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# Sorting

Sorting denotes the action of rearranging a sequence, such as a list of numbers, so that the elements are put in a specific order (e.g. ascending or descending). It is the opposite of shuffling. In computer science sorting enjoys the status of a widely spanning, quite curious topic. There are probably hundreds of sorting algorithms and their slight modifications, each with pros and cons -- attributes of these algorithms are being studied, e.g. their time complexity, memory efficiency, stability etc. Sorting algorithms are also a favorite subject of programming classes as they provide a good exercise for programming and analysis of algorithms and can be nicely put on a test :) Sorting algorithms are like Pokemon: some are big and heavy but winning in raw "strength", others are small and cute, some are just funny... and everyone probably has a favorite. { Gotta implement them all? ~drummyfish }

Some celebrities among sorting algorithms are the bubble sort (a simple KISS algorithm), quick sort (a super fast one), merge sort (also lightning fast) and stupid sort (just tries different permutations until it hits the jackpot). In case you're wondering, joke sorts exist too: Staling sort for example traverses the array and eliminates any element that's not in correct order.

In our day to day lives we commonly get away with some of the simplest, uncomplicated sorting algorithms (such as bubble sort or insertion sort) anyway, unless we're programming a database or otherwise treating enormous amounts of data. If we need to sort just a few hundred of items and/or the sorting doesn't occur very often, a simple algorithm does the job well, sometimes even faster due to a potential initial overhead of a very complex algorithm. So always consider the KISS approach first.

Attributes of sorting algorithms we're generally interested in are the following:

- time and space complexity: Time and space complexity hints on how fast the algorithm will run and how much memory it will need, specifically we're interested in the best, worst and average case depending on the length of the input sequence. Indeed we ideally want the fastest algorithm, but it has to be known that a better time complexity doesn't have to imply a faster run time in practice, especially with shorter sequences. An algorithm that's extremely fast in best case scenario may be extremely slow in non-ideal cases. With memory, we are often interested whether the algorithm works in place; such an algorithm only needs a constant amount of memory in addition to the memory that the sorted sequence takes, i.e. the sequence is sorted in the memory where it resides.
- implementation complexity: A simple algorithm is better if it's good enough. It may lead to e.g. smaller code size which may be a factor e.g. in embedded.
- stability: A stable sorting algorithm preserves the order of the elements that are considered equal. With pure numbers this of course doesn't matter, but if we're sorting more complex data structures (e.g. sorting records about people by their names), this attribute may become important.
- comparative vs non-comparative: A comparative sort only requires a single operation that compares any two elements and says which one has a higher value -- such an algorithm is general and can be used for sorting any data, but its time complexity of the average case can't be better than O(n log(n))*. Non-comparison sorts can be faster as they may take advantage of other possible integer operations.
- recursion and parallelism: Some algorithms are recursive in nature, some are not. Some algorithms can be parallelised e.g. with a GPU which will greatly increase their speed.
- other: There may be other specific, e.g. some algorithms are are slow if sorting an already sorted sequence (which is addressed by adaptive sorting), so we may have to also consider the nature of data we'll be sorting. Other times we may be interested e.g. in what machine instructions the algorithm will compile to etc.

In practice not only the algorithm but also details of its implementation matters. For instance if we have a sequence of very large data structures to sort, we may want to avoid physically rearranging these structures in memory, this could be slow. In such scenario we may want to use indirect sorting: we create an additional list whose elements are indices to the main sequence, and we only sort this list of indices.

And now a little note on comparing: in order to put given sequence in order (pun intended) it's required that the elements of the sequence be comparable with each other -- it sounds evident enough but the fact may demand a little closer attention, especially when we're sorting something more complex than plain numbers (e.g. records of people in a database, text strings and so on). Mathematically speaking we must ensure there exists an ordering relation on the set of things we are sorting AND we'll usually want the ordering to be total (as opposed to partial ordering), i.e. ensuring that any two items will be comparable, or at least be aware of whether we have a total ordering and what the implications will be. And the algorithm of HOW to compare the items doesn't have to be obvious either, in which case we should give it careful thought. Take comparing text strings as an example: do digits precede letters? Do we compare strings with numbers alphabetically or numerically (consider e.g. `12345` vs `234`)? Do lowercase letters precede uppercase or vice versa OR will we be ignoring case? How do we treat spaces? Ignore them or give them some value (and what)? What about special characters? What about accents? And so on. For an inspiration you can take a look in a dictionary or encyclopedia in your library, they typically have the exact method of their ordering explained somewhere.

