Maximum Trailing Zeros In A Cornered Path - Leetcode Solution

Problem Statement:

A two-dimensional grid with n rows and m columns is given. Each cell of the grid contains a non-negative integer. You want to go from the top-left corner to the bottom-right corner with minimum possible effort. The effort of a path is defined as the maximum difference between two consecutive cells of the path. You can only move from a cell to another if they share a side.

Return the maximum trailing zeros in the effort of such a path. The effort of a path is defined as the maximum difference between two consecutive cells of the path.

Solution:

This problem seems difficult at first glance, but we can break it down into simpler sub-problems. First, let's define some useful terms:

1. Connected components: In graph theory, a connected component is a subgraph in which every pair of vertices is connected by a path.

2. MST: A minimum spanning tree (MST) is a tree in a connected graph that connects all the vertices together with the smallest possible total edge weight.

3. Maximum edge weight: The maximum edge weight in a path is the difference between the maximum and minimum values of the cells in that path.

Now, let's consider the following algorithm:

1. Find all the connected components in the graph. A connected component is a subgraph in which every pair of vertices is connected by a path.

2. For each connected component, find its minimum spanning tree (MST). The MST is a tree in a connected graph that connects all the vertices together with the smallest possible total edge weight.

3. For each edge in the MST, compute its weight as the maximum edge weight in the path between the two vertices it connects.

4. Sort the edges of the MST by weight in descending order.

5. For each edge in the sorted list, add it to the path if it does not create a cycle. Keep track of the maximum edge weight in the path.

6. Count the number of trailing zeros in the maximum edge weight of the final path.

The algorithm works by finding the minimum cost path from the top-left corner to the bottom-right corner, where the cost of a path is defined as the maximum difference between two consecutive cells of the path.

The algorithm first finds all the connected components in the graph and computes their MSTs. It then iterates over the edges of the MSTs in descending order of their weights. For each edge, it adds it to the path if it does not create a cycle. Finally, it counts the number of trailing zeros in the maximum edge weight of the final path.

Time Complexity:

The time complexity of the algorithm is O(N^2 log N), where N is the number of cells in the grid. This is because we have to find the MST for each connected component, which takes O(N log N) time. Sorting the edges of the MSTs takes O(N log N) time, and checking for cycles takes O(N log N) time. Counting the trailing zeros in the maximum edge weight takes O(log W) time, where W is the maximum weight. Since W is bounded by the maximum value in the grid, the time complexity of the algorithm is O(N^2 log N).

Space Complexity:

The space complexity of the algorithm is O(N), where N is the number of cells in the grid. This is because we have to store the connected components and MSTs, which take O(N) space.

Conclusion:

In this problem, we have found the maximum trailing zeros in the effort of a path from the top-left corner to the bottom-right corner of a two-dimensional grid. We have developed an algorithm that works by finding the minimum cost path from the top-left corner to the bottom-right corner, where the cost of a path is defined as the maximum difference between two consecutive cells of the path. The time complexity of the algorithm is O(N^2 log N), and the space complexity is O(N).

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