RadixSort

RadixSort was developed in 1887 by Herman Hollerith and was used for sorting numbered punched cards. The algorithm is stable and fast.

How RadixSort works
BucketSort in its natural variant is the predecessor of RadixSort. If there were for example bills to be sorted by date, one started by building stacks with bills from the same year. Lets assume there were bills from ten years so we have ten stacks with bills now. Then each of these stacks was subdivided into twelve stacks representing the month (we have 120 stack now on the table). Now each of them was subdivided into 30/31 stacks representing the days of the months. We end up with about 3.600 stacks that have to be recollected in correct order (nobody will ever do it in this way; it shall only show the complexity).

The brilliant idea of Hollerith was to reverse the sorting order. In order to show what his idea was, we sort 20 three digit numbers, presented here.

387   310     529     174     011     227     020     901     522     771 993    348     974     213     226     004     918     098     107     109

In the first run the numbers are sorted by the last digit; since there are 10 different digits we use ten buckets. The result of this sort we have here: B-0   B-1     B-2     B-3     B-4     B-5     B-6     B-7     B-8     B-9 -- 310   011     522     993     174             226     387     348     529 020    901             213     974                     227     918     109        771                     004                     107     098

Now we recollect the numbers from the buckets without changing their order.

310   020     011     901     771     522     993     213     174     974 004    226     387     227     107     348     918     098     529     109

Sorting by the middle digit gives this result:

B-0   B-1     B-2     B-3     B-4     B-5     B-6     B-7     B-8     B-9 -- 901   310     020             348                     771     387     993 004    011     522                                     174             098 107    213     226                                     974 109    918     227                529

It is obvious that the content of the buckets is still sorted ascending by the last digit. We recollect again the content of the buckets without changing their order.

901   004     107     109     310     011     213     918     020     522 226    227     529     348     771     174     974     387     993     098

The third and last run gives this distribution to the buckets:

B-0   B-1     B-2     B-3     B-4     B-5     B-6     B-7     B-8     B-9 -- 004   107     213     310             522             771             901 011    109     226     348             529                             918 020    174     227     387                                             974 098                                                                    993

After recollecting the numbers the sort is finished.

004   011     020     098     107     109     174     213     226     227 310    348     387     522     529     771     901     918     974     993

The result of Hollerith idea was, that one only needs as many buckets as there are different symbols. Instead of one thousand buckets for a naive BucketSort a set of ten buckets is used three times in order to get the same result.

But what exactly is the trick for this enormous reduction? We have a look at the stack of numbers after the first recollection. We see in the stack a layer with numbers ending on 0, followed by a layer ending on 1 and so on. After the second recollection we have a layer where the second last digit is 0 and this layer could have 10 sub-layers with the last digit being 0, 1 etc. The trick of Hollerith was to substitute the sub-buckets by sub-layers which are automatically in perfect order all the time. And there is no effort necessary for administration of sub-layers. If a sub-layer exists, fine, if not, fine as well; it is not necessary for the algorithm to care about it.

And we see directly the reason why RadixSort is stable. When a layer is divided into sub-layers, the order of the data records is not changed. Maybe they do not have the same neighbors any more, but the position numbers (from the original stack) are still ascending.

We go back to the example with the bills. We want them to be sorted by year, within the years sorted by month and within the months by day. Since RadixSort is stable it becomes very easy. We first sort (in two runs) by days, then we sort (in two runs) by month and at the end (in two or four runs) by year. Sorting done!

Timing
It is obvious that any digit of any key is only processed once. Hence the number of key handlings is


 * A = n * d

where n is the number of data records and d is the maximum number of digits in the key. Since the processing of any digit of any key takes the same time RadixSort has a linear characteristic.

In order to be able to compare RadixSort with other algorithms it is better to describe the characteristic by


 * T = B * n * lg256(N)

where B is a constant reflecting the implementation and the term lg256(N) gives the number of repetitions where N is the maximum number of different key-values.