## List Of Sorting Algorithms

TODO

## Example And Code

For starters let's take a look at one of the simplest sorting algorithms, bubble sort. Its basic version looks something like this (pseudocode):

    for j from 0 to N - 2 (inclusive)
      for i from 0 to to N - 2 - j (inclusive)
        if array[i] > array[i + 1]
          swap array[i] and array[i + 1]

How does this work? Firstly notice there are two loops. The outer loop, with counter variable j, runs N - 1 times -- in each iteration of this loop we will ensure one value gets to its correct place; specifically the values will be getting to their correct places from the top -- highest values will be sorted first (you can also implement the algorithm the other way around too, to sort the lowest values first, try it as an exercise). This makes sense, imagine that we have e.g. a sequence of length N = 4 -- then the outer loop will run N - 1 = 3 times (j will have values 0, 1 and 2); after fist iteration 1 value will be in its correct place, after 2 iterations 3 values will be in place and after 3 iterations 3 values will be in place which also means the last (forth) value has to be in place too, i.e. the array must be sorted. Now for the inner loop (with variable i): this one ensures actually getting the value in its place. Notice it goes from 0 to the top and always compares two adjacent values in the array -- if the bottom neighbor is higher than the top neighbor, the loop swaps them, ensuring that the highest value will get to the top (it kind of "bubbles" up, hence the algorithm name). Also notice this loop doesn't always go to the very end of the array! It subtracts the value j from its top boundary because there the values that are already in place reside, so we don't need to sort them anymore; the inner loop can end earlier and earlier as the outer loop progresses. The algorithm would still work if we went through the whole array every time (try it), but its time complexity would suffer, i.e. by noticing the inner loop can get progressively shorter we greatly optimize the algorithm. Anyway, how the algorithm actually works is best seen on an example, so let's now try to use the algorithm to sort the following sequence:

    3 7 8 3 2

The length of the sequence is N = 5, so j (the outer loop) will go from 0 to 3. The following shows how the array changes (`/\` shows comparison of neighbors, read top to bottom and left to right):

             j = 0        j = 1        j = 2        j = 3

                                                    swapped
    i = 0    /\           /\           /\           /\
             37832        37328        33278        23378     <-- SORTED
                                                     """"
                          swapped      swapped
    i = 1     /\           /\           /\
             37832        33728        32378  <--- last 3 items are in their places
                                         """
             swapped      swapped
    i = 2      /\           /\
             37382        33278  <--- last 2 items are in their places
                             ""
             swapped
    i = 3       /\
             37328   <--- last item is in its place
                 "

Hopefully it's at least a bit clear -- if not, try to perform the algorithm by hand, that's a practically guaranteed way of gaining understanding of the algorithm.

Now let's see other algorithms and some actual runnable code. The following is a C program that shows implementations of some of the common sorting algorithms and also measures their speed:

    #include <stdio.h>
    #include <time.h>
    #include <stdlib.h>

    #define N 64

    char array[N + 1]; // extra 1 for string terminating zero

    void swap(int i1, int i2)
    {
      int tmp = array[i1];
      array[i1] = array[i2];
      array[i2] = tmp;
    }

    void setupArray(void) // fills the array with pseudorandom ASCII letters
    {
      array[N] = 0;
      srand(123);

      for (int i = 0; i < N; ++i)
        array[i] = 'A' + rand() % 26;
    }

    void test(void (*sortFunction)(void), const char *name)
    {
      int timeTotal = 0;

