Is Minimum Operations to Remove Adjacent Ones in Matrix easy or hard?

The problem is classified as Hard because it requires recognizing that a grid adjacency constraint forms a bipartite graph. Applying maximum matching and understanding the connection to minimum vertex cover makes it more advanced than typical matrix traversal problems.

Minimum Operations to Remove Adjacent Ones in Matrix Python/Java solution

Implement the solution by building a bipartite graph from the matrix and applying a maximum matching algorithm such as Hungarian or Kuhn. The same logic works across Python, Java, C++, Go, and TypeScript by storing neighbors in adjacency lists and repeatedly searching for augmenting paths.

How to solve Minimum Operations to Remove Adjacent Ones in Matrix in O(n)?

An O(n) solution is not feasible because the problem requires resolving conflicts between adjacent cells across a graph structure. The optimal approach builds a bipartite graph and runs maximum matching, which typically takes O(V*E) or O(V^3) depending on the implementation.

What is the best approach for Minimum Operations to Remove Adjacent Ones in Matrix?

The best approach models the matrix as a bipartite graph and computes a maximum matching. Each cell with value 1 becomes a node, and edges connect adjacent 1s. Using the Hungarian algorithm or another bipartite matching method, the maximum matching size equals the minimum number of cells that must be removed to eliminate all adjacent conflicts.

Is Minimum Operations to Remove Adjacent Ones in Matrix asked at Google/Amazon/Meta?

Graph matching and bipartite modeling problems frequently appear in interviews at companies like Google, Amazon, and Meta. While the exact problem may vary, recognizing grid problems that convert into bipartite graphs is a common interview pattern.

What data structure is used in Minimum Operations to Remove Adjacent Ones in Matrix?

The solution uses a bipartite graph representation built from the matrix. Adjacency lists store edges between neighboring 1s, and matching arrays track pairings during the Hungarian or DFS-based matching process.

What is the time complexity of Minimum Operations to Remove Adjacent Ones in Matrix?

Using the Hungarian algorithm for bipartite matching gives O(V^3) time complexity, where V is the number of cells containing 1. Space complexity is O(V + E) for storing the graph and matching arrays.

Description Hints 4 Solution Video 25:47

2123. Minimum Operations to Remove Adjacent Ones in Matrix

Hard42.5% acceptance1 companyPremium elsewhere, free here

You are given a 0-indexed binary matrix grid. In one operation, you can flip any 1 in grid to be 0.

A binary matrix is well-isolated if there is no 1 in the matrix that is 4-directionally connected (i.e., horizontal and vertical) to another 1.

Return the minimum number of operations to make grid well-isolated.

Example 1:

Input: grid = [[1,1,0],[0,1,1],[1,1,1]]
Output: 3
Explanation: Use 3 operations to change grid[0][1], grid[1][2], and grid[2][1] to 0.
After, no more 1's are 4-directionally connected and grid is well-isolated.

Example 2:

Input: grid = [[0,0,0],[0,0,0],[0,0,0]]
Output: 0
Explanation: There are no 1's in grid and it is well-isolated.
No operations were done so return 0.

Example 3:

Input: grid = [[0,1],[1,0]]
Output: 0
Explanation: None of the 1's are 4-directionally connected and grid is well-isolated.
No operations were done so return 0.

Constraints:

m == grid.length
n == grid[i].length
1 <= m, n <= 300
grid[i][j] is either 0 or 1.

Asked at 1 company

Amazon

Topics

Hints open one at a time

Hint 1

Consider each cell containing a 1 as a vertex whose neighbors are the cells 4-directionally connected to it. The grid then becomes a bipartite graph.

Hint 2

You want to find the smallest set of vertices such that every edge in the graph has an endpoint in this set. If you remove every vertex in this set from the graph, then all the 1’s will be disconnected. Are there any well-known algorithms for finding this set?

Hint 3

This set of vertices is called a minimum vertex cover. You can find the size of a minimum vertex cover by finding the size of a maximum matching (Konig’s theorem).

Hint 4

There are well-known algorithms such as Kuhn’s algorithm and Hopcroft-Karp-Karzanov algorithm which can find a maximum matching in a bipartite graph quickly.

25:47

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