Given a grid of numbers, find maximum length Snake sequence and print it. If multiple snake sequences exists with the maximum length, print any one of them.

A snake sequence is made up of adjacent numbers in the grid such that for each number, the number on the right or the number below it is +1 or -1 its value. For example, if you are at location (x, y) in the grid, you can either move right i.e. (x, y+1) if that number is ± 1 or move down i.e. (x+1, y) if that number is ± 1.

For example,

**9**, 6, 5, 2

**8, 7, 6, 5**

7, 3, 1, **6**

1, 1, 1, **7**

In above grid, the longest snake sequence is: (9, 8, 7, 6, 5, 6, 7)

Below figure shows all possible paths –

**We strongly recommend you to minimize your browser and try this yourself first.**

The idea is to use Dynamic Programming. For each cell of the matrix, we keep maximum length of a snake which ends in current cell. The maximum length snake sequence will have maximum value. The maximum value cell will correspond to tail of the snake. In order to print the snake, we need to backtrack from tail all the way back to snake’s head.

Let** T[i][i]** represent maximum length of a snake which ends at cell (i, j), then for given matrix M, the DP relation is defined as –

T[0][0] = 0

T[i][j] = max(T[i][j], T[i][j – 1] + 1) if M[i][j] = M[i][j – 1] ± 1

T[i][j] = max(T[i][j], T[i – 1][j] + 1) if M[i][j] = M[i – 1][j] ± 1

Below is C++ implementation of the idea –

## CPP

// C++ program to find maximum length // Snake sequence and print it #include <bits/stdc++.h> using namespace std; #define M 4 #define N 4 struct Point { int x, y; }; // Function to find maximum length Snake sequence path // (i, j) corresponds to tail of the snake list<Point> findPath(int grid[M][N], int mat[M][N], int i, int j) { list<Point> path; Point pt = {i, j}; path.push_front(pt); while (grid[i][j] != 0) { if (i > 0 && grid[i][j] - 1 == grid[i - 1][j]) { pt = {i - 1, j}; path.push_front(pt); i--; } else if (j > 0 && grid[i][j] - 1 == grid[i][j - 1]) { pt = {i, j - 1}; path.push_front(pt); j--; } } return path; } // Function to find maximum length Snake sequence void findSnakeSequence(int mat[M][N]) { // table to store results of subproblems int lookup[M][N]; // initialize by 0 memset(lookup, 0, sizeof lookup); // stores maximum length of Snake sequence int max_len = 0; // store cordinates to snake's tail int max_row = 0; int max_col = 0; // fill the table in bottom-up fashion for (int i = 0; i < M; i++) { for (int j = 0; j < N; j++) { // do except for (0, 0) cell if (i || j) { // look above if (i > 0 && abs(mat[i - 1][j] - mat[i][j]) == 1) { lookup[i][j] = max(lookup[i][j], lookup[i - 1][j] + 1); if (max_len < lookup[i][j]) { max_len = lookup[i][j]; max_row = i, max_col = j; } } // look left if (j > 0 && abs(mat[i][j - 1] - mat[i][j]) == 1) { lookup[i][j] = max(lookup[i][j], lookup[i][j - 1] + 1); if (max_len < lookup[i][j]) { max_len = lookup[i][j]; max_row = i, max_col = j; } } } } } cout << "Maximum length of Snake sequence is: " << max_len << endl; // find maximum length Snake sequence path list<Point> path = findPath(lookup, mat, max_row, max_col); cout << "Snake sequence is:"; for (auto it = path.begin(); it != path.end(); it++) cout << endl << mat[it->x][it->y] << " (" << it->x << ", " << it->y << ")" ; } // Driver code int main() { int mat[M][N] = { {9, 6, 5, 2}, {8, 7, 6, 5}, {7, 3, 1, 6}, {1, 1, 1, 7}, }; findSnakeSequence(mat); return 0; }

## Python3

# Python program to find maximum length # Snake sequence and print it M = 4 N = 4 class Point: def __init__(self, x, y): self.x = x self.y = y # Function to find maximum length Snake sequence path # (i, j) corresponds to tail of the snake def findPath(grid, mat, i, j): path = list() pt = Point(i, j) path.append(pt) while (grid[i][j] != 0): if (i > 0 and grid[i][j]-1 == grid[i-1][j]): pt = Point(i-1, j) path.append(pt) i -= 1 elif (j > 0 and grid[i][j]-1 == grid[i][j-1]): pt = Point(i, j-1) path.append(pt) j -= 1 return path # Function to find maximum length Snake sequence def findSnakeSequence(mat): # table to store results of subproblems # initialize by 0 lookup = [[0 for i in range(N)] for j in range(M)] # stores maximum length of Snake sequence max_len = 0 # store cordinates to snake's tail max_row = 0 max_col = 0 # fill the table in bottom-up fashion for i in range(M): for j in range(N): # do except for (0, 0) cell if (i or j): # look above if (i > 0 and abs(mat[i-1][j] - mat[i][j]) == 1): lookup[i][j] = max(lookup[i][j], lookup[i-1][j] + 1) if (max_len < lookup[i][j]): max_len = lookup[i][j] max_row = i max_col = j # look left if (j > 0 and abs(mat[i][j-1] - mat[i][j]) == 1): lookup[i][j] = max(lookup[i][j], lookup[i][j-1] + 1) if (max_len < lookup[i][j]): max_len = lookup[i][j] max_row = i max_col = j print("Maximum length of Snake sequence is:", max_len) # find maximum length Snake sequence path path = findPath(lookup, mat, max_row, max_col) print("Snake sequence is:") for ele in reversed(path): print(mat[ele.x][ele.y], " (", ele.x, ", ", ele.y, ")", sep = "") # Driver code mat = [[9, 6, 5, 2], [8, 7, 6, 5], [7, 3, 1, 6], [1, 1, 1, 7]] findSnakeSequence(mat) # This code is contributed # by Soumen Ghosh

Output :

Maximum length of Snake sequence is: 6 Snake sequence is: 9 (0, 0) 8 (1, 0) 7 (1, 1) 6 (1, 2) 5 (1, 3) 6 (2, 3) 7 (3, 3)

Time complexity of above solution is O(M*N). Auxiliary space used by above solution is O(M*N). If we are not required to print the snake, space can be further reduced to O(N) as we only uses the result from last row.

Reference: Stack Overflow

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