diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,5 @@ +# Revision history for random-cycle + +## 0.1.0.0 -- 2023-04-17 + +* First version. Released on an unsuspecting world. diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. By contrast, +the GNU General Public License is intended to guarantee your freedom to +share and change all versions of a program--to make sure it remains free +software for all its users. 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If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. 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 +. + + 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 +. diff --git a/README.md b/README.md new file mode 100644 --- /dev/null +++ b/README.md @@ -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. + + + +## 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 + + +Man pages for [sending patches upstream](https://man.sr.ht/git.sr.ht/#sending-patches-upstream). diff --git a/benches/Main.hs b/benches/Main.hs new file mode 100644 --- /dev/null +++ b/benches/Main.hs @@ -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) diff --git a/benches/RunTimes.hs b/benches/RunTimes.hs new file mode 100644 --- /dev/null +++ b/benches/RunTimes.hs @@ -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 diff --git a/random-cycle.cabal b/random-cycle.cabal new file mode 100644 --- /dev/null +++ b/random-cycle.cabal @@ -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 diff --git a/src/RandomCycle/List.hs b/src/RandomCycle/List.hs new file mode 100644 --- /dev/null +++ b/src/RandomCycle/List.hs @@ -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 diff --git a/src/RandomCycle/List/Partition.hs b/src/RandomCycle/List/Partition.hs new file mode 100644 --- /dev/null +++ b/src/RandomCycle/List/Partition.hs @@ -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 diff --git a/src/RandomCycle/Vector.hs b/src/RandomCycle/Vector.hs new file mode 100644 --- /dev/null +++ b/src/RandomCycle/Vector.hs @@ -0,0 +1,10 @@ +module RandomCycle.Vector + ( uniformPartition, + partitionFromBits, + uniformCyclePartition, + uniformCyclePartitionThin, + ) +where + +import RandomCycle.Vector.Cycle +import RandomCycle.Vector.Partition diff --git a/src/RandomCycle/Vector/Cycle.hs b/src/RandomCycle/Vector/Cycle.hs new file mode 100644 --- /dev/null +++ b/src/RandomCycle/Vector/Cycle.hs @@ -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 diff --git a/src/RandomCycle/Vector/Partition.hs b/src/RandomCycle/Vector/Partition.hs new file mode 100644 --- /dev/null +++ b/src/RandomCycle/Vector/Partition.hs @@ -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 diff --git a/test/Main.hs b/test/Main.hs new file mode 100644 --- /dev/null +++ b/test/Main.hs @@ -0,0 +1,8 @@ +module Main where + +import Test.Tasty (defaultMain, testGroup) +import TestCycles +import TestPartitions + +main :: IO () +main = defaultMain $ testGroup "RandomCycle" [testPartitions, testCycles] diff --git a/test/TestCycles.hs b/test/TestCycles.hs new file mode 100644 --- /dev/null +++ b/test/TestCycles.hs @@ -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 diff --git a/test/TestPartitions.hs b/test/TestPartitions.hs new file mode 100644 --- /dev/null +++ b/test/TestPartitions.hs @@ -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)