packages feed

random-cycle (empty) → 0.1.0.0

raw patch · 14 files changed

+1593/−0 lines, 14 filesdep +algebraic-graphsdep +basedep +criterion

Dependencies added: algebraic-graphs, base, criterion, mwc-random, primitive, random, random-cycle, tasty, tasty-hunit, tasty-quickcheck, vector, vector-algorithms

Files

+ CHANGELOG.md view
@@ -0,0 +1,5 @@+# Revision history for random-cycle++## 0.1.0.0 -- 2023-04-17++* First version. Released on an unsuspecting world.
+ LICENSE view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++  You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<https://www.gnu.org/licenses/>.++  The GNU General Public License does not permit incorporating your program+into proprietary programs.  If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library.  If this is what you want to do, use the GNU Lesser General+Public License instead of this License.  But first, please read+<https://www.gnu.org/licenses/why-not-lgpl.html>.
+ README.md view
@@ -0,0 +1,72 @@++[![builds.sr.ht status](https://builds.sr.ht/~brendanrbrown/random-cycle/.svg)](https://builds.sr.ht/~brendanrbrown/random-cycle/?)++## Summary+A Haskell library for efficient uniform random sampling of cycle partition+graphs on sets of vertices, and partitions of lists or vectors. Selection can+be subject to conditions.++## Cycle partitions+A cycle partition graph on a set of vertices V is a directed graph G = (E,+V) such that for each i in V there exists a unique j in V such that (i,+j) in E. In other words, it is a partition of V into a graph with disjoint+[cycle graphs](https://en.wikipedia.org/wiki/Cycle_graph).++Define C(V) to be the set of cycle partitions graphs of V.+`uniformCyclePartition` samples from the uniform distribution on C(V), in+O(|V|) time.++To do so, it relies on the fact that ++σ -> (i, σ(i))  , for i = 1..|V|++defines a bijective map between the permutations σ  on |V| distinct elements+and the edge sets of C(V).++Note self-loops are allowed in the possible configurations.++<!--TODO: sampling with conditions-->++## List or vector partitions+This package provides functions to draw uniform samples from all 2^(n-1)+possible partitions of an ordered list (or vector). `uniformPartition` selects+a single element uniformly across all possible partitions in O(n) time, and+`uniformPartitionThin` samples uniformly conditional on a predicate in O(n/p)+time on average, where `p` is the proportion of elements for which the+predicate is `True`.++Only the partitioning is randomized: Input list order is preserved.++At the moment, the `uniformPartitionThin` is implemented only for lists: It+works by rejection sampling and hence will not terminate if `p` is effectively+zero.++The samplers randomize the placement of each breakpoint in the partition. In+other words the sample space is viewed as a perfect binary tree, and random+selection is a random walk from root to leaf. The implementation samples a bit+array to determine the walk path instead of creating an actual tree structure,+for efficiency.++### Short-circuiting+The predicate provided to `uniformPartitionThin` checks each partition element,+a chunk of elements in the original list, in turn. Since partitions are built+lazily, the sampler will short-circuit and start sampling a new partition after+the first partition element for which the predicate is `False`. This is just a+consequence of the short-circuiting in `base` package function `all`.++In addition, if the predicate itself is short-circuiting, the sampler will+short-circuit.++## Contributing++### Tickets++Send by email, without need for an account, to ~brendanrbrown/random-cycle@todo.sr.ht++[Man pages](https://man.sr.ht/todo.sr.ht/) for tickets on SourceHut,+particularly the "Email access" section.++### Patches+<!--TODO: link to git send-email and imap resources-->++Man pages for [sending patches upstream](https://man.sr.ht/git.sr.ht/#sending-patches-upstream).
