# Tabasco Sort: A Super-optimal Merge Sort

EDIT: 2012-08-29: I added a section to compare comparison counts with known bounds for general comparison sorts and sorting networks.

In an earlier post, I noted how tedious coding unrolled sorts can be. Frankly, that’s the main reason I stopped at leaf sorts of size three. Recently, Neil Toronto wrote a nice post on the generation of size-specialised merge sorts. The post made me think about that issue a bit more, and I now have a neat way to generate unrolled/inlined merge sorts that are significantly smaller than the comparison and size “-optimal” inlined merge sorts.

The code is up as a single-file library, and sorts short vectors faster than SBCL’s inlined heapsort by a factor of two to three… and compiles to less machine code. The generator is a bit less than 100 LOC, so I’m not sure I want to include it in the mainline yet. If someone wants to add support for more implementations, I’d be happy to extend Tabasco sort, and might even consider letting it span multiple files ;)

## Differently-optimal sorts

The inlined merge sort for three values (a, b, and c) is copied below. It has to detect between $$3! = 6$$ permutation, and does so with an optimal binary search tree. That scheme leads to code with $$n! - 1$$ comparisons to sort n values, and for which each execution only goes through two or three comparisons ($$\approx \lg n!$$).

An optimal sorting network for three values needs only three comparisons, and always executes those three comparisons.

Finally, the leaf sort I used in SBCL is smaller than the inlined merge sort (three comparisons), but sometimes executes fewer than three comparisons. It’s superoptimal ;)

The optimal merge sort is larger than the optimal sorting network, and the optimal sorting network performs potentially more comparisons than the optimal merge sort…

Each implementation is optimal for different search spaces: the optimal merge sort never merges continuations, and the sorting network’s only control dependencies are in the conditional swaps.

The “super-optimal” merge sort does better by allowing itself both assignments (or tail-calls) and non-trivial control flow: it’s smaller than the inlined merge sort (but performs more data movement), and potentially executes fewer comparisons than the sorting network (with a larger total code size). And, essential attribute in practice, it’s easy to generate. This contrasts with optimal sorting networks, for which we do not have any generation method short of brute force search; in fact, in practice, sorting networks tend to exploit suboptimal (by a factor of $$\log n$$) schemes like bitonic sort or odd-even merge sort. Then again, we’re only concerned with tiny sorts, and asymptotics can be misleading: Batcher’s odd-even merge sort happens to be optimal for $$n\leq 8$$. The issue with sorting networks remains: data-oblivious control flow pessimises their comparison count.

## Generalising from size 3

What the last merge sort does is to first sort both halves of the values (a is trivially sorted, and b c needs one conditional swap), and then, assuming that each half (a and b c) is sorted, find the right permutation with which to merge them. Rather than $$n!$$ permutations, a merge only needs to distinguish between $$C(n, \lfloor n/2\rfloor) = \frac{n!}{\lfloor n/2\rfloor!\lceil n/2\rceil!}$$ permutations, and the recursive sorts are negligible compared to the merge step. That’s a huge reduction in code size!

A simple merge generator fits in half a screenful.

Given two lists of sorted variables, emit-merge calls emit-merge-1 to generate code that finds the right permutation, and executes it at the leaf. A binary search tree is generated by keeping track of the merged list in a reverse-order (to enable tail-sharing) accumulator of variable names. As expected, when merging a list of length one with another of length two, we get pretty much the code I wrote by hand earlier.

CL-USER> (emit-merge '(a) '(b c))
(if (< b a)
(if (< c a)
(setf (values a b c) (values b c a))
(setf (values a b c) (values b a c)))
(setf (values a b c) (values a b c)))


There’s one striking weakness: we generate useless code for the identity permutation. We could detect that case, or, more generally, we could find the cycle decomposition of each permutation and use it to minimise temporary values; that’d implicitly take care of cases like (setf (values a b c) (values b a c)), in which some values are left unaffected.

