# 3-d DP : Ninja and his friends (DP-13)

Problem Link: Ninja and his friends

Problem Description:

We are given an ‘N*M’ matrix. Every cell of the matrix has some chocolates on it, mat[i][j] gives us the number of chocolates. We have two friends ‘Alice’ and ‘Bob’. initially, Alice is standing on the cell(0,0) and Bob is standing on the cell(0, M-1). Both of them can move only to the cells below them in these three directions: to the bottom cell (↓), to the bottom-right cell(↘), or to the bottom-left cell(↙).

When Alica and Bob visit a cell, they take all the chocolates from that cell with them. It can happen that they visit the same cell, in that case, the chocolates need to be considered only once.

They cannot go out of the boundary of the given matrix, we need to return the maximum number of chocolates that Bob and Alice can together collect.

Examples
```Example:  ```

Disclaimer: Don’t jump directly to the solution, try it out yourself first.

Memorization approach Tabulation approach Space Optimization
Memorization Approach
Algorithm / Intuition

In this question, there are two fixed starting and variable ending points, but as per the movement of Alice and Bob, we know that they will end in the last row. They have to move together at a time to the next row.

Why a Greedy Solution doesn’t work?

As we have to return the maximum chocolates collected, the first approach that comes to our mind is to take a greedy approach and always form a path by locally choosing the option that gives us more chocolates. But there is no ‘uniformity’ in the values of the matrix, therefore it can happen that whenever we are making a local choice that gives us a better path, we actually take a path that in the later stages is giving us fewer chocolates.

As a greedy solution doesn’t work, our next choice will be to try out all the possible paths. To generate all possible paths we will use recursion.

Steps to form the recursive solution:

We will first form the recursive solution by the three points mentioned in Dynamic Programming Introduction

Step 1: Express the problem in terms of indexes.

This question is slightly different from all the previous questions, here we are given two starting points from where Alice and Bob can move.

We are given an ‘N*M’ matrix. We need to define the function with four parameters  i1,j1,i2, and j2 to describe the positions of Alice and Bob at a time. If we observe, initially Alice and Bob are at the first row, and they always move to the row below them every time, so they will always be in the same row. Therefore two different variables i1 and i2, to describe their positions are redundant. We can just use single parameter i, which tells us in which row of the grid both of them are.

Therefore, we can modify the function. It now takes three parameters: i,j1, and j2. f(i,j1,j2) will give us the maximum number of chocolates collected by Alice and Bob from their current positions to the last position. Base Case:

There will be the following base cases:

• When i == N-1, it means we are at the last row, so we need to return from here. Now it can happen that at the last row, both Alice and Bob are at the same cell, in this condition we will return only chocolates collected by Alice, mat[i][j1]( as question states that the chocolates cannot be doubly calculated), otherwise we return sum of chocolates collected by both, mat[i][j1] + mat[i][j1][j2].

At every cell, we have three options to go: to the bottom cell (↓), to the bottom-right cell(↘) or to the bottom-left cell(↙)

As we are moving to the bottom cell (↓), at max we will reach the last row, from where we return, so we will never go out of the bounding index.

To move to the bottom-right cell(↘) or to the bottom-left cell(↙), it can happen that we may go out of bound as shown in the figure(below). So we need to handle it, we can return -1e9, whenever we go out of bound, in this way this path will not be selected by the calling function as we have to return the maximum chocolates. • If j1<0 or j1>=M or j2<0 or j2>=M  , then we return -1e9

The pseudocode till this step will be: Step 2: Try out all possible choices at a given index.

At every cell, we have three options to go: to the bottom cell (↓), to the bottom-right cell(↘) or to the bottom-left cell(↙)

Now, we need to understand that we want to move Alice and Bob together. Both of them can individually move three moves but say Alice moves to bottom-left, then Bob can have three different moves for Alice’s move, and so on. The following figures will help to understand this: Hence we have a total of 9 different options at every f(i,j1,j2) to move Alice and Bob. Now we can manually write these 9 options or we can observe a pattern in them, first Alice moves to one side and Bob tries all three choices, then again Alice moves, then Bob, and so on. This pattern can be easily captured by using two nested loops that change the column numbers(j1 and j2).

Note: if (j1===j2), as discussed in the base case, we will only consider chocolates collected by one of them otherwise we will consider chocolates collected by both of them. Step 3:  Take the maximum of all choices

As we have to find the maximum chocolates collected of all the possible paths, we will return the maximum of all the choices(the 9 choices of step 2). We will take a maxi variable( initialized to INT_MIN). We will update maxi to the maximum of the previous maxi and the answer of the current choice. At last, we will return maxi from our function as the answer.

The final pseudocode after steps 1, 2, and 3: Steps to memoize a recursive solution:

Before moving to the memoization steps, we need to understand the dp array we are taking. The recursive function has three parameters: i,j1, and j2. Therefore, we will also need to take a 3D DP Array. Its dimensions will be [N][M][M] because when we are moving, i can go from 0 to N-1, and j1 and j2 can go from 0 to M-1.

If we draw the recursion tree, we will see that there are overlapping subproblems. In order to convert a recursive solution the following steps will be taken:

1. Create a dp array of size [N][M][M], initialized to -1.
2. Whenever we want to find the answer of a particular row and column (say f(i,j1,j2)), we first check whether the answer is already calculated using the dp array(i.e dp[i][j1][j2]!= -1 ). If yes, simply return the value from the dp array.
3. If not, then we are finding the answer for the given values for the first time, we will use the recursive relation as usual but before returning from the function, we will set dp[i][j1][j2] to the solution we get.
Code
``````
#include <bits/stdc++.h>
using namespace std;

