## 54. Spiral Matrix

Given a matrix of m x n elements (m rows, n columns), return all elements of the matrix in spiral order.

For example,
Given the following matrix:

[
[ 1, 2, 3 ],
[ 4, 5, 6 ],
[ 7, 8, 9 ]
]


You should return [1,2,3,6,9,8,7,4,5].

b'
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#### Approach #1: Simulation [Accepted]

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Intuition

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Draw the path that the spiral makes. We know that the path should turn clockwise whenever it would go out of bounds or into a cell that was previously visited.

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Algorithm

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Let the array have rows and columns. denotes that the cell on the-th row and -th column was previously visited. Our current position is , facing direction , and we want to visit x total cells.

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As we move through the matrix, our candidate next position is . If the candidate is in the bounds of the matrix and unseen, then it becomes our next position; otherwise, our next position is the one after performing a clockwise turn.

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Complexity Analysis

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Time Complexity: , where is the total number of elements in the input matrix. We add every element in the matrix to our final answer.

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Space Complexity: , the information stored in seen and in ans.

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#### Approach #2: Layer-by-Layer [Accepted]

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Intuition

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The answer will be all the elements in clockwise order from the first-outer layer, followed by the elements from the second-outer layer, and so on.

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Algorithm

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We define the -th outer layer of a matrix as all elements that have minimum distance to some border equal to . For example, the following matrix has all elements in the first-outer layer equal to 1, all elements in the second-outer layer equal to 2, and all elements in the third-outer layer equal to 3.

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[[1, 1, 1, 1, 1, 1, 1],\n [1, 2, 2, 2, 2, 2, 1],\n [1, 2, 3, 3, 3, 2, 1],\n [1, 2, 2, 2, 2, 2, 1],\n [1, 1, 1, 1, 1, 1, 1]]\n
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For each outer layer, we want to iterate through its elements in clockwise order starting from the top left corner. Suppose the current outer layer has top-left coordinates and bottom-right coordinates .

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Then, the top row is the set of elements for , in that order. The rest of the right side is the set of elements for , in that order. Then, if there are four sides to this layer (ie., and ), we iterate through the bottom side and left side as shown in the solutions below.

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Complexity Analysis

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Time Complexity: , where is the total number of elements in the input matrix. We add every element in the matrix to our final answer.

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Space Complexity: , the information stored in ans.

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Analysis written by: @awice

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