The Island Perimeter problem asks you to compute the perimeter of an island in a grid. The grid is made of land cells and water cells, and the island is formed by land cells connected horizontally or vertically. You can assume the island is a single connected component with no separate islands. The perimeter is the total length of the island’s boundary where land touches water or the grid boundary.
This problem is deceptively simple because it sounds like geometry, but it is really about local adjacency. You are not tracing an outline in a complicated way. You are counting exposed edges.
Why tracing the boundary is unnecessary
A common instinct is to “walk around” the island boundary the way you might outline a shape. That approach can work, but it adds complexity: you must find a starting boundary cell, choose a direction, avoid revisiting edges, and handle turns correctly.
The key insight is that you do not need to trace anything. The perimeter can be computed by looking at each land cell and counting how many of its sides are exposed to water or to the outside of the grid. This turns the problem into a straightforward scan.
The core idea: each land cell contributes up to four edges
Every land cell is a square with four sides. If it were isolated, it would contribute four units to the perimeter. However, if a land cell touches another land cell, the shared side is not part of the perimeter because it lies inside the island.
So the perimeter is essentially the sum of exposed sides across all land cells. This local rule is enough to compute the global boundary length.
Counting exposed edges directly
When scanning the grid, for each land cell, you check its four neighbors: up, down, left, and right.
- If a neighboring position is outside the grid or contains water, that side is exposed and contributes one unit to the perimeter.
- If the neighbor is land, that side contributes nothing because it is an internal edge.
This method is robust because it works regardless of the island’s shape. Whether the island is long and thin, compact, or full of bends, exposed edges are still exposed edges.
An equivalent viewpoint: subtract shared borders
Another helpful way to interpret the same logic is to start with four perimeter units per land cell and then subtract for shared edges.
Every time two land cells touch, they share a border that removes two units from the total perimeter count, because that border would have been counted once by each cell if you started from four.
This “add fours, subtract twos” interpretation leads to the same result and can make correctness feel more intuitive. Both explanations reflect the same underlying adjacency principle.
Why this approach is correct
The perimeter of a grid-based island is exactly the number of land edges that border water or the grid boundary. The scan-based method counts those edges directly. It never counts internal edges because those edges always have land on both sides.
Because every edge belongs to exactly one of these two categories, exposed or internal, counting exposed edges across all land cells yields the true perimeter with no gaps and no double counting.
Time and space complexity considerations
The algorithm runs in linear time with respect to the number of cells in the grid because you visit each cell once and perform a constant amount of neighbor-checking work.
Space usage is constant beyond the input grid and the running perimeter total.
This efficiency is part of why the problem is popular. It rewards candidates who can simplify a geometric-sounding task into a clean, local counting solution.
Why this problem is common in interviews
Island Perimeter shows up often in interviews because it tests whether candidates can reason locally about grid problems instead of overengineering a traversal. It also checks basic comfort with boundary conditions, since edges of the grid contribute to the perimeter in the same way water neighbors do.
The problem is also a good signal of clarity. Candidates who explain the “exposed edges” idea usually implement it correctly with fewer bugs.
What this problem teaches beyond perimeters
Beyond islands, the key takeaway is that many grid geometry problems reduce to counting local relationships. Instead of simulating a global process, you can often compute the answer by inspecting each cell and applying a consistent rule.
If you can clearly explain why each land cell starts with four sides, how shared edges stop counting toward the perimeter, and why checking neighbors captures the entire boundary, you demonstrate strong grid reasoning. That depth of understanding makes Island Perimeter an excellent exercise in local-invariant thinking and efficient grid scanning.
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