## A smarter permutation generator

I’ll represent permutations as associative lists, from source to destination. Finding a cycle is easy: just walk the permutation from an arbitrary value until we loop back.

To generate the code corresponding to a permutation, I can extract all the cycles, execute each cycle with a rotatef.

The merge step for a sort of size three is now a bit more explicit, but likely compiles to code that uses fewer registers as well. It probably doesn’t matter on good SSA-based backends, but those are the exception rather than the norm in the Lisp world.

CL-USER> (emit-merge '(a) '(b c))
(if (< b a)
(if (< c a)
(progn (rotatef a b c))
(progn (rotatef a b)))
(progn))


The only thing missing for a merge sort is to add base cases and recursion. The base case is easy: lists of length one are sorted. Inlining recursion is trivial, as is usually the case when generating Lisp code.

The resulting three-value sorter looks good; there are some redundancies with nested or empty progns, but any half-decent compiler will take care of that. Python certainly does a goob job on that code.

CL-USER> (emit-sort '(a b c))
(progn
nil
(progn
nil
nil
(if (< c b)
(progn (rotatef b c))
(progn)))
(if (< b a)
(if (< c a)
(progn (rotatef a b c))
(progn (rotatef a b)))
(progn)))
CL-USER> (disassemble (lambda (a b c)
(declare (type fixnum a b c))
(inline-sort a b c)))
; disassembly for (lambda (a b c))
; 0E88F150:       498BD0           mov rdx, r8                ; no-arg-parsing entry point
;       53:       498BC9           mov rcx, r9
;       56:       498BDA           mov rbx, r10
;       59:       4D39CA           cmp r10, r9
;       5C:       7C30             jl L3
;       5E: L0:   4C39C1           cmp rcx, r8
;       61:       7D0B             jnl L1
;       63:       4C39C3           cmp rbx, r8
;       66:       7C1B             jl L2
;       68:       488BD1           mov rdx, rcx
;       6B:       498BC8           mov rcx, r8
;       6E: L1:   488BF9           mov rdi, rcx
;       71:       488BF3           mov rsi, rbx
;       74:       488D5D10         lea rbx, [rbp+16]
;       78:       B906000000       mov ecx, 6
;       7D:       F9               stc
;       7E:       488BE5           mov rsp, rbp
;       81:       5D               pop rbp
;       82:       C3               ret
;       83: L2:   488BD1           mov rdx, rcx
;       86:       488BCB           mov rcx, rbx
;       89:       498BD8           mov rbx, r8
;       8C:       EBE0             jmp L1
;       8E: L3:   498BCA           mov rcx, r10
;       91:       498BD9           mov rbx, r9
;       94:       EBC8             jmp L0


I thought about generating calls to values in the final merge, rather than permuting, but decided against: I know SBCL doesn’t generate clever code when permuting registers, and that’d result in avoidable spills. I also considered generating code in CPS rather than emitting assignments; again, I decided against because I can’t depend on SBCL to emit clever permutation code. The transformation would make sense in a dialect with weaker support (both compiler and social) for assignment.

## How good is the generated code?

Both this inline merge sort and the original, permutation-free (except at the leaves), one actually define the exact same algorithms. For any input, both (should) execute the same comparisons in the same order: the original inline merge sort simply inlines the whole set of execution traces, without even merging control flow.

The permutation-free sort guarantees that it never performs redundant comparisons. Whether it performs the strict minimum number of comparisons, either on average or in the worst case, is another question. At first, I thought that $$\lg n!$$ should be a good estimate, since the search tree seems optimally balanced. The problem is $$n!$$ tends to have many other prime factors than 2, and we can thus expect multiple comparisons to extract less than 1 bit of information, for each execution. The lower bound can thus be fairly far from the actual value… Still, this question is only practically relevant for tiny sorts, so the discrepancy shouldn’t be too large.

A simple way to get the minimum, average or maximum comparison count would be to annotate the permutation-free generator to compute the shortest, average and longest path as it generates the search tree.