+ benches/Main.hs view
@@ -0,0 +1,112 @@+module Main where++import Criterion.Main+import Data.Vector (Vector, fromList)+import RunTimes++main :: IO ()+main = defaultMain [rtPartitionBench, rtCycleBench]++{- Partitions -}++rtPartitionBench :: Benchmark+rtPartitionBench = bgroup "partitions: Run times" $ concat [rtPartitionRL, rtPartitionWithLocRL, rtPartitionRV]++rtPartitionRL :: [Benchmark]+rtPartitionRL =+  map (\(n, xs) -> bench (msg ++ show n) $ nf uniformPartitionRL xs) $+    orderList 4+  where+    msg = "uniformPartitionRL: m = 0, n = "++rtPartitionWithLocRL :: [Benchmark]+rtPartitionWithLocRL =+  map (\(n, xs) -> bench (msg ++ show n) $ nf uniformPartitionThinRL xs) $+    orderListSmall 4+  where+    msg = "uniformPartitionThinRL: m = 0, n = "++rtPartitionRV :: [Benchmark]+rtPartitionRV =+  map (\(n, xs) -> bench (msg ++ show n) $ nf uniformPartitionRV xs) $+    orderVecs 4+  where+    msg = "uniformPartitionRV: m = 0, n = "++-- rtPartitionList :: [Benchmark]+-- rtPartitionList = map (\m -> bench (msg ++ show m ++ ", n = 100") $ nf (`uniformPartitionList` list100) m) $ take 4 $ orderSeq 100+--  where+--    msg = "uniformPartitionList: m = "++{- Cycles -}++-- TODO: return it when Vector impl is done.++rtCycleBench :: Benchmark+rtCycleBench =+  bgroup+    "cycles: Run times"+    $ concat+      [ rtCycleRV,+        rtCycleThinRV1,+        rtCycleThinRV2+      ]++rtCycleRV :: [Benchmark]+rtCycleRV =+  map+    ( \n ->+        bench (msg ++ show n) $+          nf uniformCyclePartitionRV n+    )+    $ take 4+    $ orderSeq 100+  where+    msg = "uniformCyclePartitionRV: n = "++rtCycleThinRV1 :: [Benchmark]+rtCycleThinRV1 =+  map+    ( \n ->+        bench (msg ++ show n) $+          nf uniformCyclePartitionThinRVnoSelf n+    )+    $ take 4+    $ orderSeq+    $ max 100 minN+  where+    msg = "uniformCyclePartitionThinRVnoSelf: n = "++rtCycleThinRV2 :: [Benchmark]+rtCycleThinRV2 =+  map+    ( \n ->+        bench (msg ++ show n) $+          nf uniformCyclePartitionThinRVsimpleEdgeRules n+    )+    $ take 4+    $ orderSeq+    $ max 100 minN+  where+    msg = "uniformCyclePartitionThinRVsimpleEdgeRules: n = "++{- UTILITIES -}++list100 = [1 .. 100]++-- IMPORTANT: You must ensure these inputs guarantee termination of+-- condition-based functions, such as uniformPartitionThinRL.+orderList :: Int -> [(Int, [Int])]+orderList k = take k $ map (\n -> (n, [1 .. n])) $ orderSeq 100++orderListSmall :: Int -> [(Int, [Int])]+orderListSmall k = take k $ map (\n -> (n, [1 .. n])) $ orderSeq 5++orderVecs :: Int -> [(Int, Vector Int)]+orderVecs = map (fmap fromList) . orderList++orderVecsSmall :: Int -> [(Int, Vector Int)]+orderVecsSmall = map (fmap fromList) . orderListSmall++orderSeq :: Int -> [Int]+orderSeq = iterate (* 2)
+ benches/RunTimes.hs view
@@ -0,0 +1,88 @@+module RunTimes where++import Data.List (maximum)+import Data.Maybe (fromJust)+import Data.Vector (Vector)+import qualified Data.Vector as V+import RandomCycle.List (uniformCyclePartition)+import qualified RandomCycle.List as RL+import qualified RandomCycle.Vector as RV+import System.Random.Stateful++gen = mkStdGen 0++{- Partitions -}++-- List++-- /O(nm)/ in uniformPartitionList, where 'm' is the number of partitions+-- "consumed."+-- uniformPartitionList :: Int -> [Int] -> [[[Int]]]+-- uniformPartitionList m xs = take m $ fst $ runStateGen gen (RL.uniformPartitionList xs)++-- /O(n)/ RL.uniformPartition+uniformPartitionRL :: [Int] -> [[Int]]+uniformPartitionRL xs = runStateGen_ gen (RL.uniformPartition xs)++-- /O(n\/p)/ RL.uniformPartitionThin where rule is a local condition. /p/+-- is the probability @all rule yss == True@ for a uniformly chosen partition+-- yss.+-- This can be deceptively expensive: For example, the condition ((>= 2) .+-- length) leads to huge runtimes, since the number of partitions of [1..n]+-- with at least one element of length 1 is large.+-- IMPORTANT: You must guarantee here the function terminates on the input.