# get all combinations

Hi Stefan.

With that sorted out, my script in post #13 is still the fastest ASObjC offering here so far, but this is just because it uses Sedgewick’s idea of replacing the three lowest levels of recursion with one written-out piece of code. It’s about to get a little faster as I’ve noted your use of the array() and addObject: methods, which are a definite improvement over what I was using before. Thanks! Thanks for all your helpful replies and suggestions, Shane ” and of course for your definitive beginners’ book about ASObjC, of which I gather there’s soon to be a second edition covering the changes in Yosemite? :rolleyes:

Yes, now that AppleScriptObjC is finally going to live up to my book’s title ;), a second edition is in the pipeline.

Hello, I feel this is the right place for having methods that also counts the number of permutations/combinations to generate, or having generated. Actually, the methods I posted above are pretty much dependent on those, so why not have them in the same thread.

The factorial handler, tells how many permutations you get out of a set of n objects.

The C handler (or binomial coeffecient) tells how many r combinations you get out a set of n elements. Another way to say this is, how many ways can I generate a subset with r elements, when order doesn’t matter, out of a set with n elements.

The P handler tells how many r permutations out of a set of n elements you can generate. (subsets with r elements out of a set of n elements).

``````on factorial(n)
if n > 170 then error "Factorial: n too large."
# Result greater than 1.797693E+308!
if n < 0 then error "Factorial: n not positive."
set a to 1
repeat with i from 1 to n
set a to a * i
end repeat
return a
end factorial

on C(n, r)
# Counts the number of r-combinations in a set of n elements.
# it is also called the binomial coeffecient.
if n = 0 or n < r or r < 0 then error "C: argument error"
return (factorial(n) div (factorial(r) * (factorial(n - r))))
end C

on P(n, r)
# counts the number of r-permutations in a set of n elements.
if n = 0 or n < r or r < 0 then error "P: argument error"

return (factorial(n) div (factorial(n - r)))

end P

``````

Edit
Made boundary tests, slightly more correct, as r is now allowed to be zero.

thanks, but it’s not considerable faster.

Meanwhile I measured the Swift version, the result is very surprising.

Based on the Heap’s algorithm with 9 items (362880 permutations)

Objective-C : 300 ms
Swift with Foundation arrays : 1,3 s
Swift with native Swift arrays : 18 s

Since we’re involving other programming languages, native C does it in 15ms, for 9 items.

Hi Stefan.

No. It’s probably a little slower. I was thinking more in terms of producing right results. Got it. When implementing my code as native C without using any AEDesc and AEDescList types (and their associate functions) but simple array of int values, it’s done in 15 ms on an old MacBook Pro from 2011. So the delay in my OSAX is really handling and copying AppleEvent descriptors. But I was cheating, I only swapped and printed them (and ignoring the results). In my second attempt I have created an int ** to lookup performance differences.

The results were that it took 30ms to actually find every permutation for a list of 9 items and copy them to an listed list of arrays (or array of pointer pointers). in that case someone could really do something with the results for later processing. I also tried to find the permutations of a list of 11 integers. It comes back with a list containing almost 40 million permutations (39,916,800 to be excactly) just under 4 seconds.

Anyway, if anyone is interested in the code:

Nigel,

Am I right in assuming you’re using a script object because that’s what you need to do with long AS lists? If so, there’s no need in the ASObjC version. I rewrote it without it, and I think it’s a little bit faster, but I’d be interested in your time:

``````use AppleScript version "2.3.1"
use framework "Foundation"

on prmt(l, workArray, permutations)
-- l is the zero-based index of the leftmost item affected by this iteration
set r to (workArray's |count|()) - 1
set m to r - 1
set n to r - l + 1 -- n is the number of list items affected by this iteration (l thru r)

if (n is 3) then
-- These six permutations are hard-coded to reduce low-level recursion

workArray's exchangeObjectAtIndex:r withObjectAtIndex:m

workArray's exchangeObjectAtIndex:r withObjectAtIndex:l

workArray's exchangeObjectAtIndex:r withObjectAtIndex:m

workArray's exchangeObjectAtIndex:r withObjectAtIndex:l

workArray's exchangeObjectAtIndex:r withObjectAtIndex:m
else
-- Precalculate some values for the repeat
set lPlus1 to l + 1 -- parameter for next-level recursions
set nIsEven to (n mod 2 = 0) -- true if n is even
set x to r -- the default index with which to swap if n is odd

-- Get all permutations of items (l +1) thru r with the current item l
prmt(lPlus1, workArray, permutations)
-- Repeat with successive values of item l
repeat with i from r to lPlus1 by -1
-- If n is even, swap items l and i, otherwise default to swapping items l and r
if (nIsEven) then set x to i
(workArray's exchangeObjectAtIndex:x withObjectAtIndex:l)
prmt(lPlus1, workArray, permutations)
end repeat
end if
end prmt

on allPermutations(theList)
set permutations to current application's NSMutableArray's array() -- the mutable array we will add to
set workArray to current application's NSMutableArray's arrayWithArray:theList -- the starting array
set r to (workArray's |count|()) - 1
if (r < 2) then
-- Special-case lists of less than three items
if (r is 1) then permutations's addObject:(workArray's reverseObjectEnumerator()'s allObjects())
else
-- Otherwise use the recursive handler
prmt(0, workArray, permutations)
end if
return permutations as list
end allPermutations
``````

