Snakes and Ladders: A Beginner’s Guide to Solving the Problem

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Difficulty: Medium

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Topics: Array, Breadth-First Search, Matrix

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Introduction

In this tutorial, we will explore the Snakes and Ladders problem, a classic game that can be modeled using a matrix. The goal is to determine the minimum number of dice rolls required to reach the last square of the board, navigating through snakes and ladders that can alter your path. We will use the Breadth-First Search (BFS) algorithm to find the solution efficiently.

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Prerequisites

Basic understanding of arrays and matrices

Familiarity with the Breadth-First Search algorithm

Knowledge of PHP programming language (for implementation)

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Understanding the Game

You are given an n x n integer matrix board where the cells are labeled from 1 to n². The labeling starts from the bottom left of the board (i.e., board[n - 1][0]) and alternates direction each row.

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Starting from square 1, you can move to a destination square next based on the result of a standard 6-sided die roll. The possible moves are in the range [curr + 1, min(curr + 6, n²)].

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If you land on a square with a snake or ladder (indicated by board[r][c] != -1), you must move to the destination specified by that square. The game ends when you reach square n².

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For example, if you roll a die and land on square 2, and there is a ladder leading to square 15, you must move to square 15.

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Problem Statement

Your task is to return the least number of dice rolls required to reach square n². If it is not possible to reach the square, return -1.

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Example 1:

Input: board = [[-1,-1,-1,-1,-1,-1],[-1,-1,-1,-1,-1,-1],[-1,-1,-1,-1,-1,-1],[-1,35,-1,-1,13,-1],[-1,-1,-1,-1,-1,-1],[-1,15,-1,-1,-1,-1]]

Output: 4

Explanation:\n
- Start at square 1 (row 5, column 0).
- Move to square 2 and take the ladder to square 15.
- Move to square 17 and take the snake to square 13.
- Move to square 14 and take the ladder to square 35.
- Finally, move to square 36, ending the game.
- This is the lowest possible number of moves to reach the last square, so return 4.
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Example 2:

Input: board = [[-1,-1],[-1,3]]

Output: 1

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Constraints

n == board.length == board[i].length

2 <= n <= 20

board[i][j] is either -1 or in the range [1, n²].

The squares labeled 1 and n² are not the starting points of any snake or ladder.

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Approach

Problem Analysis: The game involves moving from square 1 to square n² on an n x n board. Each move consists of rolling a 6-sided die, moving to one of the next 6 squares (if available), and following any snake or ladder encountered.

Key Insight: BFS is suitable here because it explores all possible moves level by level, ensuring the first time we reach the destination square is via the shortest path.

Board Conversion: Convert the square number to its corresponding board coordinates (row, column) considering the Boustrophedon labeling.

Snake/Ladder Handling: If a square (after the die roll) has a snake or ladder, the player must move to the destination square specified by the board value.

Visited Tracking: Maintain a visited array to avoid reprocessing the same square, optimizing the solution and preventing cycles.

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Implementation

Let’s implement this solution in PHP:

<?php\n/**\n * @param Integer[][] $board\n * @return Integer\n */\nfunction snakesAndLadders($board) {\n    // Implementation goes here\n}\n\n// Example usage:\n$board1 = [[-1,-1,-1,-1,-1,-1],[-1,-1,-1,-1,-1,-1],[-1,-1,-1,-1,-1,-1],[-1,35,-1,-1,13,-1],[-1,-1,-1,-1,-1,-1],[-1,15,-1,-1,-1,-1]];\necho snakesAndLadders($board1); // Output: 4\n\n$board2 = [[-1,-1],[-1,3]];\necho snakesAndLadders($board2); // Output: 1\n?>

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Explanation of the Algorithm

Initialization: The BFS starts from square 1 with 0 moves. A queue is used to manage the squares to be processed, and a visited array tracks processed squares to avoid cycles.

Processing Moves: For each square dequeued, the algorithm checks if it’s the target square (n²). If so, it returns the current move count.

Die Roll Simulation: For each possible die roll (1 to 6), the next square is calculated. If this square exceeds n², the loop breaks early.

Board Conversion: The next square is converted to board coordinates (row, column) considering the alternating row directions.

Snake/Ladder Check: If the converted square has a snake or ladder, the next square is updated to the destination value.

Enqueueing Valid Moves: If the next square (after handling snakes/ladders) hasn’t been visited, it is marked as visited and enqueued with an incremented move count.

Termination: If the queue is exhausted without reaching the target square, -1 is returned, indicating it’s unreachable.

This approach efficiently explores all possible paths using BFS, ensuring the shortest path is found while handling the game’s rules for snakes and ladders. The complexity is O(n²) as each square is processed at most once.

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Conclusion

In this tutorial, we learned how to solve the Snakes and Ladders problem using the Breadth-First Search

Sources:

Source: Original Article