// Function to find the maximum chocolates that can be collected recursively
int maxChocoUtil(int i, int j1, int j2, int n, int m, vector<vector<int>> &grid, vector<vector<vector<int>>> &dp) {
// Check if the positions (j1, j2) are valid
if (j1 < 0 || j1 >= m || j2 < 0 || j2 >= m)
return -1e9; // A very large negative value to represent an invalid position

// Base case: If we are at the last row, return the chocolate at the positions (j1, j2)
if (i == n - 1) {
if (j1 == j2)
return grid[i][j1];
else
return grid[i][j1] + grid[i][j2];
}

// If the result for this state is already computed, return it
if (dp[i][j1][j2] != -1)
return dp[i][j1][j2];

int maxi = INT_MIN;

// Try all possible moves (up, left, right) for both positions (j1, j2)
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;

if (j1 == j2)
ans = grid[i][j1] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);
else
ans = grid[i][j1] + grid[i][j2] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);

// Update the maximum result
maxi = max(maxi, ans);
}
}

// Store the maximum result for this state in dp
return dp[i][j1][j2] = maxi;
}

// Function to find the maximum chocolates that can be collected
int maximumChocolates(int n, int m, vector<vector<int>> &grid) {
// Create a 3D DP array to store computed results
vector<vector<vector<int>>> dp(n, vector<vector<int>>(m, vector<int>(m, -1)));

// Call the recursive utility function to find the maximum chocolates starting from the first row
return maxChocoUtil(0, 0, m - 1, n, m, grid, dp);
}

int main() {
// Define the grid as a 2D vector
vector<vector<int>> matrix{
{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5},
};

int n = matrix.size();
int m = matrix.size();

