## Introduction

Given a Knapsack/Bag with W weight capacity and a list of N items with given v_{i} value and w_{i} weight. Put these items in the knapsack in order to maximise the value of all the placed items without exceeding the limit of the Knapsack.

**0-1 Knapsack Problem**

The problem remains the same but one cannot break the items you can either select it fully ( 1) or don’t select it (0 ).

**Example of 0-1 Knapsack :**

## Method 1 (Using Bruteforce Recursion):

Our approach with recursion will be to try and create all the subsets of items with total weight less than that of the given capacity W. From the result we will return the subset with maximum value.

For every element we can,

- either select it or,
- ignore and move forward.

If we select an item then its value will be added to our current value and weight will be subtracted from the current available space.

Thus, if we take the maximum value out of both the calculated result for nth element i.e. result after selecting that element and after ignoring it, we can get to our desired answer.

**Code for Above Implementation:**

### C++ Code

int knapsack01(int W,int N,vector<int> &v,vector<int> &w) { /* Base case 0 items left or knapsack is full */ if(N == 0 || W == 0) return 0; /* if weight of current element is less than or equal to capacity we can either include or exclude the item. */ if(w[N] <= W){ return max(v[N]+knapsack01(W-w[N],N-1,v,w),knapsack01(W,N-1,v,w)); } /* if weight of current element is greater than the capacity we will not include it thus returning just the ignoring part. */ return knapsack01(W,N-1,v,w); }

### Java Code

// maximum Function for max of two integers static int max(int a, int b) { if(a>b) return a; else return b; } static int knapSack(int W,int N,int v[],int w[]) { /* Base case 0 items left or knapsack is full */ if(N == 0 || W == 0) return 0; /* if weight of current element is less than or equal to capacity we can either include or exclude the item. */ if(w[N] <= W){ return max(v[N]+knapSack(W-w[N],N-1,v,w),knapSack(W,N-1,v,w)); } /* if weight of current element is greater than the capacity we will not include it thus returning just the ignoring part. */ return knapSack(W,N-1,v,w); }

### Python Code

def knapsack01(W,N,v,w): # Base case 0 items left or knapsack is full if N == 0 or W == 0: return 0 # if weight of current element is less than or equal to capacity we can # either include or exclude the item. if w[N] <= W: return max(v[N]+knapsack01(W-w[N],N-1,v,w),knapsack01(W,N-1,v,w)) # if weight of current element is greater than the capacity we will # not include it thus returning just the ignoring part. else: return knapsack01(W,N-1,v,w)

Time Complexity of the above approach is O(2^{n}).

## Method 2 (Using Dynamic Programming):

In the above approach we can observe that we are calling recursion for same sub problems again and again thus resulting in overlapping subproblems thus we can make use of **Dynamic programming** to solve 0-1 Knapsack problem.

**For Example :**

## Approach 1: (Using memoization)

In this approach we’ll store all the computed answers in a 2 dimensional Array with indices of items in rows and weights in columns and use it further for overlapping subproblems.

**Code for Above Implementation**:

### C++ Code

int knapSackrecur(int W,int N,vector<int> &v,vector<int> &w,vector<vector<int>>& dp){ /* Base case 0 items left or knapsack is full */ if(N == 0 || W == 0) return 0; /* Checking if the result exists in DP */ if(dp[N][W]!=-1) return dp[N][W]; /* if weight of current element is less than or equal to capacity we can either include or exclude the item and store it to DP. */ if(w[N] <= W){ return dp[N][W] = max(v[N]+knapSackrecur(W-w[N],N-1,v,w,dp),knapSackrecur(W,N-1,v,w,dp)); } /* if weight of current element is greater than the capacity we will not include it thus returning just the ignoring part and storing it to DP. */ return dp[N][W] = knapSackrecur(W,N-1,v,w,dp); } int knapsack01(int W,int N,vector<int> &v,vector<int> &w) { // Defining Dp and initializing with -1. vector<vector<int>> dp(N+1,vector<int> (W+1,-1)); return knapSackrecur(W,N-1,v,w,dp); }

