Largest sum Zigzag sequence in a matrix

Given a matrix of size n x n, find sum of the Zigzag sequence with the largest sum. A zigzag sequence starts from the top and ends at the bottom. Two consecutive elements of sequence cannot belong to same column.
Examples:

Input : mat[][] = 3  1  2
                  4  8  5
                  6  9  7
Output : 18
Zigzag sequence is: 3->8->7
Another such sequence is 2->4->7

Input : mat[][] =  4  2  1
                   3  9  6
                  11  3 15
Output : 28

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

This problem has Optimal Substructure.

Maximum Zigzag sum starting from arr[i][j] to a 
bottom cell can be written as :
zzs(i, j) = arr[i][j] + max(zzs(i+1, k)), 
               where k = 0, 1, 2 and k != j
zzs(i, j) = arr[i][j], if i = n-1 

We have to find the largest among all as
Result = zzs(0, j) where 0 <= j < n

C++

// C++ program to find the largest sum zigzag sequence 
#include <bits/stdc++.h> 
using namespace std; 
  
const int MAX = 100; 
  
// Returns largest sum of a Zigzag sequence starting 
// from (i, j) and ending at a bottom cell. 
int largestZigZagSumRec(int mat[][MAX], int i, 
                                int j, int n) 
{ 
   // If we have reached bottom 
   if (i == n-1) 
     return mat[i][j]; 
  
   // Find the largest sum by considering all 
   // possible next elements in sequence. 
   int zzs = 0; 
   for (int k=0; k<n; k++) 
     if (k != j) 
       zzs = max(zzs, largestZigZagSumRec(mat, i+1, k, n)); 
  
   return zzs + mat[i][j]; 
} 
  
// Returns largest possible sum of a Zizag sequence 
// starting from top and ending at bottom. 
int largestZigZag(int mat[][MAX], int n) 
{ 
   // Consider all cells of top row as starting point 
   int res = 0; 
   for (int j=0; j<n; j++) 
     res = max(res, largestZigZagSumRec(mat, 0, j, n)); 
  
   return res; 
} 
  
// Driver program to test above 
int main() 
{ 
    int n = 3; 
    int  mat[][MAX] = { {4, 2, 1}, 
                        {3, 9, 6}, 
                        {11, 3, 15}}; 
    cout << "Largest zigzag sum: " << largestZigZag(mat, n); 
    return 0; 
} 

Java

// Java program to find the largest sum  
// zigzag sequence 
import java.io.*; 
  
class GFG { 
  
    static int MAX = 100; 
      
    // Returns largest sum of a Zigzag  
    // sequence starting from (i, j)  
    // and ending at a bottom cell. 
    static int largestZigZagSumRec(int mat[][], 
                            int i, int j, int n) 
    { 
          
        // If we have reached bottom 
        if (i == n-1) 
            return mat[i][j]; 
          
        // Find the largest sum by considering all 
        // possible next elements in sequence. 
        int zzs = 0; 
          
        for (int k=0; k<n; k++) 
            if (k != j) 
            zzs = Math.max(zzs,  
               largestZigZagSumRec(mat, i+1, k, n)); 
          
        return zzs + mat[i][j]; 
    } 
      
    // Returns largest possible sum of a Zizag 
    // sequence starting from top and ending  
    // at bottom. 
    static int largestZigZag(int mat[][], int n) 
    { 
        // Consider all cells of top row as starting 
        // point 
        int res = 0; 
        for (int j=0; j<n; j++) 
            res = Math.max(res,  
                   largestZigZagSumRec(mat, 0, j, n)); 
          
        return res; 
    } 
      
    // Driver program to test above 
    public static void main (String[] args) 
    { 
        int n = 3; 
          
        int mat[][] = { {4, 2, 1}, 
                        {3, 9, 6}, 
                        {11, 3, 15} }; 
        System.out.println( "Largest zigzag sum: " 
                       + largestZigZag(mat, n)); 
    } 
} 
  
// This code is contributed by anuj_67. 

Python 3

# Python3 program to find the largest 
# sum zigzag sequence 
MAX = 100
  
# Returns largest sum of a Zigzag  
# sequence starting from (i, j) and 
# ending at a bottom cell. 
def largestZigZagSumRec( mat, i, j, n): 
      
    # If we have reached bottom 
    if (i == n-1): 
        return mat[i][j] 
      
    # Find the largest sum by considering all 
    # possible next elements in sequence. 
    zzs = 0
    for k in range(n): 
        if (k != j): 
            zzs = max(zzs, largestZigZagSumRec(mat, i + 1, k, n)) 
      
    return zzs + mat[i][j] 
  
# Returns largest possible sum of a  
# Zizag sequence starting from top  
# and ending at bottom. 
def largestZigZag(mat, n): 
          
    # Consider all cells of top row as 
    # starting point 
    res = 0
    for j in range(n): 
        res = max(res, largestZigZagSumRec(mat, 0, j, n)) 
      
    return res 
  
# Driver Code 
if __name__ == "__main__": 
    n = 3
    mat = [ [4, 2, 1], 
            [3, 9, 6], 
            [11, 3, 15]] 
    print("Largest zigzag sum: " ,  
           largestZigZag(mat, n)) 
  
# This code is contributed by ChitraNayal 

C#

// C# program to find the largest sum  
// zigzag sequence 
using System; 
class GFG { 
  
    // static int MAX = 100; 
      
    // Returns largest sum of a Zigzag  
    // sequence starting from (i, j)  
    // and ending at a bottom cell. 
    static int largestZigZagSumRec(int [,]mat, 
                          int i, int j, int n) 
    { 
          
