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    <p>This is quite simple. If you have <code>n</code> squares, you can count the rectangles in <code>O(n)</code>.</p> <p>Assumptions:</p> <ul> <li>The borders of each valid rectangle don't share any cells with a non-valid path.</li> <li>If one rectangle is inside another, you would be happy finding them</li> </ul> <p>You would need extra memory as big as the input. Let's call this <code>visited</code> and initialize with 0.</p> <p>Let's first construct a helper function:</p> <pre><code>is_rectangle(square s) from s, go right while you see 1s and visited is 0 mark them as 1 in visited if less than 3 steps return 0 then, go down while you see 1s and visited is 0 mark them as 1 in visited if less than 3 steps return 0 then, go left while you see 1s and visited is 0 mark them as 1 in visited if less than 3 steps return 0 then, go up while you see 1s and visited is 0 mark them as 1 in visited if then one above is not s return 0 return 1 </code></pre> <p>This function basically traces the 1s in the directions of right-down-left-up and checks if the conditions are met (length at least 3 and reaching the starting position). It also marks the visited squares.</p> <p>The important thing to notice about this function, is that it works correctly only if the initial square is the top-left corner.</p> <p>Now, the solution to the problem is easy:</p> <pre><code>num_rectangles = 0 initialize visited to 0 (rows x columns) for r in rows for c in columns if visitied[r][c] or image[r][c] == 0 continue num_rectangles += is_rectangle(&lt;r,c&gt;) return num_rectangles </code></pre> <p>Here is how the algorithm executes:</p> <p><img src="https://i.stack.imgur.com/At6yQ.png" alt="enter image description here"><br> 1. Failed (and marked) part of a bad rectangle</p> <p><img src="https://i.stack.imgur.com/j4kXd.png" alt="enter image description here"><br> 2. Found (and marked) a rectangle.</p> <p><img src="https://i.stack.imgur.com/VYj4K.png" alt="enter image description here"><br> 3. Failed on a single square (of a vertical line)</p> <p><img src="https://i.stack.imgur.com/PwfS4.png" alt="enter image description here"><br> 4. Failed on a single square (of a vertical line)</p> <p><img src="https://i.stack.imgur.com/daWVu.png" alt="enter image description here"><br> 5. Failed on a single square (of a vertical line)</p> <p><img src="https://i.stack.imgur.com/esXv4.png" alt="enter image description here"><br> 6. After many similar steps, found the next rectangle.</p> <p><img src="https://i.stack.imgur.com/HTopH.png" alt="enter image description here"><br> 7. And the next rectangle</p> <p><img src="https://i.stack.imgur.com/QAcgR.png" alt="enter image description here"><br> 8. Algorithm end</p>
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    1. POFinding complete rectangles enclosing 0
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    1. COI like this, but depending on how bad the input is, when you realize that a rectangle is only partially complete, you might need to go back and remove its cells from the visited array.
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    2. CO@NateKohl, on the contrary! Given the conditions of the problem, keeping them as visited helps skip future non-rectangles.
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    3. CO@NateKohl, ok I see what you mean. I added an assumption that says your comment on the question cannot happen (so rectangles don't share border with non-rectangles)
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