That makes sense. If you modify Nigel’s script to remove the coercion of the final array to an AS list, the script runs in less than half the time. Put another way, the single act of coercing the array of arrays to an AS list accounts for more than 50% of the running time. And assuming that you can’t do that coercion to a descriptor at your end any quicker, that means the best overall result you can achieve is roughly half the time of the ASObjC version, even if you build the array in a nanosecond.

The other question is what someone is going to do with such a list. I mean, rather than say repeating through a long list in AS, it may be quicker to leave the main list as an array, and just coerce each item as you use it. That’s just moving the job elsewhere in code, but it may save time by skipping a large AS repeat loop.

Hi Shane.

Your assumption’s party right. As you probably noticed, my first ASObjC script (post #13) is just a revamp of my right-to-left vanilla script from post #10, with ObjC arrays and methods used instead of vanilla where there are more than two items.

In the vanilla scripts, the script-object-within-a-handler idea offers three advantages (to my way of thinking):

1. The script object’s properties can be “referenced” to allow faster access to the vanilla list items.
2. The recursive part of the only-partially recursive process can be contained within the main handler.
3. The lists and the precalculated r and m values can be held in local properties instead of having to be passed or recalculated with each recursion or held in globals or global properties outside the main handler.

Point 1, as you say, doesn’t apply with the ObjC arrays. Point 2 is a largely a matter of my own personal preference. Point 3 is connected with point 2, but is also relevant with regard to the amount of work the script has to do. Your rewrite passes two extra parameters AND recalculates r and m on every call to the recursive handler. I’d therefore expect it to be a little slower than my script. And it is, on my machine. With nine items, it’s taking about six seconds longer than mine this morning. But mine’s taking a fair bit longer than it did yesterday, so the difference between the scripts’ times may actually be less under optimal conditions.

Now that’s interesting! But the difference is far less dramatic for me. 117 seconds as opposed to 134 (this morning). Still, that’s a lot more than I’d have expected until you mentioned it.

Yes, I found it a bit slower here too, although I got a bit excited when I accidentally left a number off the initial list That coercion time is the killer. I found when I wrote ASObjC Runner that with even modest lists of lists, virtually all the time was taken converting AS stuff to Cocoa and vice-versa; the work performed was almost irrelevant most of the time. One of the advantages of having ASObjC everywhere is that it makes it easier to do the conversions only when you need to.

Exactly! Nigel’s code took 45 seconds on my MBP when an AppleScript list is returned but when I return a pointer to the NSMutableArray it took only 40 seconds. That means that AppleScriptObjC bridge will take up to 5 seconds to coerce to a NSMutableArray containing 362,880 items into an AppleScript list. That fits right with my OSAX “problem” because when thinking about it, the AEPutParamDesc does an extra copy of the object meaning it will take two times 5 seconds (and probably some more copying when the event is handled by the event manager).

It’s getting nicer and nicer. But . if you make a list of 362,880, I have to assume that you are going to use that list for something other than just generating it, and then, those 10 seconds, won’t take up that much time proportionally. You can for all I know, execute an Osascript in the background, from a do shell script for all I know.

I have another approach, I generate permutations, or combinations on the fly, paying a penalty with each invocation, I live well with that, at least until I have timed it, because I have a practical need for testing if vectors in matrix are orthogonal to each other, by testing them accordingly to generated combinations. If a test fails, then I must start over from scratch. So I pay a penalty in total time, but saves some time per invocation, and some memory too! I have no idea how I would create a list with 362880 items in Applescript however. Have the boundaries changed?

There are however a lot of things lists with permutations and combinations that are generated up front can be used for. Today I have understood how mikerickson’s algorithm for generating permutations with repeats works, it is pretty clever I think.