+-- You+rule = (>= 2) . sum++uniformPartitionThinRL :: [Int] -> [[Int]]+uniformPartitionThinRL xs = runStateGen_ gen (RL.uniformPartitionThin rule xs)++-- Vector++-- /O(n)/ RV.uniformPartition+uniformPartitionRV :: Vector Int -> [Vector Int]+uniformPartitionRV xs = runStateGen_ gen (RV.uniformPartition xs)++{- Cycles -}++-- List+--+-- NOTE: The list impl. for now is just a convenience wrapper around+-- the vector impl. No need to benchmark it.++-- Vector++-- /O(n)/. In effect a demonstration of linear runtime for 'uniformPermutation'+-- from mwc-random.+uniformCyclePartitionRV :: Int -> Vector (Int, Int)+uniformCyclePartitionRV n = runSTGen_ gen (RV.uniformCyclePartition n)++-- max iterations allowed for cycle sampler.+maxit = 100000++-- We want this to fail if no matches are found in maxit tries+uniformCyclePartitionThinRVnoSelf :: Int -> Vector (Int, Int)+uniformCyclePartitionThinRVnoSelf n =+  fromJust $+    runSTGen_ gen (RV.uniformCyclePartitionThin maxit noSelf n)++uniformCyclePartitionThinRVsimpleEdgeRules :: Int -> Vector (Int, Int)+uniformCyclePartitionThinRVsimpleEdgeRules n =+  fromJust $+    runSTGen_ gen (RV.uniformCyclePartitionThin maxit simpleEdgeRules n)++-- NOTE: copied from tests.+-- You must ensure the predicate check 'all' of them has non-empty support.+minN :: Int+minN = 3++noSelf :: (Int, Int) -> Bool+noSelf = uncurry (/=)++-- NOTE: This list requires the number of vertices to be > 2+-- for there to exist a solution.+simpleEdgeRules :: (Int, Int) -> Bool+simpleEdgeRules e = all ($ e) [noSelf, no12]+  where+    no12 (1, 2) = False+    no12 _ = True
+ random-cycle.cabal view
@@ -0,0 +1,84 @@+cabal-version:      2.4+name:               random-cycle+version:            0.1.0.0+synopsis: Uniform draws of partitions and cycle-partitions, with thinning.++description: +    A Haskell library for efficient uniform random sampling of cycle partition+    graphs on sets of vertices, and partitions of lists or vectors. Selection+    can be subject to conditions.++homepage: https://sr.ht/~brendanrbrown/random-cycle+bug-reports: https://todo.sr.ht/~brendanrbrown/random-cycle+license: GPL-3.0-or-later+author:             Brendan Brown+maintainer:         brendanrbrown@runbox.com++-- A copyright notice.+copyright: Brendan R. Brown, 2023+category: Math, Graphs+extra-source-files: CHANGELOG.md, README.md, LICENSE++library+    exposed-modules:  +      RandomCycle.List+      RandomCycle.Vector+    other-modules: +      RandomCycle.List.Partition+      RandomCycle.Vector.Partition+      RandomCycle.Vector.Cycle+    build-depends:    +      base ^>=4.16.3.0+      , primitive+      , mwc-random+      , random+      , vector+    hs-source-dirs:   src+    default-language: Haskell2010+    ghc-options:+      -Wall+      -Wcompat+      -Widentities+      -Wincomplete-record-updates+      -Wincomplete-uni-patterns+      -Wpartial-fields+      -Wredundant-constraints++benchmark benches+    type: exitcode-stdio-1.0+    main-is: Main.hs  +    -- Modules included in this library but not exported.+    other-modules: RunTimes++    -- LANGUAGE extensions used by modules in this package.+    -- other-extensions:+    build-depends:    +      base ^>=4.16.3.0+      , random-cycle+      , criterion+      , random+      , vector+    hs-source-dirs:   benches+    default-language: Haskell2010++test-suite random-cycle-test+    type: exitcode-stdio-1.0+    main-is: Main.hs  +    -- Modules included in this library but not exported.+    other-modules: +      TestPartitions+      TestCycles+    -- LANGUAGE extensions used by modules in this package.+    -- other-extensions:+    build-depends:    +      base ^>=4.16.3.0+      , random-cycle+      , algebraic-graphs+      , tasty+      , tasty-hunit+      , tasty-quickcheck+      , random+      , vector+      , vector-algorithms+    hs-source-dirs:   test+    default-language: Haskell2010
+ src/RandomCycle/List.hs view
