int val[], int n){    // double pointer to declare the    // table dynamically    int** dp;    dp = new int*[n];      // loop to create the table dynamically    for (int i = 0; i < n; i++)        dp[i] = new int[W + 1];      // loop to initially filled the    // table with -1    for (int i = 0; i < n; i++)        for (int j = 0; j < W + 1; j++)            dp[i][j] = -1;    return knapSackRec(W, wt, val, n - 1, dp);}  // Driver Codeint main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);    cout << knapSack(W, wt, val, n);    return 0;}
// Here is the top-down approach of  // dynamic programmingclass GFG{      // A utility function that returns  // maximum of two integers     static int max(int a, int b)     {         return (a > b) ? a : b;     }   // Returns the value of maximum profit   static int knapSackRec(int W, int wt[],                        int val[], int n,                       int [][]dp){             // Base condition    if (n == 0 || W == 0)           return 0;              if (dp[n][W] != -1)        return dp[n][W];             if (wt[n - 1] > W)                 // Store the value of function call           // stack in table before return        return dp[n][W] = knapSackRec(W, wt, val,                                       n - 1, dp);                                            else               // Return value of table after storing          return dp[n][W] = max((val[n - 1] +                               knapSackRec(W - wt[n - 1], wt,                                          val, n - 1, dp)),                              knapSackRec(W, wt, val,                                           n - 1, dp));             }  static int knapSack(int W, int wt[], int val[], int N){            // Declare the table dynamically    int dp[][] = new int[N + 1][W + 1];           // Loop to initially filled the     // table with -1     for(int i = 0; i < N + 1; i++)           for(int j = 0; j < W + 1; j++)               dp[i][j] = -1;              return knapSackRec(W, wt, val, N, dp);     }  // Driver Code public static void main(String [] args){           int val[] = { 60, 100, 120 };       int wt[] = { 10, 20, 30 };             int W = 50;      int N = val.length;                   System.out.println(knapSack(W, wt, val, N));   }     }  // This Code is contributed By FARAZ AHMAD
# This is the memoization approach of# 0 / 1 Knapsack in Python in simple# we can say recursion + memoization = DP  # driver codeval = [60, 100, 120 ]wt = [10, 20, 30 ]W = 50n = len(val)  # We initialize the matrix with -1 at first.t = [[-1 for i in range(W + 1)] for j in range(n + 1)]    def knapsack(wt, val, W, n):      # base conditions    if n == 0 or W == 0:        return 0    if t[n][W] != -1:        return t[n][W]      # choice diagram code    if wt[n-1] <= W:        t[n][W] = max(            val[n-1] + knapsack(                wt, val, W-wt[n-1], n-1),            knapsack(wt, val, W, n-1))        return t[n][W]    elif wt[n-1] > W:        t[n][W] = knapsack(wt, val, W, n-1)        return t[n][W]    print(knapsack(wt, val, W, n))  # This code is contributed by Prosun Kumar Sarkar
// Here is the top-down approach of  // dynamic programmingusing System;public class GFG {      // A utility function that returns    // maximum of two integers    static int max(int a, int b) { return (a > b) ? a : b; }      // Returns the value of maximum profit    static int knapSackRec(int W, int[] wt, int[] val,                           int n, int[, ] dp)    {          // Base condition        if (n == 0 || W == 0)            return 0;        if (dp[n, W] != -1)            return dp[n, W];        if (wt[n - 1] > W)              // Store the value of function call            // stack in table before return            return dp[n, W]                = knapSackRec(W, wt, val, n - 1, dp);          else              // Return value of table after storing            return dp[n, W]                = max((val[n - 1]                       + knapSackRec(W - wt[n - 1], wt, val,                                     n - 1, dp)),                      knapSackRec(W, wt, val, n - 1, dp));    }      static int knapSack(int W, int[] wt, int[] val, int N)    {          // Declare the table dynamically        int[, ] dp = new int[N + 1, W + 1];          // Loop to initially filled the        // table with -1        for (int i = 0; i < N + 1; i++)            for (int j = 0; j < W + 1; j++)                dp[i, j] = -1;          return knapSackRec(W, wt, val, N, dp);    }      // Driver Code    static public void Main()    {          int[] val = new int[]{ 60, 100, 120 };        int[] wt = new int[]{ 10, 20, 30 };          int W = 50;        int N = val.Length;          Console.WriteLine(knapSack(W, wt, val, N));    }}  // This Code is contributed By Dharanendra L V.
<script>// A utility function that returns// maximum of two integers    function max(a,  b)    {        return (a > b) ? a : b;    }  // Returns the value of maximum profitfunction knapSackRec(W, wt, val, n,dp){          // Base condition    if (n == 0 || W == 0)        return 0;              if (dp[n][W] != -1)        return dp[n][W];          if (wt[n - 1] > W)              // Store the value of function call        // stack in table before return        return dp[n][W] = knapSackRec(W, wt, val,                                    n - 1, dp);                                          else              // Return value of table after storing        return dp[n][W] = max((val[n - 1] +                            knapSackRec(W - wt[n - 1], wt,                                        val, n - 1, dp)),                            knapSackRec(W, wt, val,                                        n - 1, dp));            }  function knapSack( W, wt,val,N){          // Declare the table dynamically    var dp = new Array(N + 1);    for(var i = 0; i < dp.length; i++)    {        dp[i]=new Array(W + 1);      }          // Loop to initially filled the    // table with -1    for(var i = 0; i < N + 1; i++)        for(var j = 0; j < W + 1; j++)            dp[i][j] = -1;          return knapSackRec(W, wt, val, N, dp);    }       var val= [ 60, 100, 120 ];    var wt = [ 10, 20, 30 ];          var W = 50;    var N = val.length;                  document.write(knapSack(W, wt, val, N));  // This code is contributed by akshitsaxenaa09.  </script>
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Complexity Analysis: 
Time Complexity: O(N*W). As redundant calculations of states are avoided.