// Call the maximumChocolates function and print the result
cout << maximumChocolates(n, m, matrix);

return 0;
}
```
```
``````
import java.util.*;

class TUF {
// Function to find the maximum number of chocolates using dynamic programming
static int maxChocoUtil(int i, int j1, int j2, int n, int m, int[][] grid,
int[][][] dp) {
// Check if j1 and j2 are valid column indices
if (j1 < 0 || j1 >= m || j2 < 0 || j2 >= m)
return (int) (Math.pow(-10, 9));

// If we are at the last row, return the sum of chocolates in the selected columns
if (i == n - 1) {
if (j1 == j2)
return grid[i][j1];
else
return grid[i][j1] + grid[i][j2];
}

// If the result for this state is already computed, return it
if (dp[i][j1][j2] != -1)
return dp[i][j1][j2];

int maxi = Integer.MIN_VALUE;
// Iterate through possible moves in the next row
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;
// If j1 and j2 are the same, add chocolates from grid[i][j1] only
if (j1 == j2)
ans = grid[i][j1] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);
else
// Add chocolates from both j1 and j2
ans = grid[i][j1] + grid[i][j2] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);
// Update maxi with the maximum result
maxi = Math.max(maxi, ans);
}
}
// Store the result in the dp array and return it
return dp[i][j1][j2] = maxi;
}

// Function to find the maximum number of chocolates
static int maximumChocolates(int n, int m, int[][] grid) {
// Create a 3D array to store computed results
int dp[][][] = new int[n][m][m];

// Initialize the dp array with -1
for (int row1[][] : dp) {
for (int row2[] : row1) {
Arrays.fill(row2, -1);
}
}

// Call the utility function to find the maximum number of chocolates
return maxChocoUtil(0, 0, m - 1, n, m, grid, dp);
}

public static void main(String args[]) {
int matrix[][] = {{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5}};
int n = matrix.length;
int m = matrix.length;

// Call the maximumChocolates function and print the result
System.out.println(maximumChocolates(n, m, matrix));
}
}
```
```
``````
import sys

# Recursive function to find the maximum chocolates collected
def maxChocoUtil(i, j1, j2, n, m, grid, dp):
# Base cases:
# - If either of the indices is out of bounds, return a large negative value
# - If we're at the last row, return the sum of chocolates in the two selected columns
if j1 < 0 or j1 >= m or j2 < 0 or j2 >= m:
return int(-1e9)
if i == n - 1:
if j1 == j2:
return grid[i][j1]
else:
return grid[i][j1] + grid[i][j2]

# If the result for these indices has already been computed, return it
if dp[i][j1][j2] != -1:
return dp[i][j1][j2]

# Initialize the maximum chocolates collected to a large negative value
maxi = -sys.maxsize

# Iterate through the adjacent cells in the next row
for di in range(-1, 2):
for dj in range(-1, 2):
ans = 0
if j1 == j2:
ans = grid[i][j1] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp)
else:
ans = grid[i][j1] + grid[i][j2] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp)
maxi = max(maxi, ans)

# Store the maximum chocolates collected in the memoization table
dp[i][j1][j2] = maxi
return maxi

# Function to find the maximum chocolates collected
def maximumChocolates(n, m, grid):
# Initialize a memoization table with -1 values
dp = [[[-1 for j in range(m)] for i in range(m)] for k in range(n)]

# Start the recursion from the first row, columns 0 and m-1
return maxChocoUtil(0, 0, m - 1, n, m, grid, dp)

def main():
# Define the input matrix and its dimensions
matrix = [[2, 3, 1, 2], [3, 4, 2, 2], [5, 6, 3, 5]]
n = len(matrix)
m = len(matrix)

