# 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++ 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 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

 `# 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# 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. `

 ` `

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

 `// Memoization based Java program to find the largest ` `// sum zigzag sequence ` `class` `GFG ` `{ ` ` `  `static` `int` `MAX = ``100``; ` `static` `int` `[][]dp = ``new` `int``[MAX][MAX]; ` ` `  `// 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` `(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 = Math.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. ` `static` `int` `largestZigZag(``int` `mat[][], ``int` `n) ` `{ ` `    ``for` `(``int` `i = ``0``; i < MAX; i++) ` `        ``for` `(``int` `k = ``0``; k < MAX; k++) ` `                ``dp[i][k] = -``1``; ` `     `  `    ``// 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(String[] args) ` `{ ` `    ``int` `n = ``3``; ` `    ``int` `mat[][] = { {``4``, ``2``, ``1``}, ` `                    ``{``3``, ``9``, ``6``}, ` `                    ``{``11``, ``3``, ``15``}}; ` `    ``System.out.print(``"Largest zigzag sum: "` `+  ` `                        ``largestZigZag(mat, n)); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

 `# Memoization based Python3 program to find the largest ` `# sum zigzag sequence ` `MAX` `=` `100``; ` ` `  `dp ``=` `[[``0` `for` `i ``in` `range``(``MAX``)] ``for` `j ``in` `range``(``MAX``)] ` ` `  `# Returns largest sum of a Zigzag sequence starting ` `# from (i, j) and ending at a bottom cell. ` `def` `largestZigZagSumRec(mat, i, j, n): ` `    ``if` `(dp[i][j] !``=` `-``1``): ` `        ``return` `dp[i][j]; ` ` `  `    ``# If we have reached bottom ` `    ``if` `(i ``=``=` `n ``-` `1``): ` `        ``dp[i][j] ``=` `mat[i][j]; ` `        ``return` `(dp[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)); ` `    ``dp[i][j] ``=` `(zzs ``+` `mat[i][j]); ` `    ``return` `(dp[i][j]); ` ` `  `# Returns largest possible sum of a Zizag sequence ` `# starting from top and ending at bottom. ` `def` `largestZigZag(mat, n): ` `    ``for` `i ``in` `range``(``MAX``): ` `        ``for` `k ``in` `range``(``MAX``): ` `            ``dp[i][k] ``=` `-``1``; ` ` `  `    ``# 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 Rajput-Ji `

 `// Memoization based C# program to find the largest ` `// sum zigzag sequence ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `static` `int` `MAX = 100; ` `static` `int` `[,]dp = ``new` `int``[MAX, MAX]; ` ` `  `// 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` `(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 = Math.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. ` `static` `int` `largestZigZag(``int` `[,]mat, ``int` `n) ` `{ ` `    ``for` `(``int` `i = 0; i < MAX; i++) ` `        ``for` `(``int` `k = 0; k < MAX; k++) ` `                ``dp[i, k] = -1; ` `     `  `    ``// 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(String[] args) ` `{ ` `    ``int` `n = 3; ` `    ``int` `[,]mat = { {4, 2, 1}, ` `                    ``{3, 9, 6}, ` `                    ``{11, 3, 15}}; ` `    ``Console.Write(``"Largest zigzag sum: "` `+  ` `                        ``largestZigZag(mat, n)); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

```Largest zigzag sum: 28
```