## Posts Tagged ‘Sorting’

### Sorting (in) embedded systems

Sunday, March 15th, 2009 Nigel Jones

Although countless PhD’s have been awarded on sorting algorithms, it’s not a topic that seems to come up much in embedded systems (or at least the kind of embedded systems that I work on). Thus it was with some surprise recently that I found myself needing to sort an array of integers. The array wasn’t very large (about twenty entries) and I was eager to move on to the real problem at hand and so I just dropped in a call to the standard C routine qsort(). I didn’t give it a great deal of thought because I ‘knew’ that a ‘Quick Sort’ algorithm is in general fast and well behaved and that with sorting so few entries I wasn’t too concerned about it being ‘optimal’. Anyway, with the main task at hand solved, on a whim I decided to take another look at qsort(), just to make sure that I wasn’t being too cavalier in my approach. Boy did I get a shock! My call to qsort() was increasing my code size by 1500 bytes and it wasn’t giving very good sort times either. For those of you programming big systems, this may seem acceptable. In my case, the target processor had 16K of memory and so 1500 bytes was a huge hit.

Surely there had to be a better solution? Well there’s always a better solution, but in my case in particular, and for embedded systems in general, what is the optimal sorting algorithm?

Well, after thinking about it for a while, I think the optimal sorting algorithm for embedded systems has these characteristics:

1. It must sort in place.
2. The algorithm must not be recursive.
3. Its best, average and worst case running times should be of similar magnitude.
4. Its code size should be commensurate with the problem.
5. Its running time should increase linearly or logarithmically with the number of elements to be sorted.
6. Its implementation must be ‘clean’ – i.e. free of breaks and returns in the middle of a loop.
##### Sort In Place

This is an important criterion not just because it saves memory, but most importantly because it obviates the need for dynamic memory allocation. In general dynamic memory allocation should be avoided in embedded systems because of problems with heap fragmentation and allocation performance. If you aren’t aware of this issue, then read this series of articles by Dan Saks on the issue.

##### Recursion

Recursion is beautiful and solves certain problems amazingly elegantly. However, it’s not fast and it can easily lead to problems of stack overflow. As a result, it should never be used in embedded systems.

##### Running Time Variability

Even the softest of real time systems have some time constraints that need to be met. As a result a function whose execution time varies enormously with the input data can often be problematic. Thus I prefer code whose execution time is nicely bounded.

##### Code Size

This is often a concern. Suffice to say that the code size should be reasonable for the target system.

##### Data Size Dependence

Sorting algorithms are usually classified using ‘Big O notation’ to denote how sensitive they are to the amount of data to be sorted. If N is the number of elements to be sorted, then an algorithm whose running time is N Log N is usually preferred to one whose running time is N2. However, as you shall see, for small N the advantage of the more sophisticated algorithms can be lost by the the overhead of the sophistication.

##### Clean Implementation

I’m a great proponent of ‘clean’ code. Thus code where one exits from the middle of a loop isn’t as acceptable as code where everything proceeds in an orderly fashion. Although this is a personal preference of mine, it is also codified in for example the MISRA C requirements, to which many embedded systems are built. Anyway to determine the optimal sorting algorithm, I went to the Wikipedia page on sorting algorithms and initially selected the following for comparison to the built in qsort: Comb, Gnome, Selection, Insertion, Shell & Heap sorts. All of these are sort in place algorithms. I originally eschewed the Bubble & Cocktail sorts as they really have nothing to commend them. However, several people posted comments asking that I include them – so I did. As predicted they have nothing to commend them. In all cases, I used the Wikipedia code pretty much as is, optimized for maximum speed. (I recognize that the implementations in Wikipedia may not be optimal – but they are the best I have). For each algorithm, I sorted arrays of 8, 32 & 128 signed integers. In every case I sorted the same random array, together with a sorted array and an inverse sorted array. First the code sizes in bytes:

```qsort()      1538
Gnome()        76
Selection()   130
Insertion()   104
Shell()       242
Comb()        190
Heap()        200
Bubble()      104
Cocktail()    140```

Clearly, anything is a lot better than the built in qsort(). However, we are not comparing apples and oranges, because qsort() is a general purpose routine, whereas the others are designed explicitly to sort integers. Leaving aside qsort(), the Gnome sort Insertion sort and Bubble sorts are clearly the code size leaders. Having said that, in most embedded systems, a 100 bytes here or there is irrelevant and so we are free to choose based upon other criteria.

#### Execution times for the 8 element array

```Name        Random  Sorted  Inverse Sorted
qsort()     3004     832    2765
Gnome()     1191     220    2047
Selection() 1120    1120    1120
Insertion() 544      287    756
Shell()     1233    1029    1425
Comb()      2460    1975    2480
Heap()      1265    1324    1153
Bubble()     875     208    1032
Cocktail()  1682     927    2056```

In this case, the Insertion sort is the clear winner. Not only is it dramatically faster in almost all cases, it also has reasonable variability and it has almost the smallest code size. Notice that the bubble sort for all its vaunted simplicity consumes as much code and runs considerably slower. Notice that the Selection sort’s running time is completely consistent – and not too bad when compared to other methods.

#### Execution times for the 32 element array

```Name        Random  Sorted  Inverse Sorted
qsort()     23004    3088   19853
Gnome()     17389     892   35395
Selection() 14392   14392   14392
Insertion()  5588    1179   10324
Shell()      6589    4675    6115
Comb()      10217    8638   10047
Heap()       8449    8607    7413
Bubble()    13664     784   16368
Cocktail()  17657    3807   27634```

In this case, the winner isn’t so clear cut. Although the insertion sort still performed well, it’s showing a very large variation in running time now. By contrast the shell sort has got decent times with small variability. The Gnome, Bubble and Cocktail sorts are showing huge variability in execution times (with a very bad worst case), while the Selection sort shows consistent execution time. On balance, I’d go with the shell sort in most cases.

#### Execution times for the 128 element array

```Name         Random  Sorted  Inverse Sorted
qsort()      120772   28411   77896
Gnome()      316550    3580  577747
Selection()  217420  217420  217420
Insertion()   88475    4731  158020
Shell()       41661   25611   34707
Comb()        50858   43523   48568
Heap()        46959   49215   43314
Bubble()     231294    3088  262032
Cocktail()   271821   15327  422266```

In this case the winner is either the shell sort or the heap sort depending on whether you want raw performance more or less when compared to performance variability. The Gnome, Bubble and Cocktail sorts are hopelessly outclassed. So what to make of all this? Well in any comparison like this there are a myriad of variables that one should take into account, and so I don’t believe these data should be treated as gospel. What is clear to me is that:

1. Being a general purpose routine, qsort() is unlikely to be the optimal solution for an embedded system.
2. For many embedded applications, a shell sort has a lot to commend it – decent code size, fast running time, well behaved and a clean implementation. Thus if you don’t want to bother with this sort of investigation every time you need to sort an array, then a shell sort should be your starting point. It will be for me henceforth.

A quick update: Check out this link for a very cool visualization of different sorting algorithms.

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