# Call the maximumChocolates function and print the result
print(maximumChocolates(n, m, matrix))

if __name__ == "__main__":
main()
```
```
``````
function maxChocoUtil(i, j1, j2, n, m, grid, dp) {
// Check if the indices are out of bounds
if (j1 < 0 || j1 >= m || j2 < 0 || j2 >= m) {
return -1e9; // A very large negative value for invalid states
}

// Base case: if we are at the last row
if (i == n - 1) {
// If both indices are the same, return the value at that position
if (j1 == j2) {
return grid[i][j1];
} else {
// If the indices are different, return the sum of values at both positions
return grid[i][j1] + grid[i][j2];
}
}

// If the result for this state is already computed, return it
if (dp[i][j1][j2] != -1) {
return dp[i][j1][j2];
}

let maxi = Number.MIN_SAFE_INTEGER; // Initialize the maximum value to a very small number

// Iterate through neighboring positions
for (let di = -1; di <= 1; di++) {
for (let dj = -1; dj <= 1; dj++) {
let ans;
if (j1 == j2) {
ans = grid[i][j1] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);
} else {
ans = grid[i][j1] + grid[i][j2] + maxChocoUtil(i + 1, j1 + di, j2 + dj, n, m, grid, dp);
}
// Update the maximum value
maxi = Math.max(maxi, ans);
}
}

// Store the maximum value in the dp array and return it
dp[i][j1][j2] = maxi;
return maxi;
}

function maximumChocolates(n, m, grid) {
// Initialize a 3D dp array with -1 values
const dp = new Array(n).fill(null).map(() =>
new Array(m).fill(null).map(() =>
new Array(m).fill(-1)
)
);

// Call the recursive utility function to find the maximum chocolates
return maxChocoUtil(0, 0, m - 1, n, m, grid, dp);
}

function main() {
const matrix = [
[2, 3, 1, 2],
[3, 4, 2, 2],
[5, 6, 3, 5]
];

const n = matrix.length;
const m = matrix.length;

console.log(maximumChocolates(n, m, matrix));
}

// Call the main function to execute the code
main();
```
```

Output: 21

Complexity Analysis

Time Complexity: O(N*M*M) * 9

Reason: At max, there will be N*M*M calls of recursion to solve a new problem and in every call, two nested loops together run for 9 times.

Space Complexity: O(N) + O(N*M*M)

Reason: We are using a recursion stack space: O(N), where N is the path length and an external DP Array of size ‘N*M*M’.

Tabulation Approach
Algorithm / Intuition

Steps to convert Recursive Solution to Tabulation one.

For the tabulation approach, it is better to understand what a cell in the 3D DP array means. As we had done in memoization, we will initialize a dp[] array of size [N][M][M].

So now, when we say dp, what does it mean? It means that we are getting the value of the maximum chocolates collected by Alice and Bob, when Alice is at (2,0) and Bob is at (2,3).

The below figure gives us a bit more clarity. Next, we need to initialize the base value conditions. In the recursive code, our base condition is when we reach the last row, therefore in our dp array, we will also initialize dp[n-1][][], i.e (the last plane of 3D Array) as the base condition. Dp[n-1][j1][j2] means Alice is at (n-1,j1) and Bob is at (n-1,j2). As this is the last row, its value will be equal to mat[i][j1], if (j1==j2) and mat[i][j1] + mat[i][j2] otherwise.

Once we have filled the last plane, we can move to the second-last plane and so on, we will need three nested loops to do this traversal.

The steps to convert to the tabular solution are given below:

• Declare a dp[] array of size [N][M][M]
• First, initialize the base condition values as explained above.
• We will then move from dp[n-2][][] to dp[][]. We will set three nested loops to do this traversal.
• Inside the three nested loops( say i,j1,j2 as loop variables), we will use the recursive relations, i.e we will again set two nested loops to try all the nine options.
• The outer three loops are just for traversal, and the inner two loops that run for 9 times mainly decide, what should be the value of the cell. If you are getting confused, please see the code.
• Inside the inner two nested loops, we will calculate an answer as we had done in the recursive relation, but this time using values from the next plane of the 3D DP Array( dp[i+1][x][y] instead of recursive calls, where x and y will vary according to inner 2 nested loops).
• At last, we will set dp[i][j1][j2] as the maximum of all the 9 options.
• After the outer three nested loops iteration has ended, we will return dp[m-1] as our answer.
Code
``````
#include <bits/stdc++.h>
using namespace std;

// Function to find the maximum chocolates that can be collected
int maximumChocolates(int n, int m, vector<vector<int>> &grid) {
// Create a 3D DP array to store computed results
vector<vector<vector<int>>> dp(n, vector<vector<int>>(m, vector<int>(m, 0)));