I’ll instead mimic the current generator.

The first step is to find the number of comparisons to perform a merge of m and n values.

If either m or n is zero, merging the sequences is trivial.

Otherwise, the minimum number of comparisons is (min m n): the sequences are pre-sorted, and the shortest sequence comes first. The maximum is (1- (+ m n)). I couldn’t find a simple expression for the average over all permutations. Instead, I iterate over all possible combinations and count the number of comparisons until either subsequence is exhausted.

Counting the number of comparisons in sorts is then trivial, with a recursive function. I didn’t even try to memoise repeated computations: the generated code is ludicrously long when sorting as few as 13 or 14 values.

CL-USER> (loop for i from 2 upto 16
do (multiple-value-bind (min avg max)
(sort-count i)
(format t "~4D ~6,2F ~6D ~6,2F ~6D~%"
i (log (! i) 2d0)
min (float avg) max)))
;; n  lg(n!)   min   avg     max ;   best   network
2   1.00      1   1.00      1 ;   1      1
3   2.58      2   2.67      3 ;   3      3
4   4.58      4   4.67      5 ;   5      5
5   6.91      5   7.17      8 ;   7      9
6   9.49      7   9.83     11 ;   10     12
7  12.30      9  12.73     14 ;   13     16
8  15.30     12  15.73     17 ;   16     19
9  18.47     13  19.17     21 ;   19     25?
10  21.79     15  22.67     25 ;   22     29?
11  25.25     17  26.29     29 ;   26     35?
12  28.84     20  29.95     33 ;   30     39?
13  32.54     22  33.82     37 ;   34     45?
14  36.34     25  37.72     41 ;   38     51?
15  40.25     28  41.69     45 ;   42     56?
16  44.25     32  45.69     49 ;   46?    60?


I annotated the output with comments (marked with semicolons). The columns are, from left to right, the sort size, the theoretical lower bound (on the average or maximum number of comparisons), the minimum number of comparisons (depending on the input permutation), the average (over all input permutations), and the maximum count. I added two columns by hand: the optimal worst-case (maximum) comparison counts (over all sorting methods, copied from the OEIS), and the optimal size for sorting networks, when known (lifted from a table here [pdf]). Inexact (potentially higher than the optimum) bounds are marked with a question mark.

For the input size the inline merge sort can reasonably tackle (up to ten or so), its worst-case is reasonably close to the best possible, and its average case tends to fall between the lower bound and the best possible. Over all these sizes, the merge sort’s worst case performs fewer comparisons than the optimal or best-known sorting networks. The current best upper bounds on the minimal worst-case comparison count seem to be based on insertion sort passes that minimise the number of comparisons with a binary search. I don’t believe that’s directly useful for the current use case, but a similar trick might be useful to reduce the number of comparisons, at the expense of reasonably more data movement.

## Making it CL-strength

That’s already a decent proof of concept. It’s also far too plain to fit in the industrial-strength Common Lisp way. An inline sorting macro worthy of CL should be parameterised on both comparator and key functions, and work with arbitrary places rather than only variables.

Parameterising the comparator is trivial. The key could be handled by calling it at each comparison, but that’s wasteful. We’re generating code; might as well go for glory. Just like in the list merge sort, I’ll cache calls to the key function in the merge tree. I’ll also use special variables instead of passing a half-dozen parameters around in the generator.

Finally, handling arbitrary places, thus letting the macro take care of writing results back to the places, is just regular macrology.

Now, the macro can be used to sort, e.g., vectors of double floats “in-place” (inasmuch as copying everything to registers can be considered in-place).

