Comb sort

Comb sort

Class	Sorting algorithm
Data structure	Array
Worst-case performance	$O(n\log n)$
Worst-case space complexity	$O(n)$
Optimal	?

In computer science, comb sort is a relatively simplistic sorting algorithm originally designed by Wlodek Dobosiewicz in 1980 and later rediscovered and popularised by Stephen Lacey and Richard Box, who described it in Byte Magazine in April 1991. Comb sort improves on bubble sort, and rivals in speed more complex algorithms like Quicksort. The basic idea is to eliminate turtles, or small values near the end of the list, since in a bubble sort these slow the sorting down tremendously. (Rabbits, large values around the beginning of the list, do not pose a problem in bubble sort.).

In bubble sort, when any two elements are compared, they always have a gap (distance from each other) of 1. The basic idea of comb sort is that the gap can be much more than one. (Shell sort is also based on this idea, but it is a modification of insertion sort rather than bubble sort.)

The gap starts out as the length of the list being sorted divided by the shrink factor (generally 1.3; see below), and the list is sorted with that value (rounded down to an integer if needed) for the gap. Then the gap is divided by the shrink factor again, the list is sorted with this new gap, and the process repeats until the gap is 1. At this point, comb sort continues using a gap of 1 until the list is fully sorted. The final stage of the sort is thus equivalent to a bubble sort, but by this time most turtles have been dealt with, so a bubble sort will be efficient.

Shrink factor

The shrink factor has a great effect on the efficiency of comb sort. In the original article, the authors suggested 1.3 after trying some random lists and finding it to be generally the most effective. A value too small slows the algorithm down because more comparisons must be made, whereas a value too large may not kill enough turtles to be practical.

Text describes an improvement to comb sort using the base value $1/\left(1-{\frac {1}{e^{\varphi }}}\right)\approx 1.247330950103979$ as the shrink factor. It also contains a pseudocode implementation with a pre-defined gap table.

Combsort11

With a shrink factor around 1.3, there are only three possible ways for the list of gaps to end: (9, 6, 4, 3, 2, 1), (10, 7, 5, 3, 2, 1), or (11, 8, 6, 4, 3, 2, 1). Only the last of those endings kills all turtles before the gap becomes 1. Therefore, significant speed improvements can be made if the gap is set to 11 whenever it would otherwise become 9 or 10. This variation is called Combsort11.

If either of the sequences beginning with 9 or 10 were used, the final pass with a gap of 1 is less likely to completely sort the data, necessitating another pass with a gap of 1. The data is sorted when no swaps were done during a pass with gap = 1.

Pseudocode example of combsort11

function combsort11(array input)
    gap := input.size //initialize gap size
    
    loop until gap <= 1 and swaps = 0
        //update the gap value for a next comb
        if gap > 1
            gap := gap / 1.3
            if gap = 10 or gap = 9
                gap := 11
            end if
        end if
        

        i := 0
        swaps := 0 //see bubblesort for an explanation
        

        //a single "comb" over the input list
        loop until i + gap >= input.size //see shellsort for similar idea
            if input[i] > input[i+gap]
                swap(input[i], input[i+gap])
                swaps := 1 // Arithmetic_overflow fixup
            end if
            i := i + 1
        end loop
       

    end loop
end function

See also

• Bubble sort, a generally slower algorithm, is the basis of comb sort.
• Cocktail sort, or bidirectional bubble sort, is a variation of bubble sort that also addresses the problem of turtles.

External links

• .NET Implementation of comb sort and several other algorithms