This is really a great thread. AppleScript allows lists up to 2^31 items, well at least in theory. Because the maximum allowed index is the maximum value of a signed 32-bit integer and negative indexes are not allowed. To create a list with 362880 items, there is no problem you just have to wait 20+ seconds.

Hello DJ.

Thanks for enlightening me, I was dead sure anything above 2^14 was futile. 20 seconds isn’t a long time, except for when in front of a computer! Hello.

For those interested, here is a generator of r-combinations. There is nothing wrong with Daehls, I just wanted something simpler to understand as the problem intrigued me, so this is probably slower. I found the pseudo code in a buggy! :mad: pdf document somwhere. (Daehls delist handler is something that seems to be a great way to translate items into indicies IMO.)

``````set comboList to {}
combination(comboList, 3, 6)
listprint(comboList)
(*
"{1, 2, 3}
{1, 2, 4}
{1, 2, 5}
{1, 2, 6}
{1, 3, 4}
{1, 3, 5}
{1, 3, 6}
{1, 4, 5}
{1, 4, 6}
{1, 5, 6}
{2, 3, 4}
{2, 3, 5}
{2, 3, 6}
{2, 4, 5}
{2, 4, 6}
{2, 5, 6}
{3, 4, 5}
{3, 4, 6}
{3, 5, 6}
{4, 5, 6}"
*)
on combination(combinations, r, n)
# print the first r combination
set S to {}
repeat with i from 1 to r
set end of S to i
end repeat
copy S to end of combinations

repeat with i from 2 to C(n, r)
set m to r
set max_val to n

repeat while (((item m) of S) = max_val)
set m to m - 1
set max_val to max_val - 1
end repeat
# increment the above rightmost element
set item m of S to (item m of S) + 1
# all others are the successors:

repeat with j from (m + 1) to r
set item j of S to (item (j - 1) of S) + 1
end repeat
copy S to end of combinations
end repeat
end combination

on listprint(theL)
try
text 0 of theL
on error e
set ofs to offset of "{" in e
set tmp to text (ofs + 1) thru -3 of e
tell (a reference to text item delimiters)
set astid to contents of it
set contents of it to "}, "
set newl to text items of tmp
set contents of it to "}" & return
set newt to newl as text
set contents of it to astid
end tell
return newt
end try
end listprint

on C(n, r)
# Counts the number of r-combinations in a set of n elements.
# it is also called the binomial coeffecient.
if n = 0 or n < r or r < 0 then error "C: argument error"
return (factorial(n) div (factorial(r) * (factorial(n - r))))
end C

on factorial(n)
if n > 170 then error "Factorial: n too large."
# Result greater than 1.797693E+308!
if n < 0 then error "Factorial: n not positive."
set a to 1
repeat with i from 1 to n
set a to a * i
end repeat
return a
end factorial

``````

Edit
Made a “fancier” listprint handler. Hello.

