Minimum Operations to Remove Adjacent Ones in Matrix

Minimum Operations to Remove Adjacent Ones in Matrix

What is this problem about?

The Minimum Operations to Remove Adjacent Ones in Matrix problem involves a binary matrix where you want to eliminate all pairs of adjacent 1s — horizontally or vertically. In each operation, you can flip a cell from 1 to 0. The goal is to compute the minimum number of such flips needed to ensure no two 1s are adjacent. This Minimum Operations to Remove Adjacent Ones in Matrix coding problem is rooted in graph theory and combinatorics.

Why is this asked in interviews?

Amazon includes this in its hard interview rounds because it requires knowledge of minimum vertex cover and bipartite graph matching — topics rarely seen in standard practice but highly valued in algorithm-heavy engineering roles. It signals mastery of graph interview patterns beyond the basics. Solving this problem demonstrates the ability to model real-world constraints (conflicts between neighboring cells) as formal graph problems.

Algorithmic pattern used

The core pattern is maximum bipartite matching (via graph modeling). Color the matrix like a chessboard — cells where (row + col) is even form one partition, odd cells form the other. Two adjacent 1-cells always belong to different partitions. The problem reduces to finding a minimum vertex cover in a bipartite graph, which by König's theorem equals the maximum matching. Standard algorithms like Hopcroft-Karp or augmenting paths are used.

Example explanation

Consider a 2x3 matrix:

1 1 0
0 1 1

Cells (0,0), (0,1), (1,1), (1,2) are all 1s. Adjacent pairs: (0,0)-(0,1), (0,1)-(1,1), (1,1)-(1,2). Model this as a bipartite graph. Running maximum matching finds 2 matches. By König's theorem, minimum vertex cover = 2. So 2 flips suffice to eliminate all adjacencies.

Common mistakes candidates make

• Treating this as a greedy problem (always flipping the most conflicted cell) — greedy fails to guarantee optimality.
• Not recognizing the bipartite structure of the grid.
• Implementing DFS-based augmenting paths incorrectly.
• Forgetting that 0-cells are irrelevant; only 1-cells need to be modeled as graph nodes.

Interview preparation tip

For hard matrix and graph interview problems, practice recognizing when a grid can be treated as a bipartite graph. Any chessboard-colored grid naturally creates two disjoint sets. Familiarize yourself with König's theorem and minimum vertex cover, as these appear in several hard combinatorics-on-graphs problems. Even if you can't recall the full proof, knowing the theorem's statement — min vertex cover = max matching in bipartite graphs — can earn you significant partial credit in interviews.