// Initialize the DP array for the last row
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
if (j1 == j2)
dp[n - 1][j1][j2] = grid[n - 1][j1];
else
dp[n - 1][j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}

// Outer nested loops for traversing the DP array from the second-to-last row up to the first row
for (int i = n - 2; i >= 0; i--) {
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
int maxi = INT_MIN;

// Inner nested loops to try out 9 options (diagonal moves)
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;

if (j1 == j2)
ans = grid[i][j1];
else
ans = grid[i][j1] + grid[i][j2];

// Check if the move is valid (within the grid boundaries)
if ((j1 + di < 0 || j1 + di >= m) || (j2 + dj < 0 || j2 + dj >= m))
ans += -1e9; // A very large negative value to represent an invalid move
else
ans += dp[i + 1][j1 + di][j2 + dj]; // Include the DP result from the next row

maxi = max(ans, maxi); // Update the maximum result
}
}
dp[i][j1][j2] = maxi; // Store the maximum result for this state in the DP array
}
}
}

// The maximum chocolates that can be collected is stored at the top-left corner of the DP array
return dp[m - 1];
}

int main() {
// Define the grid as a 2D vector
vector<vector<int>> matrix{
{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5},
};

int n = matrix.size(); // Number of rows
int m = matrix.size(); // Number of columns

// Call the maximumChocolates function and print the result
cout << maximumChocolates(n, m, matrix);

return 0;
}
```
```
``````
import java.util.*;

class TUF {
// Function to find the maximum number of chocolates using dynamic programming
static int maximumChocolates(int n, int m, int[][] grid) {
// Create a 3D array to store computed results
int dp[][][] = new int[n][m][m];

// Initialize the dp array with values from the last row of the grid
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
if (j1 == j2)
dp[n - 1][j1][j2] = grid[n - 1][j1];
else
dp[n - 1][j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}

// Outer nested loops to traverse the DP array from the second last row to the first row
for (int i = n - 2; i >= 0; i--) {
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
int maxi = Integer.MIN_VALUE;

// Inner nested loops to try out 9 options
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;

if (j1 == j2)
ans = grid[i][j1];
else
ans = grid[i][j1] + grid[i][j2];

// Check if the indices are valid
if ((j1 + di < 0 || j1 + di >= m) || (j2 + dj < 0 || j2 + dj >= m))
ans += (int) Math.pow(-10, 9);
else
ans += dp[i + 1][j1 + di][j2 + dj];

// Update maxi with the maximum result
maxi = Math.max(ans, maxi);
}
}
// Store the result in the dp array
dp[i][j1][j2] = maxi;
}
}
}

// The final result is stored at the top row (first row) of the dp array
return dp[m - 1];
}

public static void main(String args[]) {
int matrix[][] = {{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5}};
int n = matrix.length;
int m = matrix.length;

// Call the maximumChocolates function and print the result
System.out.println(maximumChocolates(n, m, matrix));
}
}
```
```
``````
import sys

# Function to find the maximum chocolates collected
def maximumChocolates(n, m, grid):
# Initialize a 3D memoization table dp with zeros
dp = [[[0 for _ in range(m)] for _ in range(m)] for _ in range(n)]

# Initialize the values for the last row of dp based on grid values
for j1 in range(m):
for j2 in range(m):
if j1 == j2:
dp[n - 1][j1][j2] = grid[n - 1][j1]
else:
dp[n - 1][j1][j2] = grid[n - 1][j1] + grid[n - 1][j2]

# Iterate through rows from the second-to-last row to the first row
for i in range(n - 2, -1, -1):
for j1 in range(m):
for j2 in range(m):
maxi = -sys.maxsize