@@ -0,0 +1,38 @@+module RandomCycle.List+  ( uniformPartition,+    uniformPartitionThin,+    uniformCyclePartition,+    uniformCyclePartitionThin,+    partitionLengths,+    partitionFromBits,+  )+where++import Control.Monad.Primitive (PrimMonad)+import qualified Data.Vector as V+import RandomCycle.List.Partition+import qualified RandomCycle.Vector as RV+import System.Random.Stateful (StatefulGen)++-- | Sample a cycle graph partition of '[0.. n-1]',+-- uniformly over the \(n!\) possibilities. The list implementation+-- is a convenenience wrapper around 'RandomCycle.Vector.uniformCyclePartition'.+uniformCyclePartition :: (PrimMonad m, StatefulGen g m) => Int -> g -> m [(Int, Int)]+uniformCyclePartition n g = do+  v <- RV.uniformCyclePartition n g+  pure $ V.toList v++-- | Sample a cycle graph partition of '[0.. n-1]',+-- uniformly over the set satisfying the conditions.+-- The list implementation is a convenenience wrapper around+-- 'RandomCycle.Vector.uniformCyclePartitionThin'.+uniformCyclePartitionThin ::+  (PrimMonad m, StatefulGen g m) =>+  Int ->+  ((Int, Int) -> Bool) ->+  Int ->+  g ->+  m (Maybe [(Int, Int)])+uniformCyclePartitionThin maxit r n g = do+  v <- RV.uniformCyclePartitionThin maxit r n g+  pure $ V.toList <$> v
+ src/RandomCycle/List/Partition.hs view
@@ -0,0 +1,160 @@+-- | Internal module whose primary exports are 'uniformPartition'+-- and 'uniformPartitionThin'. Import @RandomCycle.List@ instead.+module RandomCycle.List.Partition where++import Control.Monad (guard)+import Data.Bits+import GHC.Natural (Natural)+import System.Random.Stateful++{- UTILITIES -}++-- | Internal. Version of @Data.List.'span'@ that uses the supplied bits @bs@+-- as a grouping variable. 'switch` flips the booleans, so that the input bit's+-- least significant digit determines the grouping. Note the case @bs == 0@ is+-- not handled specially here, since termination is guaranteed whenever 'xs' is+-- finite. Compare to @RandomCycle.Vector.Partitions.'commonSubseqBits'@.+--+-- Note this would be simpler to implement with `countTrailingZeros`, but that+-- would limit the input list to some length, e.g. 64 if using @Word@, which is+-- too restrictive.+spanBits :: (Bool -> Bool) -> Natural -> [a] -> ([a], (Natural, [a]))+spanBits _ bs xs@[] = (xs, (bs, xs))+spanBits switch bs (x : xs)+  | switch (bs `testBit` 0) = let (zs, (bs', zzs)) = spanBits switch (bs `shiftR` 1) xs in (x : zs, (bs', zzs))+  | otherwise = ([], (bs, x : xs))++-- | Utility to generate a list partition using the provided 'Natural'+-- as grouping variable, viewed as 'Bits'. The choice of grouping variable is to+-- improve performance since the number of partitions grows exponentially in the+-- input list length.+--+-- This can be used to generate a list of all possible partitions of the input list+-- as shown in the example. See 'RandomCycle.Vector.partitionFromBits' for other examples.+--+-- >>> import GHC.Natural+-- >>> :{+-- >>> allPartitions n | n < 0 = []+-- >>> allPartitions n = map (`partitionFromBits` [0..n]) [0::Natural .. 2^(n-1) - 1]+-- >>> }+-- >>> allPartitions 4+-- [[[0,1,2,3,4]],[[0],[1,2,3,4]],[[0],[1],[2,3,4]],+-- [[0,1],[2,3,4]],[[0,1],[2],[3,4]],[[0],[1],[2],[3,4]],+-- [[0],[1,2],[3,4]],[[0,1,2],[3,4]]]+partitionFromBits :: Natural -> [a] -> [[a]]+partitionFromBits _ [] = []+partitionFromBits bs xs =+  -- NOTE: Grouping is determined by the first bit. This is important for+  -- correctness of grouping based on spanBits implementation, but also to+  -- ensure uniformPartition is uniform over 2^(n-1) partitions.+  let switch = if bs `testBit` 0 then id else not+      (ys, (bs', yss)) = spanBits switch bs xs+   in ys : partitionFromBits bs' yss++-- | Primarily a testing utility, to compute directly the lengths of each+-- partition element for a list of size 'n', using 'countTrailingZeros'. Note+-- this uses 'Word'.+partitionLengths :: Word -> Int -> [Int]+partitionLengths bs = op bs (countTrailingZeros bs)+  where+    op b 0 m = let b' = complement b in op b' (countTrailingZeros b') m+    op b z m =+      if z > m+        then [m | m > 0]+        else+          let b' = b `shiftR` z+           in z : op b' (countTrailingZeros b') (m - z)++{- PARTITIONING WITH THINNING -}++-- | Internal. Partition a list as determined by bits @bs@, but shortcircuit if+-- the local condition 'r' is false for some partition element.