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In this article at OpenGenus, we have explored the time and space complexity of im2row and im2col algorithms that are frequently, used for GEMM based Convolution algorithms.

## Time Complexity of im2row and im2col

In short:

- Time Complexity of im2row = Output Width x Kernel Height x Kernel Width x Number of Channels
- Time Complexity of im2col = Output Height x Kernel Height x Kernel Width x Number of Channels

```
im2row = OW x KH x KW x C
im2col = OH x KH x KW x C
```

Note the difference between im2row and im2col in terms of time complexity is the first parameter. im2row has Output Width (OW) whereas im2col has Output Height (OH).

## Space Complexity of im2row and im2col

The space complexity of both im2row and im2col is O(Input Height x Input Width x Channels) provided im2row and im2col are not in-place algorithms.

If you do not want to consider the output patch matrix/ data, then space complexity of im2row and im2col is O(1).

## im2row and im2col in Convolution

When im2row and im2col is called in Convolution, the function (im2row or im2col) is called B times where B is the batch size or the number of images to be processed.

Due to this, the time complexity for im2row or im2col function in Convolution becomes:

```
im2row = B x OW x KH x KW x C
im2col = B x OH x KH x KW x C
```

With this article at OpenGenus, you must have the complete idea of Time and Space Complexity of im2row and im2col.