Auxiliary Space: O(N*W). The use of 2D array data structure for storing intermediate states-:
[Note: For 32bit integer use long instead of int.]References: 
http://www.es.ele.tue.nl/education/5MC10/Solutions/knapsack.pdf
http://www.cse.unl.edu/~goddard/Courses/CSCE310J/Lectures/Lecture8-DynamicProgramming.pdf
https://youtu.be/T4bY72lCQac?list=PLqM7alHXFySGMu2CSdW_6d2u1o6WFTIO- 
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Method 4 :-  Again we use the dynamic programming approach with even more optimized space complexity .
C++
Java
Python3
C#
Javascript
#include <bits/stdc++.h>using namespace std;int knapSack(int W, int wt[], int val[], int n){    // making and initializing dp array    int dp[W + 1];    memset(dp, 0, sizeof(dp));      for (int i = 1; i < n + 1; i++) {        for (int w = W; w >= 0; w--) {              if (wt[i - 1] <= w)                // finding the maximum value                dp[w] = max(dp[w],                            dp[w - wt[i - 1]] + val[i - 1]);        }    }    return dp[W]; // returning the maximum value of knapsack}int main(){    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = sizeof(val) / sizeof(val[0]);    cout << knapSack(W, wt, val, n);    return 0;}
import java.util.*;  class GFG{  static int knapSack(int W, int wt[], int val[], int n)  {    // making and initializing dp array    int []dp = new int[W + 1];        for (int i = 1; i < n + 1; i++) {      for (int w = W; w >= 0; w--) {          if (wt[i - 1] <= w)                      // finding the maximum value          dp[w] = Math.max(dp[w],                           dp[w - wt[i - 1]] + val[i - 1]);      }    }    return dp[W]; // returning the maximum value of knapsack  }      // Driver code  public static void main(String[] args)  {    int val[] = { 60, 100, 120 };    int wt[] = { 10, 20, 30 };    int W = 50;    int n = val.length;    System.out.print(knapSack(W, wt, val, n));  }}  // This code is contributed by gauravrajput1 
# code# A Dynamic Programming based Python# Program for 0-1 Knapsack problem# Returns the maximum value that can# be put in a knapsack of capacity W    def knapSack(W, wt, val, n):    dp = [0 for i in range(W+1)]  # Making the dp array      for i in range(1, n+1):  # taking first i elements        for w in range(W, 0, -1):  # starting from back,so that we also have data of                                # previous computation when taking i-1 items            if wt[i-1] <= w:                # finding the maximum value                dp[w] = max(dp[w], dp[w-wt[i-1]]+val[i-1])      return dp[W]  # returning the maximum value of knapsack    # Driver codeval = [60, 100, 120]wt = [10, 20, 30]W = 50n = len(val)# This code is contributed by Suyash Saxenaprint(knapSack(W, wt, val, n))
using System;public class GFG {    static int knapSack(int W, int []wt, int []val, int n)    {                // making and initializing dp array        int[] dp = new int[W + 1];          for (int i = 1; i < n + 1; i++)        {            for (int w = W; w >= 0; w--)            {                  if (wt[i - 1] <= w)                      // finding the maximum value                    dp[w] = Math.Max(dp[w], dp[w - wt[i - 1]] + val[i - 1]);            }        }        return dp[W]; // returning the maximum value of knapsack    }      // Driver code    public static void Main(String[] args) {        int []val = { 60, 100, 120 };        int []wt = { 10, 20, 30 };        int W = 50;        int n = val.Length;        Console.Write(knapSack(W, wt, val, n));    }}  // This code is contributed by gauravrajput1 
<script>    function knapSack(W , wt , val , n)    {              // making and initializing dp array        var dp = Array(W + 1).fill(0);          for (i = 1; i < n + 1; i++) {            for (w = W; w >= 0; w--) {                  if (wt[i - 1] <= w)                      // finding the maximum value                    dp[w] = Math.max(dp[w], dp[w - wt[i - 1]] + val[i - 1]);            }        }        return dp[W]; // returning the maximum value of knapsack    }      // Driver code        var val = [ 60, 100, 120 ];        var wt = [ 10, 20, 30 ];        var W = 50;        var n = val.length;        document.write(knapSack(W, wt, val, n));  // This code is contributed by Rajput-Ji </script>
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Complexity Analysis:
Time Complexity: O(N*W). As redundant calculations of states are avoided.
Auxiliary Space: O(W) As we are using 1-D array instead of 2-D array.
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