+partitionFromBitsThin :: ([a] -> Bool) -> Natural -> [a] -> Maybe [[a]]+partitionFromBitsThin _ _ [] = Just []+partitionFromBitsThin r bs xs =+  let ps = partitionFromBits bs xs+   in guard (all r ps) >> pure ps++-- | Internal. Inner logic of 'uniformPartitionThin' that carries around+-- the input list length to avoid recomputation. It is the callers job to+-- ensure @n == length xs@.+uniformPartitionThinN ::+  (StatefulGen g m) =>+  Int ->+  Int ->+  ([a] -> Bool) ->+  [a] ->+  g ->+  m (Maybe [[a]])+uniformPartitionThinN maxit _ _ _ _ | maxit <= 0 = pure Nothing+uniformPartitionThinN maxit n r xs g = do+  bs <- uniformRM (0, 2 ^ n - 1) g+  case partitionFromBitsThin r bs xs of+    Nothing -> uniformPartitionThinN (maxit - 1) n r xs g+    Just ys -> pure $ Just ys++{- RANDOM -}++-- | Draw a random partition of the input list 'xs' from the uniform+-- distribution on partitions. This proceeds by randomizing the placement of+-- each breakpoint, in other words by walking a random path in a perfect binary+-- tree. /O(n)/ for a vector length /n/.+--+-- This function preserves the order of the input list.+--+-- ==== __Examples__+--+-- >>> import System.Random.Stateful+-- >>> pureGen = mkStdGen 0+-- >>> runStateGen_ pureGen $ uniformPartition [1..5::Int]+-- [[1,2,3],[4],[5]]+-- >>> runStateGen_ pureGen $ uniformPartition ([] :: [Int])+-- []+uniformPartition :: (StatefulGen g m) => [a] -> g -> m [[a]]+uniformPartition xs g = do+  let d = length xs+  -- Drawing w.p. 1/2^d, but first bit determines grouping.+  bs <- uniformRM (0, 2 ^ d - 1) g+  pure $ partitionFromBits bs xs++-- TODO: be more precise in the statement below about exponential growth in the+-- length condition case.++-- | Generate a partition with a local condition @r@ on each partition element.+-- Construction of a partition shortcircuits to failure as soon as the local+-- condition is false.+--+-- Since this is a rejection sampling method, the user is asked to provide+-- a counter for the maximum number of sampling attempts in order to guarantee+-- termination in cases where the edge predicate has probability of success close+-- to zero.+--+-- Run time on average is /O(n\/p)/ where /p/ is the probability @all r yss+-- == True@ for a uniformly generated partition @yss@, assuming @r@ has run+-- time linear in the length of its argument. This can be highly non-linear+-- because /p/ in general is a function of /n/.+--+-- Some cases can perhaps be deceptively expensive: For example, the condition ((>=+-- 2) .  length) leads to huge runtimes, since the number of partitions with at+-- least one element of length 1 is exponential in /n/.+--+-- ==== __Examples__+--+-- >>> import System.Random.Stateful+-- >>> maxit = 1000+-- >>> pureGen = mkStdGen 0+-- >>> r = (>= 2) . length+-- >>> runStateGen_ pureGen $ uniformPartitionThin maxit r [1..5::Int]+-- Just [[1,2],[3, 4, 5]]+-- >>> runStateGen_ pureGen $ uniformPartitionThin maxit (const False) ([] :: [Int])+-- Just []+-- >>> runStateGen_ pureGen $ uniformPartitionThin maxit r [1::Int]+-- Nothing+uniformPartitionThin ::+  (StatefulGen g m) =>+  Int ->+  ([a] -> Bool) ->+  [a] ->+  g ->+  m (Maybe [[a]])+uniformPartitionThin maxit r xs = uniformPartitionThinN maxit (length xs) r xs
+ src/RandomCycle/Vector.hs view
@@ -0,0 +1,10 @@+module RandomCycle.Vector+  ( uniformPartition,+    partitionFromBits,+    uniformCyclePartition,+    uniformCyclePartitionThin,+  )+where++import RandomCycle.Vector.Cycle+import RandomCycle.Vector.Partition
+ src/RandomCycle/Vector/Cycle.hs view