      for (int i = 0; i < 64; ++i) // run the sort many times to average it a bit
      {
        setupArray();
        long int t = clock();
        sortFunction();
        timeTotal += clock() - t;
      }

      printf("%-10s: %s, CPU ticks: %d\n",name,array,(int) timeTotal);
    }

    // ============================ sort algorithms ================================

    void sortBubble(void)
    {
      for (int j = 0; j < N - 1; ++j)
        for (int i = 0; i < N - 1 - j; ++i)
          if (array[i] > array[i + 1])
            swap(i,i + 1);
    }

    void sortBubble2(void) // simple bubble s. improvement, faster if already sorted
    {
      for (int j = 0; j < N - 1; ++j)
      {
        int swapped = 0;

        for (int i = 0; i < N - 1 - j; ++i)
          if (array[i] > array[i + 1])
          {
            swap(i,i + 1);
            swapped = 1;
          }

        if (!swapped) // if no swap happened, the array is already sorted
          break;
      }
    }

    void sortInsertion(void)
    {
      for (int j = 1; j < N; ++j)
        for (int i = j; i > 0; --i)
          if (array[i] < array[i - 1]) // keep moving down until we find its place
            swap(i,i - 1);
          else
            break;
    }

    void sortSelection(void)
    {
      for (int j = 0; j < N - 1; ++j)
      {
        int min = j;

        for (int i = j + 1; i < N; ++i) // find the minimum
          if (array[i] < array[min])
            min = i;

        swap(j,min);
      }
    }

    void sortStupid(void)
    {
      while (1)
      {
        for (int i = 0; i < N; ++i) // check if the array is sorted
          if (i == (N - 1))
            return;
          else if (array[i] > array[i + 1])
            break; // we got to the end, the array is sorted

        for (int i = 0; i < N; ++i) // randomly shuffle the array
          swap(i,rand() % N);
      }
    }

    void _sortQuick(int a, int b) // helper recursive function for the main quick s.
    {
      if (b <= a || a < 0)
        return;

      int split = a - 1;

      for (int i = a; i < b; ++i)
        if (array[i] < array[b])
        {
          split++;
          swap(split,i);
        }

      split++;
      swap(split,b);

      _sortQuick(a,split - 1);
      _sortQuick(split + 1,b);
    }

    void sortQuick(void)
    {
      _sortQuick(0,N - 1);
    }

    int main(void)
    {
      setupArray();
      printf("array     : %s\n",array);

    #if 0 // stupid sort takes too long, only turn on while decreasing N to like 10
      test(sortStupid,"stupid");
    #endif
      test(sortBubble,"bubble");
      test(sortBubble2,"bubble2");
      test(sortInsertion,"insertion");
      test(sortBubble2,"selection");
      test(sortQuick,"quick");

      return 0;
    }

    // TODO: let's add more algorithms in the future :-)

It may output for example:

    array     : RLPALFTOCFWGVJYPLLUNEPDBSOMIBMXSXMVLROZUWXARHAIUNCJTUNVMDHWHTTZT
    bubble    : AAABBCCDDEFFGHHHIIJJLLLLLMMMMNNNOOOPPPRRRSSTTTTTUUUUVVVWWWXXXYZZ, CPU ticks: 1191
    bubble2   : AAABBCCDDEFFGHHHIIJJLLLLLMMMMNNNOOOPPPRRRSSTTTTTUUUUVVVWWWXXXYZZ, CPU ticks: 1164
    insertion : AAABBCCDDEFFGHHHIIJJLLLLLMMMMNNNOOOPPPRRRSSTTTTTUUUUVVVWWWXXXYZZ, CPU ticks: 665
    selection : AAABBCCDDEFFGHHHIIJJLLLLLMMMMNNNOOOPPPRRRSSTTTTTUUUUVVVWWWXXXYZZ, CPU ticks: 1217
    quick     : AAABBCCDDEFFGHHHIIJJLLLLLMMMMNNNOOOPPPRRRSSTTTTTUUUUVVVWWWXXXYZZ, CPU ticks: 365

## See Also

- searching
- pathfinding

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