CL-USER> (macroexpand-1 (inline-sort (#'< :key #'-)
(aref array 0) (aref array 1) (aref array 2)))
(let* ((#:comparator1184 #'<)
(#:comparator1184
(if (functionp #:comparator1184)
#:comparator11184
(symbol-function #:comparator1184)))
(#:key1185 #'-)
(#:key1185
(if (functionp #:key1185)
#:key1185
(symbol-function #:key1185)))
(#:overwrite1186 t)
(#:array1195 array)
(#:array1192 array)
(#:array1189 array)
(#:temp1193 (aref #:array1195 0))
(#:temp1190 (aref #:array1192 1))
(#:temp1187 (aref #:array1189 2)))
(declare (ignorable #:comparator1184 #:key1185 #:overwrite1186))
(progn
nil
(progn
nil
nil
(progn (rotatef #:temp1190 #:temp1187))
(progn))))
(progn (rotatef #:temp1193 #:temp1190 #:temp1187))
(progn (rotatef #:temp1193 #:temp1190))))
(progn))))
(when #:overwrite1186
(let ((#:new1194 #:temp1193))
(sb-kernel:%aset #:array1195 0 #:new1194))
(let ((#:new1191 #:temp1190))
(sb-kernel:%aset #:array1192 1 #:new1191))
(let ((#:new1188 #:temp1187))
(sb-kernel:%aset #:array1189 2 #:new1188)))
(values #:temp1193 #:temp1190 #:temp1187))
t
CL-USER> (disassemble (lambda (array)
(declare (type (simple-array double-float (3)) array))
(inline-sort (#'< :key #'-)
(aref array 0) (aref array 1) (aref array 2))
array))
; disassembly for (lambda (array))
; 0C5A5661:       F20F105201       movsd XMM2, [rdx+1]        ; no-arg-parsing entry point
;      666:       F20F104209       movsd XMM0, [rdx+9]
;      66B:       F20F104A11       movsd XMM1, [rdx+17]
;      670:       660F28E0         movapd XMM4, XMM0
;      674:       660F5725A4000000 xorpd XMM4, [rip+164]
;      67C:       660F28D9         movapd XMM3, XMM1
;      680:       660F571D98000000 xorpd XMM3, [rip+152]
;      688:       660F2FDC         comisd XMM3, XMM4
;      68C:       7A02             jp L0
;      68E:       7267             jb L3
;      690: L0:   660F28DA         movapd XMM3, XMM2
;      694:       660F571D84000000 xorpd XMM3, [rip+132]      ; negate double-float
;      69C:       660F28E0         movapd XMM4, XMM0
;      6A0:       660F572578000000 xorpd XMM4, [rip+120]
;      6A8:       660F2FE3         comisd XMM4, XMM3
;      6AC:       7A26             jp L1
;      6AE:       7324             jnb L1
;      6B0:       660F28E1         movapd XMM4, XMM1
;      6B4:       660F572564000000 xorpd XMM4, [rip+100]
;      6BC:       660F2FE3         comisd XMM4, XMM3
;      6C0:       7A27             jp L2
;      6C2:       7325             jnb L2
;      6C4:       660F28DA         movapd XMM3, XMM2
;      6C8:       660F28D0         movapd XMM2, XMM0
;      6CC:       660F28C1         movapd XMM0, XMM1
;      6D0:       660F28CB         movapd XMM1, XMM3
;      6D4: L1:   F20F115201       movsd [rdx+1], XMM2
;      6D9:       F20F114209       movsd [rdx+9], XMM0
;      6DE:       F20F114A11       movsd [rdx+17], XMM1
;      6E3:       488BE5           mov rsp, rbp
;      6E6:       F8               clc
;      6E7:       5D               pop rbp
;      6E8:       C3               ret
;      6E9: L2:   660F28DA         movapd XMM3, XMM2
;      6ED:       660F28D0         movapd XMM2, XMM0
;      6F1:       660F28C3         movapd XMM0, XMM3
;      6F5:       EBDD             jmp L1
;      6F7: L3:   660F28D8         movapd XMM3, XMM0
;      6FB:       660F28C1         movapd XMM0, XMM1
;      6FF:       660F28CB         movapd XMM1, XMM3
;      703:       EB8B             jmp L0


## Bonus: Hooking in SBCL

The inline sort supports the same options as CL:SORT, so it’d be really interesting to opportunistically compile calls to the latter into size-specialised inline sort. The usual, portable, way to code that sort of macro qua source-to-source optimiser in CL is with compiler macros; compiler macros have access to all the usual macroexpansion-time utility, but the function definition is left in place. That way the user can still use the function as a first-class function, and the compiler-macro can decline the transformation if a regular call would work better (and the compiler can ignore any compiler macro). That’s not enough for our needs, though… and there can only be one compiler macro per function, so adding one to code we don’t own is a bad idea.