### Java Code

// maximum of two integers static int max(int a, int b) { if(a>b) return a; else return b; } static int knapSack(int W,int N,int v[],int w[],int [][]dp) { /* Base case 0 items left or knapsack is full */ if(N == 0 || W == 0) return 0; if(dp[N][W] == -1) return dp[N][W]; /* if weight of current element is less than or equal to capacity we can either include or exclude the item. */ if(w[N] <= W){ return dp[N][W] = max(v[N]+knapSack(W-w[N],N-1,v,w),knapSack(W,N-1,v,w)); } /* if weight of current element is greater than the capacity we will not include it thus returning just the ignoring part. */ return dp[N][W] = knapSack(W,N-1,v,w); } static int knapsack01(int W,int N,int v[],int w[]){ // Defining Dp and initializing with -1. int DP[][] = new int [N+1][W+1]; for(int i=0;i<=N;i++){ for(int j=0;j<=W;j++){ DP[i][j] = -1; } } return knapSack(W,N,v,w,DP); }

### Python Code

# Globally defining the DP and initialising as -1 dp = [[-1 for i in range(W+1)] for j in range(N+1)] def knapSack(W,N,v,w): # Base case 0 items left or knapsack is full if N == 0 or W == 0: return 0 # checking if the result is precalculated and returning it. if dp[N][W] != -1: return dp[N][W] # if weight of current element is less than or equal to capacity we can # either include or exclude the item. if w[N] <= W: return dp[N][W] = max(v[N]+knapSack(W-w[N],N-1,v,w),knapSack(W,N-1,v,w)) # if weight of current element is greater than the capacity we will # not include it thus returning just the ignoring part. else: return dp[N][W] = knapSack(W,N-1,v,w)

Time Complexity of the above approach is O(N*W).

Space Complexity of the above approach is O(N*W).

## Approach 2: (Using Iterative DP)

In this approach we’ll define 2 dimensional DP of index for items defined on rows whereas weights from 1 to W on columns and for every weight we can compute the answer for placing items till nth item.

Similar to the recursive approach we can define two cases that is,

- Use this item or
- Ignore it.

Thus for every, Dp[i][j] we can calculate values for these two cases and store out the maximum of those two ,

Therefore we can get our answer for {i,j} as ,

DP[i][j] = max(v[i] + DP[i-1][j-w[i]],DP[i-1][j])

**Code for Above Implementation**

### C++ Code

int knapsack01(int W,int N,vector<int> &v,vector<int> &w) { int DP[N+1][W+1]; // Defining DP /* If there is no space left that is W reaches 0 then DP[i][0] for every i will be 0.*/ for(int i=0;i<N+1;i++) DP[i][0] = 0; /* If there are no items left that is N reaches 0 then DP[0][i] for every i will be 0.*/ for(int i=0;i<W+1;i++) DP[0][i] = 0; for(int i=1;i<N+1;i++){ for(int j=1;j<W+1;j++){ if(w[i-1] <= j){ /* Taking max of both the cases i.e to take that item or to ignore it.*/ DP[i][j] = max(v[i-1]+DP[i-1][j-w[i-1]],DP[i-1][j]); } else{ /* If the weight of current element is greater than the space left in the bag we'll ignore it.*/ DP[i][j] = DP[i-1][j]; } } } /* returning answer for W space and N items */ return DP[N][W]; }

