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The permutation vectors in
ser_permutation
are suitable if the number of permutation vectors matches the numberof dimensions of x
and if the length of each permutation vectorhas the same length as the corresponding dimension of x
.For 1-dimensional/1-mode data (list, vector,
dist
),order
can also be a singlepermutation vector of class ser_permutation_vector
or data which can beautomatically coerced to this class(e.g. a numeric vector).For matrix-like objects, the additional parameter
margin
can be specified to permuteonly a single dimension. In this case, order
can be a single permutation vector or a completeliis with pemutations for all dimensions. In the latter case, all permutations but the one specified inmargin
are ignored.For
dendrograms
and hclust
, subtrees are rotated torepresent the order best possible. If the order is not achieved perfectlythen the user is warned. This behavior can be changed with the extraparameter incompatible
which can take the values'warn'
(default), 'stop'
or 'ignore'
.Permute 3 0 501
Array order is an important consideration when looping over data in MATLAB/Octave. Choosing which dimension to loop over can actually have a significant impact on computational efficiency. In this tutorial, I introduce and illustrate the concept of array order, create a simple experiment to demonstrate the effects of array order when looping, and explain how the permute() command in MATLAB can be used to write efficient loops every time.
All of the source code for the examples can be found in the GitHub repository.
Array Order
When dealing with multi-dimensional or N-dimensional (ND) arrays it is important to be aware of how the data are stored in memory. This is referred to as array ordering. There are two general conventions: row-major and column-major ordering. These two different methods simply refer to which elements of the ND array area contiguous in memory, as well as, how such arrays should be linearly indexed. In column-major order, the elements of columns are contiguous in memory. and in row-major order, the elements of rows are contiguous in memory. So basically, we count elements down the columns for column-major and along the rows for row-major.
The following code snippet shows an example of a 3 x 3 matrix where each element shows what the linear index is. Similarly, you can think of the numbers in the matrix as the order of the data in memory (e.g. 1 is the first memory address, 2 is the second memory address, etc…).
We can trivially see how MATLAB stores the data by using the ‘:’ operator, which will unwrap the data. Since MATLAB is column-major, you can see that the colMaj matrix returns the index in proper order, but the rowMaj matrix returns the elements out of order.
To make this a little more clear, we can build a simple animation. The animation shows how a matrix is unwrapped using both conventions. The linear index at the bottom shows how the individual elements are actually stored in memory.
Looping Over Data
So why does array order matter? Well, when you are looping over data, array order can actually have a significant impact on the run time. It is much more efficient to access data that is contiguous in memory. So having the knowledge of how our data is organized in memory can also help us write more computationally efficient code.
Let’s demonstrate this with a simple example. Suppose we have a collection of images that we want to operate on. The images are of size (nX, nY) pixels and we have nI images in our data set. There are two likely ways in which these images might be arranged in memory: [nX, nY, nI] or [nI, nX, nY]. In the first orientation, the images will be arranged contiguously in memory, while in the second orientation it will not.
To see what kind of difference array order makes, we can write the following test code. Note, the tic() and toc() functions allow us to time parts of our code execution. If you are not familiar with this functions, tic() essentially starts a timer and toc() records how much time has elapsed since you started the timer. You can find more documentation in the MATLAB help.
To illustrate the effects of array order, the code was run with nX = nY = 128 and nI taking on values of 1,000, 10,000, and 50,000. The brave and curious are encouraged to download the source code and try out more interesting combinations!
Starting with just 1,000 images, we can see that the array order doesn’t seem to have a big impact, however, there is still about a 10% increase in run time.
Permute 3 0 5 4a+8 9 2a
Increasing the number of images to 10,000, we start to see an actual difference in run times. There is 2.4 factor increase in run time when the images are not stored contiguously in memory.
Further increasing the number of images to 50,000, we continue to see larger increase in the run time. There is now a factor 8 increase in the run time!
3 0google
It is important to note that this was a simple example with a single loop and only three dimensions. This effect can actually be much worse for arrays with more dimensions or when we are nesting multiple loops. Sometimes it is worthwhile to add some tic() / toc() calls to your code or to run Profiler to where you might want to make some updates to your code.
Permute() in MATLAB
MATLAB includes a function called permute(), which is a generalization of the transpose function but for ND arrays. Permute() takes in an ND-array and the desired array order and then returns the rearranged data. The syntax looks this: newArray = permute( oldArray, [oldDim1, oldDim2, oldIm3, etc…] ). In this syntax the variables oldDim# represent array order index of the old array and the position in the function input represent the new array order position.
For example, suppose we have an ND-array, A, of shape [200, 450, 120, 680], and that we called the permute() function as such: B = permute( A, [3, 1, 4, 2]). The resulting shape of array B would be [120, 200, 680, 450]. Here we have moved dimension 3 to dimension 1, dimension 1 to dimension 2, dimension 4 to dimension 3, and dimension 2 to dimension 4.
Using the permute() function, we can rearrange any ND-array based on what dimension we want to loop over. Returning to our previous image example, we can take our array imageData of shape [nI, nX, nY], and rearrange this to an array of shape [nI, nX, nY] by using the following command.
This would now allow us to loop over the last dimension, which we just showed to be more computational efficient. Adding this simple line to our previous example, we can re-run our test. The updated code will look as follows.
Running this experiment with nI = 50,000, results in the following output.
Using the permute() command has significantly reduced our looping time. There is, of course, a small overhead for using the permute function, but in almost all practical cases it is best to include it. As I mentioned previously, this is especially true with high-dimensional arrays and when we are nesting multiple loops.
Bleach Vs Naruto 3.0
Summary
In this tutorial we covered the importance of array order when looping over ND-arrays in MATLAB/Octave. Using some simple MATLAB code, we demonstrated the benefit of looping over data that is contiguous in memory. We further showed that the permute() command allows us to rearrange the array dimensions before looping so we can always ensure that we are looping efficiently.
Here are a few points to remember:
- Data are arranged in column-major order in MATLAB
- We should always loop over the outermost dimensions to make our loops as efficient as possible.
- Permute() can be used before (and after) any loop to ensure that we loop over ND-arrays efficiently.
- For nested loops, the innermost loop should loop over the outermost dimension.
Permute 3 0 5 6 Installer
Happy coding!