Question

I was reading the Dutch national flag problem, but couldn’t understand what the low and high arguments are in the threeWayPartition function in the C++ implementation.

If I assume them as min and max elements of the array to be sorted, then the if and else if statements doesn’t makes any sense since (data[i] < low) and (data[i] > high) always returns zero.

Where am I wrong?

Answer 1

low and high are the values you have defined to do the three-way partition i.e. to do a three-way partition you only need two values:

[bottom] <= low  < [middle] < high <= [top]

In the C++ program what you are moving are the positions where the partitions occurred. A step-by-step example:

data = [ 3, 1, 4, 9, 8, 2, 6, 9, 0 ]
low  = 4
high = 8

   [ 3 , 1 , 4 , 9 , 8 , 2 , 6 , 9 , 0 ]
p^  i^                                  q^

   [ 3 , 1 , 4 , 9 , 8 , 2 , 6 , 9 , 0 ]
    p^  i^                              q^

   [ 3 , 1 , 4 , 9 , 8 , 2 , 6 , 9 , 0 ]
        p^  i^                          q^

   [ 3 , 1 , 4 , 9 , 8 , 2 , 6 , 9 , 0 ]
        p^      i^                      q^

   [ 3 , 1 , 4 , 0 , 8 , 2 , 6 , 9 , 9 ]
        p^      i^                  q^

   [ 3 , 1 , 0 , 4 , 8 , 2 , 6 , 9 , 9 ]
            p^      i^              q^

   [ 3 , 1 , 0 , 4 , 9 , 2 , 6 , 8 , 9 ]
            p^      i^          q^

   [ 3 , 1 , 0 , 4 , 6 , 2 , 9 , 8 , 9 ]
            p^      i^      q^

   [ 3 , 1 , 0 , 4 , 6 , 2 , 9 , 8 , 9 ]
            p^          i^  q^

   [ 3 , 1 , 0 , 2 , 6 , 4 , 9 , 8 , 9 ]
                p^         iq^

As the algorithm says you:

• Swap the element above the bottom (i.e. p + 1) because everything below the bottom has been already checked, or
• Swap the element below the top (i.e. q - 1) because everything above the top has been already checked, or
• Leave the element where it is because it belongs to middle.

You get [3, 1, 0, 2], [6, 4] and [9, 8, 9] as bottom, middle and top partitions respectively.