### Java Code

static int knapSack(int W,int N,int v[],int w[]) { int DP[][] = new int[N+1][W+1]; // Defining DP /* If there is no space left that is W reaches 0 then DP[i][0] for every i will be 0.*/ for(int i=0;i<N+1;i++) DP[i][0] = 0; /* If there are no items left that is N reaches 0 then DP[0][i] for every i will be 0.*/ for(int i=0;i<W+1;i++) DP[0][i] = 0; for(int i=1;i<N+1;i++){ for(int j=1;j<W+1;j++){ if(w[i-1] <= j){ /* Taking max of both the cases i.e to take that item or to ignore it.*/ int a = v[i-1]+DP[i-1][j-w[i-1]]; int b = DP[i-1][j]; if(a>b) DP[i][j] = a; else DP[i][j] = b; } else{ /* If the weight of current element is greater than the space left in the bag we'll ignore it.*/ DP[i][j] = DP[i-1][j]; } } } /* returning answer for W space and N items */ return DP[N][W]; }

### Python Code

def knapsack01(W,N,v,w) : DP = [[0 for i in range(W+1)] for j in range(N+1)] # Defining DP for i in range(1,N+1) : for j in range(1,W+1) : if w[i-1] <= j : # Taking max of both the cases i.e to take that # item or to ignore it. DP[i][j] = max(v[i-1]+DP[i-1][j-w[i-1]],DP[i-1][j]) else : # If the weight of current element is greater # than the space left in the bag we'll ignore it. DP[i][j] = DP[i-1][j] # returning answer for W space and N items return DP[N][W]

Time Complexity of the above approach is O(N*W).

Space Complexity of the above approach is O(N*W).

## Approach 3: (Space optimized solution)

In the above approach we used a 2D array to store the computed results in our DP but we can observe one thing that to compute the result for the n^{th} element we just need the results for the (n-1)^{th} element thus we do not need the rest of the result.

Therefore, if we just get the results for the (n-1)^{th} element we can reach the n^{th} result.

Thus we can reduce the size of our DP to 1D just storing the results on different weights till the previous element.

**Code For above Implementation :**

### C++ Code

int knapsack01(int W,int N,vector<int> &v,vector<int> &w) { int DP[W+1]; // Defining DP /* For N = 0 defining DP[i] = 0.*/ for(int i=0;i<W+1;i++) DP[i] = 0; for(int i=1;i<N+1;i++){ for(int j=W;j>=w[i-1];j--){ /* Taking max of both the cases i.e to take that item or to ignore it.*/ DP[j] = max(v[i-1]+DP[j-w[i-1]],DP[j]); } } /* returning answer for W space and N items */ return DP[W]; }

### Java Code

static int knapSack(int W,int N,int v[],int w[]) { int DP[] = new int[W+1]; // Defining DP /* For N = 0 defining DP[i] = 0.*/ for(int i=0;i<W+1;i++) DP[i] = 0; for(int i=1;i<N+1;i++){ for(int j=W;j>=w[i-1];j--){ /* Taking max of both the cases i.e to take that item or to ignore it.*/ int a = v[i-1]+DP[j-w[i-1]]; int b = DP[j]; if(a>b) DP[j] = a; else DP[j] = b; } } /* returning answer for W space and N items */ return DP[W]; }

### Python Code

def knapsack01(W,N,v,w) : DP = [0 for i in range(W+1)] # Defining DP for i in range(1,N+1) : for j in range(W,w[i-1]-1,-1) : # Taking max of both the cases i.e to take that # item or to ignore it. DP[j] = max(v[i-1]+DP[j-w[i-1]],DP[j]) # returning answer for W space and N items return DP[W]

**Time Complexity** of the above approach is O(N*W).

**Space Complexity** of the above approach is O(W).

**Practice Question**

## FAQs

**Can we solve the 0/1 Knapsack Problem using Backtracking?**

Yes, the recursive DP approach itself is the backtracking approach for 0/1 knapsack.

**What is the Time Complexity of 0/1 Knapsack Problem?**

Time complexity for 0/1 Knapsack problem solved using DP is O(N*W) where N denotes number of items available and W denotes the capacity of the knapsack.

**Can we solve the 0/1 Knapsack Problem using Greedy Algorithm?**

No, 0/1 Knapsack Problem cannot be solved using a greedy approach.