        // If we have reached bottom 
        if (i == n-1) 
            return mat[i,j]; 
          
        // Find the largest sum by considering all 
        // possible next elements in sequence. 
        int zzs = 0; 
          
        for (int k = 0; k < n; k++) 
            if (k != j) 
            zzs = Math.Max(zzs, largestZigZagSumRec(mat,  
                                           i + 1, k, n)); 
          
        return zzs + mat[i,j]; 
    } 
      
    // Returns largest possible 
    // sum of a Zizag sequence  
    // starting from top and ending  
    // at bottom. 
    static int largestZigZag(int [,]mat, int n) 
    { 
          
        // Consider all cells of  
        // top row as starting 
        // point 
        int res = 0; 
        for (int j = 0; j < n; j++) 
            res = Math.Max(res,  
                largestZigZagSumRec(mat, 0, j, n)); 
          
        return res; 
    } 
      
    // Driver Code 
    public static void Main () 
    { 
        int n = 3; 
        int [,]mat = {{4, 2, 1}, 
                      {3, 9, 6}, 
                      {11, 3, 15}}; 
        Console.WriteLine("Largest zigzag sum: "
                           + largestZigZag(mat, n)); 
    } 
} 
  
// This code is contributed by anuj_67. 

PHP

<?php 
// PHP program to find the  
// largest sum zigzag sequence 
  
$MAX = 100; 
  
// Returns largest sum of a  
// Zigzag sequence starting 
// from (i, j) and ending at 
// a bottom cell. 
function largestZigZagSumRec($mat, $i, 
                              $j, $n) 
{ 
    // If we have reached bottom 
    if ($i == $n - 1) 
        return $mat[$i][$j]; 
      
    // Find the largest sum 
    // by considering all 
    // possible next elements 
    // in sequence. 
    $zzs = 0; 
    for ($k = 0; $k < $n; $k++) 
        if ($k != $j) 
        $zzs = max($zzs, largestZigZagSumRec($mat,  
                                $i + 1, $k, $n)); 
      
    return $zzs + $mat[$i][$j]; 
} 
  
// Returns largest possible 
// sum of a Zizag sequence 
// starting from top and  
// ending at bottom. 
function largestZigZag( $mat, $n) 
{ 
      
    // Consider all cells of top 
    // row as starting point 
    $res = 0; 
    for ($j = 0; $j < $n; $j++) 
        $res = max($res, largestZigZagSumRec( 
                            $mat, 0, $j, $n)); 
      
    return $res; 
} 
  
    // Driver Code 
    $n = 3; 
    $mat = array(array(4, 2, 1), 
                 array(3, 9, 6), 
                 array(11, 3, 15)); 
    echo "Largest zigzag sum: " , largestZigZag($mat, $n); 
      
// This code is contributed by anuj_67. 
?> 

Output:

Largest zigzag sum: 28

Overlapping Subproblems
Considering the above implementation, for a matrix mat[][] of size 3 x 3, to find zigzag sum(zzs) for an element mat(i,j), the following recursion tree is formed.

Recursion tree for cell (0, 0)
             zzs(0,0)                                
           /         \                               
    zzs(1,1)           zzs(1,2)                      
    /     \            /      \                      
zzs(2,0)  zzs(2,2)  zzs(2,0)  zzs(2,1)               


Recursion tree for cell (0, 1)
            zzs(0,1)
           /         \              
    zzs(1,0)          zzs(1,2)
    /     \            /      \    
zzs(2,1)  zzs(2,2)  zzs(2,0)  zzs(2,1)

Recursion tree for cell (0, 2)
             zzs(0,2)
           /         \                                             
    zzs(1,0)           zzs(1,1)                             
    /     \            /      \                             
 zzs(2,1)  zzs(2,2)  zzs(2,0)  zzs(2,2)

We can see that there are many subproblems which are solved again and again. So this problem has Overlapping Substructure property and recomputation of same subproblems can be avoided by either using Memoization or Tabulation. Following is a tabluated implementation for the LIS problem.

// Memoization based C++ program to find the largest 
// sum zigzag sequence 
#include <bits/stdc++.h> 
using namespace std; 
  
const int MAX = 100; 
int dp[MAX][MAX]; 
  
// Returns largest sum of a Zigzag sequence starting 
// from (i, j) and ending at a bottom cell. 
int largestZigZagSumRec(int mat[][MAX], int i, 
                                int j, int n) 
{ 
   if (dp[i][j] != -1) 
      return dp[i][j]; 
  
   // If we have reached bottom 
   if (i == n-1) 
     return (dp[i][j] = mat[i][j]); 
  
   // Find the largest sum by considering all 
   // possible next elements in sequence. 
   int zzs = 0; 
   for (int k=0; k<n; k++) 
     if (k != j) 
       zzs = max(zzs, largestZigZagSumRec(mat, i+1, k, n)); 
  
   return (dp[i][j] = (zzs + mat[i][j])); 
} 
  
// Returns largest possible sum of a Zizag sequence 
// starting from top and ending at bottom. 
int largestZigZag(int mat[][MAX], int n) 
{ 
   memset(dp, -1, sizeof(dp)); 
  
   // Consider all cells of top row as starting point 
   int res = 0; 
   for (int j=0; j<n; j++) 
     res = max(res, largestZigZagSumRec(mat, 0, j, n)); 
  
   return res; 
} 
  
// Driver program to test above 
int main() 
{ 
    int n = 3; 
    int  mat[][MAX] = { {4, 2, 1}, 
                        {3, 9, 6}, 
                        {11, 3, 15}}; 
    cout << "Largest zigzag sum: " << largestZigZag(mat, n); 
    return 0; 
} 

Output:

References: Asked in Directi

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