Understanding the Problem
We are given a binary grid where 1
represents land and 0
represents water. An island is formed by connecting adjacent 1s either vertically or horizontally. Our goal is to flip at most one 0 to 1 and determine the size of the largest island possible after this operation.
To solve this efficiently, we use the Disjoint Set Union (DSU) data structure to track connected components (i.e., islands) and their sizes. We simulate flipping each 0 to a 1 and check which islands it could connect to, to maximize the resulting island size.
Step-by-Step Solution with Example
Step 1: Group All Existing Islands Using DSU
We iterate through each cell in the grid. Whenever we find a 1
, we perform a union operation with its adjacent 1s (up, down, left, right). Each island is treated as a connected component in DSU.
Step 2: Track the Size of Each Island
We maintain a map that stores the size of each component, using the root representative as the key. This way, we know how big each island is, which is critical when we later evaluate the impact of flipping a 0.
Step 3: Evaluate Each 0 for Maximum Island Size
Now, we scan the grid again. For every 0
cell, we:
- Look in the four directions (up, down, left, right).
- Collect the root IDs of neighboring
1
s (i.e., neighboring islands).
- Make sure not to count the same island twice (use a set).
- Sum up the sizes of these unique neighboring islands and add 1 (for the current 0 being flipped to 1).
This gives us the size of the island we can potentially form by flipping this specific 0.
Step 4: Track the Maximum Island Size
During the above evaluations, we keep updating the maximum island size encountered. This ensures we return the largest possible island size that can be formed with one flip.
Step 5: Handle the Case Where No 0 Exists
If the grid has no 0s, the largest island is the entire grid of 1s. In that case, we return the size of the whole grid.
Edge Cases
- All 1s: No 0 to flip. The answer is the total count of 1s.
- All 0s: Flipping any one 0 gives an island of size 1.
- Disconnected 1s: Flipping a 0 between them can connect multiple islands.
- Flipping a border 0: Still needs to check only valid neighbors within grid bounds.
Finally
This problem is a great example of how Disjoint Set Union can be used to manage groups of connected elements efficiently. Instead of recalculating connected components multiple times, we preprocess the island structure once and reuse the information for every possible 0 we might flip. The use of sets to avoid duplicate island counting ensures correctness, and edge cases are naturally handled through the logic flow.
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