Minimum Number of Days to Disconnect Island
A hard Graph problem included in Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 1
- Core for
- 11 roles
- Platform
- LeetCode
The problem
Given an m x n binary grid where 1 represents land and 0 represents water, find the minimum number of days to disconnect the island. Each day, any water cell adjacent to a land cell turns into land. Return the minimum days needed so that no single connected island remains.
Example 1
- Input
- grid = [[0,1,1,0],[0,1,1,0],[0,0,0,0]]
- Output
- 2
- Why
- It takes 2 days for water to spread enough to split the island.
Example 2
- Input
- grid = [[1,1]]
- Output
- 2
- Why
- Both cells must become water, which takes 2 days.
Constraints
- 1 <= grid.length, grid[i].length <= 10
- grid[i][j] is 0 or 1
How to think about it
Updated 2026-09-09The question asks for the minimum operations to disconnect an island. Observe that any 2D connected shape on a grid can be disconnected by removing at most two cells (for instance, clearing the two neighbors of any corner cell). Testing three possibilities in increasing order: check if the graph is already disconnected (0), check if removing any single 1 disconnects it (1), otherwise the answer is 2.
Approaches, worst first
Exhaustive pair subset removal simulation
time O((m * n)^3) · space O(m * n)
Test disconnectivity after removing each pair of land cells in addition to single cells. Simulating component counts across all pairs of cells takes quartic grid time, whereas the maximum answer can never exceed 2 so checking pairs is unnecessary once single removals fail.
Sequential component count checkingWrite this one
time O((m * n)^2) · space O(m * n)
Write a helper that counts connected components of 1s in the grid using BFS or DFS. If the helper returns anything other than 1, return 0. Iterate over all cells: if a cell is 1, change it to 0, recount components, and restore it; if the count is not 1, return 1. If no single cell works, return 2.
Where people lose marks · 3
- Empty grid or disconnected components at the outset require 0 operations; returning 1 or 2 without testing the initial state fails base tests.
- Grids with 1 or 2 total land cells cannot be split into two disjoint non-empty components; removing cells reduces count to 0 or 1, which represents disconnected state.
- Always restore the mutated cell back to 1 before moving to the next candidate cell in the simulation loop.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
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