@@ -0,0 +1,122 @@+{-# LANGUAGE FlexibleContexts #-}++-- | Internal module for sampling of cycle graph partitions.+-- Import 'RandomCycle.Vector' instead.+module RandomCycle.Vector.Cycle where++import Control.Monad (when)+import Control.Monad.Primitive (PrimMonad, PrimState, liftPrim)+import Data.STRef+import qualified Data.Vector as V+import System.Random.MWC.Distributions (uniformPermutation, uniformShuffleM)+import System.Random.Stateful++{- INTERNAL -}++-- | Internal. Helper for uniformCyclePartitionThin so as to avoid+-- re-allocating the input vector for each rejected sample.+-- IMPORTANT: Caller's responsibility to ensure proper+-- management of the 'chk' for match found.+uniformCyclePartitionThinM ::+  (StatefulGen g m, PrimMonad m) =>+  STRef (PrimState m) Bool ->+  STRef (PrimState m) Int ->+  ((Int, Int) -> Bool) ->+  V.MVector (PrimState m) Int ->+  g ->+  m ()+uniformCyclePartitionThinM chk maxit r v gen = do+  maxitVal <- liftPrim $ readSTRef maxit++  when (maxitVal <= 0) (pure ())++  uniformShuffleM v gen+  -- TODO: Repeated calls to freeze, indexed+  -- a possible opportunity for optimization,+  -- e.g. with imap or a check that takes 'chk'+  -- reference and shortcircuits.+  vVal <- V.freeze v+  if V.all r (V.indexed vVal)+    then do+      liftPrim $ modifySTRef' chk (const True)+    else do+      liftPrim $ modifySTRef' maxit (\x -> x - 1)+      uniformCyclePartitionThinM chk maxit r v gen+  pure ()++{- RANDOM -}++-- TODO: uniform (full) cycle with [Sattolo's algorithm](https://algo.inria.fr/seminars/summary/Wilson2004b.pdf)+-- uniformCycle++-- | Select a partition of '[0..n-1]' into disjoint+--  [cycle graphs](https://en.wikipedia.org/wiki/Cycle_graph),+--  uniformly over the \(n!\) possibilities. The sampler relies on the fact that such+--  partitions are isomorphic with the permutations of '[0..n-1]' via the map sending+--  a permutation \(\sigma\) to the edge set \(\{(i, \sigma(i))\}_0^{n-1}\). In other words,+--  the cycle partition graphs are isomorphic with the rotation matrices.+--+--  Therefore, this function simply calls 'uniformPermutation' and tuples the result with its+--  indices. The returned value is a vector of edges. \(O(n)\), since 'uniformPermutation'+--  implements the [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle).+--+--  'uniformPermutation' uses in-place mutation, so this function must be run in a 'PrimMonad'+--  context.+--+-- ==== __Examples__+--+-- >>> import System.Random.Stateful+-- >>> import RandomCycle.Vector+-- >>> import Data.Vector (Vector)+-- >>> runSTGen_ (mkStdGen 1305) $ RV.uniformCyclePartition 4 :: Vector (Int, Int)+-- [(0,1),(1,3),(2,2),(3,0)]+uniformCyclePartition ::+  (StatefulGen g m, PrimMonad m) =>+  Int ->+  g ->+  m (V.Vector (Int, Int))+uniformCyclePartition n gen = V.indexed <$> uniformPermutation n gen++-- TODO: apply short-circuiting behavior by creating  modification+-- of 'uniformSuffleM' that carries a validity state and short-circuits as+-- soon as some edge does not satisfy the predicate.+-- current implementation is the lazy one (as in human-lazy). note that would require+-- posting a notice in this module in accordance with the BSD2 license of mwc-random.++-- | Uniform selection of a cycle partition graph of '[0..n-1]' elements,+-- conditional on an edge-wise predicate. See 'uniformCyclePartition' for+-- details on the sampler.+--+-- /O(n\/p)/, where /p/ is the probability that a uniformly sampled+-- cycle partition graph (over all /n!/ possible) satisfies the conditions.+-- This can be highly non-linear since /p/ in general is a function of /n/.+--+-- Since this is a rejection sampling method, the user is asked to provide+-- a counter for the maximum number of sampling attempts in order to guarantee+-- termination in cases where the edge predicate has probability of success close+-- to zero.+--+-- Note this will return 'pure Nothing' if given a number of vertices that is+-- non-positive, in the third argument, unlike 'uniformCyclePartition' which+-- will throw an error.+uniformCyclePartitionThin ::+  (StatefulGen g m, PrimMonad m) =>+  Int ->+  ((Int, Int) -> Bool) ->+  Int ->+  g ->+  m (Maybe (V.Vector (Int, Int)))+uniformCyclePartitionThin maxit _ n _en | maxit <= 0 || n <= 0 = pure Nothing+uniformCyclePartitionThin maxit r n gen = do+  let v = V.generate n id+  mv <- V.thaw v+  chk' <- liftPrim $ newSTRef False+  maxit' <- liftPrim $ newSTRef maxit++  uniformCyclePartitionThinM chk' maxit' r mv gen++  chk <- liftPrim $ readSTRef chk'+  if chk+    then do+      Just . V.indexed <$> V.freeze mv+    else pure Nothing