I have made a handler for generating combinations with repetitions, for completion, since we then have all kinds of handlers here. You’ll have to allow for every element being chosen as many times as there are rooms in your subset.

``````set comboList to {}

set donuts to {"iced", "jam", "plain", "something completely different"}
set chosen to {0, 0, 0, 0}
# the number of items in the list chosen, must coincide with the n_chosen parameter.

choose(comboList, chosen, 1, 4, 1, 3)
--> 15 (combinations

listprint(comboList)
(*
"{\"iced\", \"iced\", \"iced\", \"iced\"}
{\"iced\", \"iced\", \"iced\", \"jam\"}
{\"iced\", \"iced\", \"iced\", \"plain\"}
{\"iced\", \"iced\", \"jam\", \"jam\"}
{\"iced\", \"iced\", \"jam\", \"plain\"}
{\"iced\", \"iced\", \"plain\", \"plain\"}
{\"iced\", \"jam\", \"jam\", \"jam\"}
{\"iced\", \"jam\", \"jam\", \"plain\"}
{\"iced\", \"jam\", \"plain\", \"plain\"}
{\"iced\", \"plain\", \"plain\", \"plain\"}
{\"jam\", \"jam\", \"jam\", \"jam\"}
{\"jam\", \"jam\", \"jam\", \"plain\"}
{\"jam\", \"jam\", \"plain\", \"plain\"}
{\"jam\", \"plain\", \"plain\", \"plain\"}
{\"plain\", \"plain\", \"plain\", \"plain\"}"

*)

(*

The calculation for the number of combinations with repetitions are:

C( maxtypes+n_chosen-1,n_chosen) or C(n+r-1,r) with other variable names.

*)

log C(3 + 4 - 1, 4)

--> 15

on choose(combinations, got, n_chosen, len, atw, maxtypes)
global donuts

set tcount to 0

if n_chosen = (len + 1) then
# log "n_chosen = len "
if got = 0 then return 1
set lineout to {}
repeat with i from 1 to len
set end of lineout to item (item i of got) of donuts
end repeat
copy lineout to end of combinations
return 1
end if

repeat with i from atw to maxtypes
if (got â‰  0) then
set item n_chosen of got to i
set tcount to tcount + choose(combinations, got, (n_chosen + 1), len, i, maxtypes)
end if
end repeat
return tcount

end choose

on listprint(theL)
try
text 0 of theL
on error e
set ofs to offset of "{" in e
set tmp to text (ofs + 1) thru -3 of e
tell (a reference to text item delimiters)
set astid to contents of it
set contents of it to "}, "
set newl to text items of tmp
set contents of it to "}" & return
set newt to newl as text
set contents of it to astid
end tell
return newt
end try
end listprint

on C(n, r)
# Counts the number of r-combinations in a set of n elements.
# it is also called the binomial coeffecient.
if n = 0 or n < r or r < 0 then error "C: argument error"
return (factorial(n) div (factorial(r) * (factorial(n - r))))
end C

on factorial(n)
if n > 170 then error "Factorial: n too large."
# Result greater than 1.797693E+308!
if n < 0 then error "Factorial: n not positive."
set a to 1
repeat with i from 1 to n
set a to a * i
end repeat
return a
end factorial
``````

Here’s an interesting one.

By means of concatenation, this vanilla version of my script prebuilds a list of the required length to hold the permutations. When used as a Library in exactly the same way as the ASObjC version in post #13, it executes in a mere 5 seconds or so as opposed to the ASObjC version’s 134 seconds! However, with both versions, it then takes nearly two and a half minutes to do anything with the result, such as setting a variable to it or getting its class!

``````on allPermutations(theList)

script o
property workList : missing value
property permutations : {}
property r : count theList -- index of the rightmost item of workList.
property m : r - 1 -- index of the middle item of the last three of workList.
property p : 1 -- index into the permutations list.

on prmt(l)
-- l is the index of the leftmost item affected by this iteration
set n to r - l + 1 -- n is the number of list items affected by this iteration (l thru r)

if (n is 3) then
-- These six permutations are hard-coded to reduce low-level recursion
copy my workList to item p of my permutations

set {v1, v2, v3} to items l thru r of my workList

set item m of my workList to v3
set item r of my workList to v2
copy my workList to item (p + 1) of my permutations

set item l of my workList to v2
set item r of my workList to v1
copy my workList to item (p + 2) of my permutations

set item m of my workList to v1
set item r of my workList to v3
copy my workList to item (p + 3) of my permutations

set item l of my workList to v3
set item r of my workList to v2
copy my workList to item (p + 4) of my permutations

set item m of my workList to v2
set item r of my workList to v1
copy my workList to item (p + 5) of my permutations

set my p to p + 6
else
-- Precalculate some values for the repeat
set lPlus1 to l + 1 -- parameter for next-level recursions
set nIsEven to (n mod 2 = 0) -- true if n is even
set x to r -- the default index with which to swap if n is odd

-- Get all permutations of items (l +1) thru r with the current item l
prmt(lPlus1)
-- Repeat with successive values of item l
repeat with i from r to lPlus1 by -1
-- If n is even, swap items l and i, otherwise default to swapping items l and r
if (nIsEven) then set x to i
tell item x of my workList
set item x of my workList to item l of my workList
set item l of my workList to it
end tell
prmt(lPlus1)
end repeat
end if
end prmt

end script

if (o's r < 3) then
-- Special-case lists of less than three items
copy theList to the beginning of o's permutations
if (o's r is 2) then set the end of o's permutations to the reverse of theList
else
-- Otherwise use the recursive handler
copy theList to o's workList
-- Prebuild a list of the required length (factorial of theList's length) to hold the permutations.
set mv to missing value
set mvList to {mv, mv, mv, mv, mv, mv} -- Minimum length is 6 with a 3-item input list.
set mvList2 to mvList
repeat with i from 3 to (o's r) - 1
repeat i times
set mvList2 to mvList2 & mvList
end repeat
set mvList to mvList2
end repeat
set o's permutations to mvList
o's prmt(1)
end if

return o's permutations

end allPermutations
``````