                          BUBBLE SORTING                                                                                                    Bubble sort is a routine designed to sort any number of elements      in an array.  The word 'BUBBLE' refers to the way in which the        data item works its way up the list until they are in ascending       or descending order.  The algorithm for sorting an array is as        follows:                                                                                                                                         STEP 1.  Examine the first two elements in the array                                                                                        STEP 2.  If they are not in order then change (SWAP) them (Do                  nothing if they are in order).                                                                                                     STEP 3.  Go to the next pair of elements (EXAMPLE: If you were                 examining elements 1 & 2, you would NOW be looking                    at elements 2 & 3).                                                                                                                STEP 4.  Repeat STEP 2 and 3 until the last two elements                       in the array have been compared.  At this point the                   largest element in the list has 'BUBBLED' up the list                 so that it is now the last item in the list.                                                                                       STEP 5.  Repeat the entire procedure for the first n-1                         elements in the array (n is the total number of                       elements in the array).  At this point the second                     largest element in the array will be second to last.                  Keep repeating this ( use n-2, n-3, n-4, ... n-[n-1])                 until the entire array is sorted.                                                                                             The bubble sort technique requires approx. « * Nư operations (N is    the total number of elements in the array).  As the array size        increases, the sorting time increases greatly.                                                                                                                                                                    The bubble sorting algorithm listed above can be easily translated    into any computer language code.  The following page displays a       bubble sorting program written in Quick BASIC (Please remember        that there are many different ways to translate any given             algorithm).  The following variables are used in the Quick BASIC      program listed on the next page:                                                                                                            ARR$       = Array of type string used to store the data items.                                                                             ARRAY.SIZE = Integer variable containing the size of the array.                                                                             I          = Integer index variable used showing the number of                     comparisons made during each pass.                                                                                             PASSES     = Integer variable used in FOR-LOOP showing the total                   number of passes made in the array.                                                                                            REM  Bubble sorting program                                                                                                                 FOR passes = 1 TO array.size-1          ' STEPS 1. & 5.                                                                                        FOR i = 1 TO array.size - passes  ' One less comparison each LOOP.                                                                             IF arr$(i) > arr$(i + 1) THEN     ' STEP 2.                                                                                                    SWAP arr$(i), arr$(i+1)        ' Swap if not in order.                                                                                   END IF                                                                                                                                   NEXT i                               ' STEP 3. & 4.                                                                                      NEXT passes                             ' STEP 5.                                                                                           REM  End of BUBBLE SORT                                               