+ src/RandomCycle/Vector/Partition.hs view
@@ -0,0 +1,60 @@+-- | Internal module whose primary export is 'uniformPartition'.  Use+-- 'RandomCycle.Vector' instead.+module RandomCycle.Vector.Partition where++import Data.Bits+import qualified Data.Vector.Generic as GV+import GHC.Natural (Natural)+import System.Random.Stateful++{- UTILITIES -}++---- | Internal. Find the first index where the bit is flipped, shifting the+-- bits as you go and returning the final shifted bit vector. The degenerate+-- case @bs == 0@ returns the otherwise unreachable point @(0, 0)@ to guarantee+-- termination, but note that case is nonsensical in 'partitionFromBits' and+-- handled explicitly there.+commonSubseqBits :: Natural -> (Natural, Int)+commonSubseqBits 0 = (0, 0)+commonSubseqBits bs = until done (\(bs', i) -> (bs' `shiftR` 1, i + 1)) (bs, 0)+  where+    done = if bs `testBit` 0 then not . (`testBit` 0) . fst else (`testBit` 0) . fst++-- | Partition a vector 'v' according to groupings provided by the bits 'bs'.+-- If the first set bit in 'bs' is at a position larger than the last index of+-- 'v', this returns @[v]@. More generally, bits set at positions after the+-- last index of 'v' do not contribute to the grouping. @bs == 0@ always+-- results in @[v]@.+--+-- See 'RandomCycle.List.partitionFromBits' for other examples.+--+-- >>> import qualified Data.Vector as V+-- >>> partitionFromBits 5 (V.fromList [0..2::Int])+-- [[0],[1],[2]]+-- >>> partitionFromBits 13 (V.fromList [0..2::Int])+-- [[0],[1],[2]]+-- >>> partitionFromBits 4 (V.fromList [0..2::Int])+-- [[0,1],[2]]+-- >>> partitionFromBits 8 (V.fromList [0..2::Int])+-- [[0,1,2]]+partitionFromBits :: (GV.Vector v a) => Natural -> v a -> [v a]+partitionFromBits _ v | GV.null v = []+partitionFromBits 0 v = [v]+partitionFromBits bs v =+  let (bs', idx) = commonSubseqBits bs+      (v1, v2) = GV.splitAt idx v+   in v1 : partitionFromBits bs' v2++{- RANDOM -}++-- | Draw a random partition of the input vector 'xs' from the uniform+-- distribution on partitions. This proceeds by randomizing the placement of+-- each breakpoint, in other words by walking a random path in a perfect binary+-- tree. /O(n)/ for a vector length /n/.+--+-- This function preserves the order of the input list.+uniformPartition :: (GV.Vector v a, StatefulGen g m) => v a -> g -> m [v a]+uniformPartition xs g = do+  let d = GV.length xs+  bs <- uniformRM (0, 2 ^ d - 1) g+  pure $ partitionFromBits bs xs
+ test/Main.hs view
@@ -0,0 +1,8 @@+module Main where++import Test.Tasty (defaultMain, testGroup)+import TestCycles+import TestPartitions++main :: IO ()+main = defaultMain $ testGroup "RandomCycle" [testPartitions, testCycles]
+ test/TestCycles.hs view
@@ -0,0 +1,77 @@+module TestCycles where++import Algebra.Graph (Graph, circuit, edges, overlays)+import Data.List (nub, permutations, sort)+import qualified Data.Vector as V+import qualified Data.Vector.Algorithms.Merge as VA+import GHC.Natural+import qualified RandomCycle.List as RL+import qualified RandomCycle.Vector as RV+import System.Random.Stateful (mkStdGen, runSTGen_)+import Test.Tasty+import Test.Tasty.HUnit (assertBool, testCase)+import Test.Tasty.QuickCheck++{- Top-level -}++testCycles :: TestTree+testCycles = testGroup "Cycles" [cpTest 5, cpThinProp]+  where+    cpTest m =+      testGroup "Cycle partition isomorphism" $+        map+          ( \n ->+              testCase (show n) $+                assertBool "Cycle partition isomorphism check failed" $+                  cyclePartitionIsomorphism n+          )+          [0 .. m]+    cpThinProp = testProperty "Cycle partition with thinning" prop_cyclePartitionThin++{- PROPERTIES -}++-- | Utility to directly create the cycle partition graph from a given partition.+-- This is just for testing and is blatantly inefficient as written here.+cyclePartitionGraph :: [[Int]] -> Graph Int+cyclePartitionGraph = overlays . map circuit++-- | Test validating the claim underlying the sampling scheme that cycle partitions+-- are isomorphic with the permutations, which the reader could also prove directly.