Sorting documentation fixups for 1.9 (#48440)

- Fix typos
- Clarify that ! means mutation, not "in-place-ness". This should be backported because sort! is even less in place in 1.9 than it already was in 1.8.
- Rewrite the section on default policy to reflect the new default policy
- Move examples and extended description of previously default sorting algorithms out of sort.md and into their respective docstrings (still rendered in sort.md)

Co-authored-by: Jeremie Knuesel <knuesel@gmail.com>

base/sort.jl

@@ -524,7 +524,7 @@ Base.size(v::WithoutMissingVector) = size(v.data)
     send_to_end!(f::Function, v::AbstractVector; [lo, hi])
 
 Send every element of `v` for which `f` returns `true` to the end of the vector and return
-the index of the last element which for which `f` returns `false`.
+the index of the last element for which `f` returns `false`.
 
 `send_to_end!(f, v, lo, hi)` is equivalent to `send_to_end!(f, view(v, lo:hi))+lo-1`
 
@@ -1242,7 +1242,7 @@ Otherwise, we dispatch to [`InsertionSort`](@ref) for inputs with `length <= 40`
 perform a presorted check ([`CheckSorted`](@ref)).
 
 We check for short inputs before performing the presorted check to avoid the overhead of the
-check for small inputs. Because the alternate dispatch is to [`InseritonSort`](@ref) which
+check for small inputs. Because the alternate dispatch is to [`InsertionSort`](@ref) which
 has efficient `O(n)` runtime on presorted inputs, the check is not necessary for small
 inputs.
 
@@ -1891,6 +1891,26 @@ Characteristics:
     ignores case).
   * *in-place* in memory.
   * *divide-and-conquer*: sort strategy similar to [`MergeSort`](@ref).
+
+  Note that `PartialQuickSort(k)` does not necessarily sort the whole array. For example,
+
+```jldoctest
+julia> x = rand(100);
+
+julia> k = 50:100;
+
+julia> s1 = sort(x; alg=QuickSort);
+
+julia> s2 = sort(x; alg=PartialQuickSort(k));
+
+julia> map(issorted, (s1, s2))
+(true, false)
+
+julia> map(x->issorted(x[k]), (s1, s2))
+(true, true)
+
+julia> s1[k] == s2[k]
+true
 """
 struct PartialQuickSort{T <: Union{Integer,OrdinalRange}} <: Algorithm
     k::T
@@ -1927,6 +1947,8 @@ Characteristics:
     case).
   * *not in-place* in memory.
   * *divide-and-conquer* sort strategy.
+  * *good performance* for large collections but typically not quite as
+    fast as [`QuickSort`](@ref).
 """
 const MergeSort     = MergeSortAlg()
 

doc/src/base/sort.md

@@ -21,7 +21,8 @@ julia> sort([2,3,1], rev=true)
  1
 ```
 
-To sort an array in-place, use the "bang" version of the sort function:
+`sort` constructs a sorted copy leaving its input unchanged. Use the "bang" version of
+the sort function to mutate an existing array:
 
 ```jldoctest
 julia> a = [2,3,1];
@@ -134,74 +135,33 @@ Base.Sort.partialsortperm!
 
 ## Sorting Algorithms
 
-There are currently four sorting algorithms available in base Julia:
+There are currently four sorting algorithms publicly available in base Julia:
 
   * [`InsertionSort`](@ref)
   * [`QuickSort`](@ref)
   * [`PartialQuickSort(k)`](@ref)
   * [`MergeSort`](@ref)
 
-`InsertionSort` is an O(n²) stable sorting algorithm. It is efficient for very small `n`,
-and is used internally by `QuickSort`.
+By default, the `sort` family of functions uses stable sorting algorithms that are fast
+on most inputs. The exact algorithm choice is an implementation detail to allow for
+future performance improvements. Currently, a hybrid of `RadixSort`, `ScratchQuickSort`,
+`InsertionSort`, and `CountingSort` is used based on input type, size, and composition.
+Implementation details are subject to change but currently available in the extended help
+of `??Base.DEFAULT_STABLE` and the docstrings of internal sorting algorithms listed there.
 
-`QuickSort` is a very fast sorting algorithm with an average-case time complexity of
-O(n log n). `QuickSort` is stable, i.e., elements considered equal will remain in the same
-order. Notice that O(n²) is worst-case complexity, but it gets vanishingly unlikely as the
-pivot selection is randomized.
-
-`PartialQuickSort(k::OrdinalRange)` is similar to `QuickSort`, but the output array is only
-sorted in the range of `k`. For example:
-
-```jldoctest
-julia> x = rand(1:500, 100);
-
-julia> k = 50:100;
-
-julia> s1 = sort(x; alg=QuickSort);
-
-julia> s2 = sort(x; alg=PartialQuickSort(k));
-
-julia> map(issorted, (s1, s2))
-(true, false)
-
-julia> map(x->issorted(x[k]), (s1, s2))
-(true, true)
-
-julia> s1[k] == s2[k]
-true
-```
-
-!!! compat "Julia 1.9"
-    The `QuickSort` and `PartialQuickSort` algorithms are stable since Julia 1.9.
-
-`MergeSort` is an O(n log n) stable sorting algorithm but is not in-place – it requires a temporary
-array of half the size of the input array – and is typically not quite as fast as `QuickSort`.
-It is the default algorithm for non-numeric data.
-
-The default sorting algorithms are chosen on the basis that they are fast and stable.
-Usually, `QuickSort` is selected, but `InsertionSort` is preferred for small data.
-You can also explicitly specify your preferred algorithm, e.g.
-`sort!(v, alg=PartialQuickSort(10:20))`.
-
-The mechanism by which Julia picks default sorting algorithms is implemented via the
-`Base.Sort.defalg` function. It allows a particular algorithm to be registered as the
-default in all sorting functions for specific arrays. For example, here is the default
-method from [`sort.jl`](https://github.com/JuliaLang/julia/blob/master/base/sort.jl):
-
-```julia
-defalg(v::AbstractArray) = DEFAULT_STABLE
-```
-
-You may change the default behavior for specific types by defining new methods for `defalg`.
+You can explicitly specify your preferred algorithm with the `alg` keyword
+(e.g. `sort!(v, alg=PartialQuickSort(10:20))`) or reconfigure the default sorting algorithm
+for custom types by adding a specialized method to the `Base.Sort.defalg` function.
 For example, [InlineStrings.jl](https://github.com/JuliaStrings/InlineStrings.jl/blob/v1.3.2/src/InlineStrings.jl#L903)
 defines the following method:
 ```julia
 Base.Sort.defalg(::AbstractArray{<:Union{SmallInlineStrings, Missing}}) = InlineStringSort
 ```
 
 !!! compat "Julia 1.9"
-    The default sorting algorithm (returned by `Base.Sort.defalg`) is guaranteed
-    to be stable since Julia 1.9. Previous versions had unstable edge cases when sorting numeric arrays.
+    The default sorting algorithm (returned by `Base.Sort.defalg`) is guaranteed to
+    be stable since Julia 1.9. Previous versions had unstable edge cases when
+    sorting numeric arrays.
 
 ## Alternate orderings