# Try out 9 possible options by changing the indices
for di in range(-1, 2):
for dj in range(-1, 2):
ans = 0
if j1 == j2:
ans = grid[i][j1]
else:
ans = grid[i][j1] + grid[i][j2]

if ((j1 + di < 0 or j1 + di >= m) or (j2 + dj < 0 or j2 + dj >= m)):
ans += int(-1e9)  # A large negative value if out of bounds
else:
ans += dp[i + 1][j1 + di][j2 + dj]  # Add the value from the next row

maxi = max(ans, maxi)

# Store the maximum chocolates collected in the memoization table
dp[i][j1][j2] = maxi

# Return the maximum chocolates collected in the top row and the last column
return dp[m - 1]

def main():
# Define the input matrix and its dimensions
matrix = [[2, 3, 1, 2], [3, 4, 2, 2], [5, 6, 3, 5]]
n = len(matrix)
m = len(matrix)

# Call the maximumChocolates function and print the result
print(maximumChocolates(n, m, matrix))

if __name__ == '__main__':
main()
```
```
``````
function maximumChocolates(n, m, grid) {
// Initialize a 3D dp array with zeros
const dp = new Array(n).fill(null).map(() =>
new Array(m).fill(null).map(() =>
new Array(m).fill(0)
)
);

// Initialize dp array for the last row based on grid values
for (let j1 = 0; j1 < m; j1++) {
for (let j2 = 0; j2 < m; j2++) {
if (j1 === j2) {
dp[n - 1][j1][j2] = grid[n - 1][j1];
} else {
dp[n - 1][j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}
}

// Iterate through rows in reverse order
for (let i = n - 2; i >= 0; i--) {
for (let j1 = 0; j1 < m; j1++) {
for (let j2 = 0; j2 < m; j2++) {
let maxi = Number.MIN_SAFE_INTEGER;

// Iterate through all possible move combinations
for (let di = -1; di <= 1; di++) {
for (let dj = -1; dj <= 1; dj++) {
let ans;

if (j1 === j2) {
ans = grid[i][j1];
} else {
ans = grid[i][j1] + grid[i][j2];
}

// Check if the move is valid (within grid bounds)
if (
j1 + di >= 0 && j1 + di < m &&
j2 + dj >= 0 && j2 + dj < m
) {
ans += dp[i + 1][j1 + di][j2 + dj];
} else {
ans += -1e9; // A very large negative value for invalid moves
}

maxi = Math.max(ans, maxi);
}
}

dp[i][j1][j2] = maxi;
}
}
}

// The maximum chocolates will be stored in dp[m - 1]
return dp[m - 1];
}

function main() {
const matrix = [
[2, 3, 1, 2],
[3, 4, 2, 2],
[5, 6, 3, 5]
];

const n = matrix.length;
const m = matrix.length;

console.log(maximumChocolates(n, m, matrix));
}

// Call the main function to execute the code
main();
```
```

Output: 21

Complexity Analysis

Time Complexity: O(N*M*M)*9

Reason: The outer nested loops run for (N*M*M) times and the inner two nested loops run for 9 times.

Space Complexity: O(N*M*M)

Reason: We are using an external array of size ‘N*M*M’. The stack space will be eliminated.

Space Optimization Approach
Algorithm / Intuition

If we look closely, to compute dp[i][j1][j2], we need values only from dp[i+1][][]. Therefore it is not necessary to store a three-dimensional array. Instead, we can store a two-dimensional array and update it as we move from one plane to the other in the 3D Array.

The Steps to space optimize the tabulation approach are as follows:

• Initially, we can take a dummy 2D Array ( say front). We initialize this 2D Array as we had done in the Tabulation Approach.
• Next, we also initialize a 2D Array( say cur), which we will need in the traversal.
• Now we set our three nested loops to traverse the 3D Array, from the second last plane.
• Following the same approach as we did in the tabulation approach, we find the maximum number of chocolates collected at each cell. To calculate it we have all the values in our ‘front’ 2D Array.
• Previously, we assigned dp[i][j1][j2] to maxi, now we will simply assign cur[j1][j2] to maxi.
• Then whenever the plane of the 3D DP(the first parameter) is going to change, we assign the front to cur.

At last, we will return front[m-1] as our answer.

Code
``````
#include <bits/stdc++.h>
using namespace std;

