Finding the longest path in a grid can be like solving a really hard problem. The snake sequence problem is a famous test of your ability to think logically and spatially for students and developers who are just starting to learn about data structures and algorithms (DSA). You have to walk across a grid and identify a sequence of numbers where each step only goes right or down and the numbers stay “connected” by a difference of one. This article goes over the logic, the code, and the visual structure of this grid-based method to help you learn it.
Also Read – Advance Data Structure and Algorithms
What Is The Snake Sequence Rule?
Before we jump into the technical implementation, let us define what actually makes a sequence “snaky”. In a two-dimensional grid, it is a chain of cells starting from any location. But there are stringent limits about how you can travel and what numbers you can choose.
- You can only move one step to the right or one step down.
- The Value Constraint says that the following cell’s value must be either the current value plus one or the current value minus one.
- The goal is to find the longest chain that follows these conditions.
Imagine a snake moving through a garden. It can’t hop across tiles; it has to relocate to a nearby area that is mathematically “close” to where it recently was.
Snake Sequence Path Logic Using Dynamic Programming
Dynamic Programming (DP) is a method we employ to tackle this problem quickly. We make a second grid of the same size instead of verifying every possible path (which would take forever on a big grid). This second grid stores the length of the longest snake path, ending at that specific cell.
Also Read – Introduction to Red-Black Tree
Step-by-Step Breakdown
- Initialise a DP Table: Create a 2D matrix (let us call it lookup) where every cell starts with a value of 0.
- Iterate Through the Grid: We move through the original grid row by row and column by column.
- Check the Left Neighbour: If the cell to the left exists and the difference between it and the current cell is 1, we update our current DP cell.
- Look at the Top Neighbour: We update our current DP cell again if the cell above it exists and the difference between it and the current cell is 1. We pick the highest value between the left and top options.
- Track the Maximum: While filling this table, keep a record of the highest number found. This represents the length of our longest snake path.
Snake Sequence DP Formula
To formalise the logic, let DP[i][j] represent the length of the longest snake path ending at cell (i, j).
The recurrence relation is:
- If top cell is valid:
DP[i][j] = max(DP[i][j], DP[i-1][j] + 1) - If left cell is valid:
DP[i][j] = max(DP[i][j], DP[i][j-1] + 1)
Condition:
- |grid[i][j] – grid[i-1][j]| = 1
- |grid[i][j] – grid[i][j-1]| = 1
This ensures that each DP cell stores the length of the longest snake ending at that position.
Snake Sequence Example
Let us look at a numerical example to see how the numbers connect. Imagine the following 3×3 grid:
| Col 0 | Col 1 | Col 2 | |
| Row 0 | 9 | 6 | 5 |
| Row 1 | 4 | 5 | 6 |
| Row 2 | 8 | 3 | 2 |
In this case, we can go from (0, 1), which is 6, to (0, 2), which is 5 (the difference being 1). We can’t go much further to the right from there. But we can proceed to (1, 1), which is 5, and then to (1, 2), which is 6, if we look at the path from (1, 0), which is 4. That makes a snake path that is 3 units long.
Python Code for Snake Sequence
The following code demonstrates how to calculate the length and the path of the snake path
Python
def find_snake_sequence(grid):
rows = len(grid)
cols = len(grid[0])
# Create a DP table filled with 0
lookup = [[0 for _ in range(cols)] for _ in range(rows)]
max_len = 0
max_row, max_col = 0, 0
for i in range(rows):
for j in range(cols):
# Check from top
if i > 0 and abs(grid[i][j] – grid[i-1][j]) == 1:
lookup[i][j] = max(lookup[i][j], lookup[i-1][j] + 1)
# Check from left
if j > 0 and abs(grid[i][j] – grid[i][j-1]) == 1:
lookup[i][j] = max(lookup[i][j], lookup[i][j-1] + 1)
if lookup[i][j] > max_len:
max_len = lookup[i][j]
max_row, max_col = i, j
return max_len, max_row, max_col
# Example usage
grid = [
[9, 6, 5, 2],
[8, 7, 6, 5],
[7, 3, 1, 6],
[1, 1, 1, 7]
]
length, r, c = find_snake_sequence(grid)
print(f”Maximum length is: {length + 1}“)
Also Read – Geometric Algorithms
Snake Sequence Output Format
Once the DP table is computed, the final output should include:
- Maximum Length of Snake Path
- Actual Path (sequence of values)
Example Output:
Maximum Length: 3
Snake path: 4 → 5 → 6
Snake Path Reconstruction
Finding the length is only half the solution. To retrieve the actual snake path, we use backtracking.
How it works
- Start from the cell where the maximum length was found
- Move backwards by checking:
- If top cell contributed to DP value
- Else if left cell contributed
- Continue until you reach a cell with value 0
Backtracking Logic
At cell (i, j):
- If
DP[i][j] == DP[i-1][j] + 1 → move UP - Else if
DP[i][j] == DP[i][j-1] + 1 → move LEFT
Collect values while moving backward, then reverse the sequence.
Python Code to Print Snake Path
def print_snake_path(grid, lookup, i, j):
path = []
while i > 0 or j > 0:
path.append(grid[i][j])
if i > 0 and lookup[i][j] == lookup[i-1][j] + 1:
i -= 1
elif j > 0 and lookup[i][j] == lookup[i][j-1] + 1:
j -= 1
else:
break
path.append(grid[i][j])
path.reverse()
return path
Snake Sequence vs Other Grid Problems
In the world of technology and digital audio, terms like sting sequencer often pop up, which can sometimes confuse students searching for programming solutions.
While a snake path in DSA refers to grid traversal, a snake sequencer in music (like the mdd snake fl studio plugin) refers to a tool that generates patterns. If you are looking for how to install mdd snake, you are likely dealing with a MIDI tool rather than a coding algorithm. However, both rely on the concept of “chains”—one being a chain of numbers and the other a chain of musical notes.
| Problem Type | Movement Allowed | Constraint | Algorithm Used |
| Snake Path | Right, Down | Difference of 1 | Dynamic Programming |
| Longest Path | Any direction | Increasing values | DFS + Memoization |
| Shortest Path | Right, Down | Minimum sum | Dynamic Programming |
| Knight’s Tour | L-shape | Visit all cells | Backtracking |
Key Points to Remember of Snake Sequence
To understand the snake path problem, remember these three pillars:
- Directionality: You only move forward (right or down), which prevents infinite loops and simplifies the DP structure.
- The Difference Rule: The absolute difference of 1 is the gatekeeper. If the difference is 0 or 2, the snake stops there.
- Backtracking: If you need to print the actual path (the values of the cells), start from the maximum value in your DP table and work backwards to find which neighbour contributed to that length.
FAQs
Can a snake path move diagonally?
No, the normal definition only lets you shift to the right or down cells that are next to you. Adding diagonals would change the logic of the DP table.
What happens if more than one sequence has the longest length?
The algorithm will find the right length. To find all paths, you would need to store all contributing neighbours during the DP process and use a recursive function to print them.
How do I handle negative numbers in the grid?
The logic is still the same. For example, -1 and -2 or -1 and 0 can be part of the snake path as long as the absolute difference between them is 1.
Can I use recursion in place of dynamic programming?
Yes, you may use memoisation with recursion. A bottom-up DP strategy is usually easier to understand for grid problems and doesn't run the risk of stack overflow on really large grids.