Stable Sorting, ie, Minimally-Disruptive Sorting

Question

Suppose I have a list of things (numbers, to keep things simple here) and I have a function I want to use to sort them by, using SortBy. For example, the following sorts a list

Answer 1

After asking around, I was given a satisfying explanation:

Short answer: You want SortBy[list, {f}] to get a stable sort.

Long answer:

SortBy[list, f] sorts list in the order determined by applying f to each element of list, breaking ties using the canonical ordering explained under Sort. (This is the second documented "More Information" note in the documentation for SortBy.)

SortBy[list, {f, g}] breaks ties using the order determined by applying g to each element.

Note that SortBy[list, f] is the same as SortBy[list, {f, Identity}].

SortBy[list, {f}] does no tie breaking (and gives a stable sort), which is what you want:

In[13]:= SortBy[{19, 301, 201, 502, 501, 101, 300}, {Mod[#, 10] &}]

Out[13]= {300, 301, 201, 501, 101, 502, 19}

Finally, sakra's solution SortBy[list, {f, tie++ &}] is effectively equivalent to SortBy[list, {f}].