// Function to find the maximum chocolates that can be collected
int maximumChocolates(int n, int m, vector<vector<int>> &grid) {
// Create two 2D vectors 'front' and 'cur' to store computed results
vector<vector<int>> front(m, vector<int>(m, 0));
vector<vector<int>> cur(m, vector<int>(m, 0));

// Initialize 'front' for the last row
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
if (j1 == j2)
front[j1][j2] = grid[n - 1][j1];
else
front[j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}

// Outer nested loops for traversing the DP array from the second-to-last row up to the first row
for (int i = n - 2; i >= 0; i--) {
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
int maxi = INT_MIN;

// Inner nested loops to try out 9 options (diagonal moves)
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;

if (j1 == j2)
ans = grid[i][j1];
else
ans = grid[i][j1] + grid[i][j2];

// Check if the move is valid (within the grid boundaries)
if ((j1 + di < 0 || j1 + di >= m) || (j2 + dj < 0 || j2 + dj >= m))
ans += -1e9; // A very large negative value to represent an invalid move
else
ans += front[j1 + di][j2 + dj]; // Include the value from the 'front' array

maxi = max(ans, maxi); // Update the maximum result
}
}
cur[j1][j2] = maxi; // Store the maximum result for this state in the 'cur' array
}
}
front = cur; // Update 'front' with the values from 'cur' for the next iteration
}

// The maximum chocolates that can be collected is stored at the top-left corner of the 'front' array
return front[m - 1];
}

int main() {
// Define the grid as a 2D vector
vector<vector<int>> matrix{
{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5},
};

int n = matrix.size(); // Number of rows
int m = matrix.size(); // Number of columns

// Call the maximumChocolates function and print the result
cout << maximumChocolates(n, m, matrix);

return 0;
}
```
```
``````
import java.util.*;

class TUF {
// Function to find the maximum number of chocolates using dynamic programming
static int maximumChocolates(int n, int m, int[][] grid) {
// Create two 2D arrays to store computed results: front and cur
int[][] front = new int[m][m];
int[][] cur = new int[m][m];

// Initialize the front array with values from the last row of the grid
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
if (j1 == j2)
front[j1][j2] = grid[n - 1][j1];
else
front[j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}

// Outer nested loops to traverse the DP array from the second last row to the first row
for (int i = n - 2; i >= 0; i--) {
for (int j1 = 0; j1 < m; j1++) {
for (int j2 = 0; j2 < m; j2++) {
int maxi = Integer.MIN_VALUE;

// Inner nested loops to try out 9 options
for (int di = -1; di <= 1; di++) {
for (int dj = -1; dj <= 1; dj++) {
int ans;

if (j1 == j2)
ans = grid[i][j1];
else
ans = grid[i][j1] + grid[i][j2];

// Check if the indices are valid
if ((j1 + di < 0 || j1 + di >= m) || (j2 + dj < 0 || j2 + dj >= m))
ans += (int) Math.pow(-10, 9);
else
ans += front[j1 + di][j2 + dj];

// Update maxi with the maximum result
maxi = Math.max(ans, maxi);
}
}
// Store the result in the cur array
cur[j1][j2] = maxi;
}
}

// Update the front array with the values from the cur array for the next row
for (int a = 0; a < m; a++) {
front[a] = cur[a].clone();
}
}

// The final result is stored at the top left corner of the front array
return front[m - 1];
}

public static void main(String args[]) {
int matrix[][] = {{2, 3, 1, 2},
{3, 4, 2, 2},
{5, 6, 3, 5}};

int n = matrix.length;
int m = matrix.length;

// Call the maximumChocolates function and print the result
System.out.println(maximumChocolates(n, m, matrix));
}
}
```
```
``````
import sys

