Beginner to Mastery series: Graphs, Leetcode 2812, Find safest path in a grid, Dijkstra's Algorithm.

Описание: LeetCode 2812 Solved: Find Safest Path in a Grid - Dijkstra's Algorithm

With a surpirse pro tip:
Define ds as type of structure you want to contain to prevent code resusability

Welcome back to the Beginner to Mastery Graphs Series! This continues our in-depth exploration of essential graph algorithms and advanced problem-solving techniques.

In this video, we are tackling an intriguing problem: LeetCode 2812: Find Safest Path in a Grid. Although officially rated as a LeetCode Medium problem, it requires combining two powerful and distinct graph algorithms in sequence: Breadth-First Search (BFS) and Dijkstra's Algorithm. Mastering this type of multi-step problem is a significant milestone on your path to algorithmic mastery.

The core challenge is to find a path from the top-left cell (0, 0) to the bottom-right cell (N-1, N-1) such that the minimum "safety factor" of any cell on that path is maximized. This is known as a minimax optimization problem.

Part 1: Calculating the Safety Factor using Multi-Source BFS

The first crucial step is to define and calculate the safety factor for every single cell in the grid. The safety factor of a cell is defined as the Manhattan distance to the nearest 'thief' cell (marked with a value of 1). Since a thief can be anywhere, we need an efficient way to find the minimum distance from every non-thief cell to any thief.

Initialization: We initialize a distance grid where all thief cells have a distance of 0, and all other cells have a large initial distance (infinity).

The Power of Multi-Source BFS: We use a Multi-Source Breadth-First Search (BFS). We enqueue all the thief cells simultaneously at the start. BFS naturally explores outwards in layers, guaranteeing that when we first visit a cell, we have found the shortest distance to the nearest source (thief). This process efficiently populates our safety factor grid. The time complexity of this step is O(N^2), where N is the dimension of the grid, as every cell is visited and processed exactly once.

Part 2: Finding the Safest Path using Modified Dijkstra's Algorithm

Once the safety factor grid is ready, the problem transforms into a graph pathfinding problem: "Find the path that maximizes the minimum weight."

Graph Adaptation: We view the grid as a graph where cells are nodes and adjacent cells have an edge between them. The weight of moving to a cell is its pre-calculated safety factor.

Dijkstra's Modification: Standard Dijkstra's minimizes the total path sum. Here, we need to maximize the bottleneck capacity (the minimum safety factor encountered so far on the path).

Priority Queue Logic: We use a max-heap priority queue. Each entry in the priority queue will store the tuple (current_min_safeness, row, col). When exploring neighbors, the new minimum safety factor will be min(current_min_safeness, safety_factor[neighbor]). By using a max-heap, we ensure that we always explore paths that offer the highest guaranteed safety first, allowing us to greedily find the globally optimal minimax path.

Termination: The algorithm terminates when the bottom-right cell (N-1, N-1) is extracted from the priority queue. The safety score associated with that cell is the maximum safest path possible.

This video provides a complete, clear, and highly efficient implementation of both the BFS pre-processing and the modified Dijkstra's pathfinding algorithm, giving you a comprehensive understanding of how to solve complex grid problems.

Key Takeaways and Summary

Problem: LeetCode 2812: Find Safest Path in a Grid

Difficulty: LeetCode Medium (High Complexity)

Techniques: Multi-Source Breadth-First Search (BFS) and Modified Dijkstra's Algorithm

Concept: Minimax Optimization and Bottleneck Capacity Pathfinding

Time Complexity: Dominated by the two main steps, resulting in O(N^2 log N) or O(N^2 log(N^2)), depending on the priority queue implementation, which is highly efficient for the given constraints.

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