+-- This is very inefficient, so it should be run on only a small 'n'.+-- Should serve as a stand-in for a property test of 'uniformCyclePartition',+-- which just applies this fact to claim uniformity in sampling.+cyclePartitionIsomorphism :: Int -> Bool+cyclePartitionIsomorphism n = and $ zipWith (==) cps gps+  where+    n' = abs n+    ids = [0 .. n']+    perms = permutations ids+    cps = sort $ map (edges . zip ids) perms+    gps =+      sort $+        nub $+          concatMap (\n -> map (cyclePartitionGraph . RL.partitionFromBits n) perms) [0 :: Natural .. 2 ^ n' - 1]++-- | Property test checking that sampling subject to a simple set of rules produces the correct result.+prop_cyclePartitionThin :: NonNegative Int -> Property+prop_cyclePartitionThin (NonNegative n) = True === chk (V.all simpleEdgeRules <$> v)+  where+    -- Min number for there to exist a solution with given rules+    nmin = n + 3+    v = runSTGen_ (mkStdGen 1305) $ RV.uniformCyclePartitionThin 1000 simpleEdgeRules nmin+    chk Nothing = False+    chk (Just b) = b++{- UTILITIES -}++-- | Some simple rules to check.+-- You must ensure the predicate check 'all' of them has non-empty support.+--+-- NOTE: This list requires the number of vertices to be > 2+-- for there to exist a solution.+simpleEdgeRules :: (Int, Int) -> Bool+simpleEdgeRules e = all ($ e) [noSelf, no12]+  where+    noSelf = uncurry (/=)+    no12 (1, 2) = False+    no12 _ = True
+ test/TestPartitions.hs view
@@ -0,0 +1,83 @@+module TestPartitions (testPartitions) where++import Data.Maybe (fromJust)+import qualified Data.Vector as V+import GHC.Natural (wordToNatural)+import RandomCycle.List as RL+import RandomCycle.Vector as RV+import System.Random.Stateful (mkStdGen, runStateGen_)+import Test.Tasty (TestTree)+import Test.Tasty.QuickCheck++{- Top-level -}++testPartitions :: TestTree+testPartitions = testProperties "Partitions" prop_partitions++{- Partition Properties -}++prop_partitions :: [(String, Property)]+prop_partitions = props1+  where+    props1 =+      [ ("isPartitionV", property prop_isPartitionV),+        ("isPartitionL", property prop_isPartitionL),+        ("breaksCorrectL", property prop_breaksCorrectL),+        ("breaksCorrectV", property prop_breaksCorrectV),+        -- With thinning+        ("isPartitionThinL", property prop_isPartitionThinL)+      ]++-- Vectors++prop_isPartitionV :: [Int] -> Property+prop_isPartitionV xs = V.toList xs' === xs+  where+    xs' = V.concat (run $ RV.uniformPartition $ V.fromList xs)++-- Are the breaks in the right places?+prop_breaksCorrectV :: Word -> NBits -> Property+prop_breaksCorrectV bs (NBits n) = pl === pl'+  where+    pl = RL.partitionLengths bs n+    pl' = map length $ RV.partitionFromBits (wordToNatural bs) $ V.fromList [1 .. n]++-- Lists++-- Is it a partition?+prop_isPartitionL :: [Int] -> Property+prop_isPartitionL xs = concat (run $ RL.uniformPartition xs) === xs++-- Are the breaks in the right places?+prop_breaksCorrectL :: Word -> NBits -> Property+prop_breaksCorrectL bs (NBits n) = pl === pl'+  where+    pl = RL.partitionLengths bs n+    pl' = map length $ RL.partitionFromBits (wordToNatural bs) [1 .. n]++-- Is it a partition and does it follow the rule?+-- NonNegative used to generate a list that with reasonable probability will+-- satisfy the rules.+prop_isPartitionThinL :: NonNegative Int -> Property+prop_isPartitionThinL (NonNegative i) = propP .&&. propR+  where+    xs = [0 .. (i + 2)]+    r = (>= 2) . sum+    -- NOTE: at the moment, we don't want to test+    -- "no match found" cases here.+    ps = fromJust $ run $ RL.uniformPartitionThin maxit r xs+    propP = concat ps === xs+    propR = all r ps === True++{- Utilities -}++-- Max iterations for all thin tests+maxit = 1000++run = runStateGen_ (mkStdGen 1305)++-- Int restricted to range of 1..64 bits+newtype NBits = NBits Int deriving (Show)++instance Arbitrary NBits where+  arbitrary = NBits <$> choose (1, 64)