# 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

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 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

## 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

## 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.

/div>

## PHP



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 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

Output:

28