# Function to find the maximum chocolates collected
def maximumChocolates(n, m, grid):
# Initialize two matrices: front (for the current row) and cur (for the next row)
front = [ * m for _ in range(m)]
cur = [ * m for _ in range(m)]

# Initialize the values for the last row of front based on grid values
for j1 in range(m):
for j2 in range(m):
if j1 == j2:
front[j1][j2] = grid[n - 1][j1]
else:
front[j1][j2] = grid[n - 1][j1] + grid[n - 1][j2]

# Iterate through rows from the second-to-last row to the first row
for i in range(n - 2, -1, -1):
for j1 in range(m):
for j2 in range(m):
maxi = -sys.maxsize

# Try out 9 possible options by changing the indices
for di in range(-1, 2):
for dj in range(-1, 2):
ans = 0
if j1 == j2:
ans = grid[i][j1]
else:
ans = grid[i][j1] + grid[i][j2]

if ((j1 + di < 0 or j1 + di >= m) or (j2 + dj < 0 or j2 + dj >= m)):
ans += int(-1e9)  # A large negative value if out of bounds
else:
ans += front[j1 + di][j2 + dj]  # Add the value from the current front row

maxi = max(ans, maxi)
cur[j1][j2] = maxi

# Update front with the values of cur for the next iteration
front = [row[:] for row in cur]

# Return the maximum chocolates collected in the top-left corner of front
return front[m - 1]

def main():
# Define the input matrix and its dimensions
matrix = [[2, 3, 1, 2], [3, 4, 2, 2], [5, 6, 3, 5]]
n = len(matrix)
m = len(matrix)

# Call the maximumChocolates function and print the result
print(maximumChocolates(n, m, matrix))

if __name__ == '__main__':
main()
```
```
``````
function maximumChocolates(n, m, grid) {
// Initialize two 2D arrays: front and cur
let front = new Array(m).fill(null).map(() => new Array(m).fill(0));
let cur = new Array(m).fill(null).map(() => new Array(m).fill(0));

// Initialize front array for the last row based on grid values
for (let j1 = 0; j1 < m; j1++) {
for (let j2 = 0; j2 < m; j2++) {
if (j1 === j2) {
front[j1][j2] = grid[n - 1][j1];
} else {
front[j1][j2] = grid[n - 1][j1] + grid[n - 1][j2];
}
}
}

// Outer nested loops for traversing the DP array
for (let i = n - 2; i >= 0; i--) {
for (let j1 = 0; j1 < m; j1++) {
for (let j2 = 0; j2 < m; j2++) {
let maxi = Number.MIN_SAFE_INTEGER;

// Inner nested loops to try out 9 options
for (let di = -1; di <= 1; di++) {
for (let dj = -1; dj <= 1; dj++) {
let ans;

if (j1 === j2) {
ans = grid[i][j1];
} else {
ans = grid[i][j1] + grid[i][j2];
}

// Check if the move is valid (within grid bounds)
if (
j1 + di >= 0 && j1 + di < m &&
j2 + dj >= 0 && j2 + dj < m
) {
ans += front[j1 + di][j2 + dj];
} else {
ans += -1e9; // A very large negative value for invalid moves
}

maxi = Math.max(ans, maxi);
}
}
cur[j1][j2] = maxi;
}
}
// Update the front array with values from the cur array
front = [...cur];
}

// The maximum chocolates will be stored in front[m - 1]
return front[m - 1];
}

function main() {
const matrix = [
[2, 3, 1, 2],
[3, 4, 2, 2],
[5, 6, 3, 5]
];

const n = matrix.length;
const m = matrix.length;

console.log(maximumChocolates(n, m, matrix));
}

// Call the main function to execute the code
main();
```
```

Output:21

Complexity Analysis

Time Complexity: O(N*M*M)*9

Reason: The outer nested loops run for (N*M*M) times and the inner two nested loops run for 9 times.

Space Complexity: O(M*M)

Reason: We are using an external array of size ‘M*M’.

Video Explanation