Python’s first internal representation (ir1) is optimised by iteratively deriving tighter type information, and (mostly) pattern matching on the type of function calls. Its DEFTRANSFORM form lets us add new rules, and there may be an arbitrary number of such rules for each function.

The two deftransforms at the end define new rules that match on calls to CL:SORT and CL:STABLE-SORT, with arbitrary argument types and return types: basic type checks are performed elsewhere, and maybe-emit-unrolled-merge-sort does the rest. Transforms are identified by the docstring (which also improve compiler notes), and the argument and return types, so the forms are reevaluation-safe.

All the logic lies in maybe-emit-unrolled-merge-sort. The policy form checks that the optimisation policy at the call node has speed greater than space, and gives up on the transformation otherwise. The next step is to make sure the sequence argument is an array, and that its dimensions are known and define a vector (its dimension list is a list of one number). The final guard makes sure we only specialise on small sorts (at most *unrolled-vector-sort-max-length*).

Finally, we get to code generation itself. A vector of length 1 or 0 is trivially pre-sorted. I could also directly emit the inner inline-sort form, but SBCL has some stuff to hoist out computations related to hairy arrays. with-array-data takes a potentially complex array (e.g. displaced, or not a vector), and binds the relevant variables to the underlying simple array of rank 1, and the start and end indices corresponding to the range we defined (defaulting to the whole range of the input array). Bound checks are eliminated because static information ensures the accesses are safe (or the user lied and asked not to insert type checks earlier), and the start index is declared to be small enough that we can add to it without overflow – Python doesn’t implement the sort of sophisticated shape analyses that could figure that out. Finally, a straight inline-sort form can be emitted.

That machinery means we’ll get quicker and shorter inline sort code when the size is known ahead of time. For example, a quick disassembly shows the following is about one fourth the size of the size-generic inline code (with (simple-array double-float (*)), and (optimize speed (space 0)).

CL-USER> (lambda (x)
(declare (type (simple-array double-float (4)) x)
(optimize speed))
(sort x #'<))
#<FUNCTION (lambda (x)) {100BFF457B}>
CL-USER> (disassemble *)
; disassembly for (lambda (x))
; 0BFF45CF:       F20F105A01       movsd XMM3, [rdx+1]        ; no-arg-parsing entry point
;      5D4:       F20F104209       movsd XMM0, [rdx+9]
;      5D9:       F20F104A11       movsd XMM1, [rdx+17]
;      5DE:       F20F105219       movsd XMM2, [rdx+25]
;      5E3:       660F2FC3         comisd XMM0, XMM3
;      5E7:       7A0E             jp L0
;      5E9:       730C             jnb L0
;      5EB:       660F28E3         movapd XMM4, XMM3
;      5EF:       660F28D8         movapd XMM3, XMM0
;      5F3:       660F28C4         movapd XMM0, XMM4
[...]
`

Even for vectors of length 8 (the default limit), the specialised merge sort is shorter than SBCL’s inlined heapsort, and about three times as fast on shuffled vectors.

## That’s it

It took me much longer to write this up than to code the generator, but I hope this can be useful to other people. One thing I’d like to note is that sorting networks are much harder to get right than this generator, and pessimise performance: without branches, there must be partially redundant computations on non-power-of-two sizes. In the absence of solid compiler support for conditional swaps, I doubt the additional overhead of optimal sorting networks can be compensated by the simpler control flow, never mind the usual odd-even or bitonic networks.