Îõ³h&%$—    None¤  random-cycleInternal. Version of  Data.List.  that uses the supplied bits bs as a grouping variable. switchí flips the booleans, so that the input bit's least significant digit determines the grouping. Note the case bs == 0Ê 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 È, but that would limit the input list to some length, e.g. 64 if using Word, which is too restrictive. random-cycle8Utility to generate a list partition using the provided " as grouping variable, viewed as Š. The choice of grouping variable is to improve performance since the number of partitions grows exponentially in the input list length.ïThis can be used to generate a list of all possible partitions of the input list as shown in the example. See  for other examples.import GHC.NaturalallPartitions n | n < 0 = []ÇallPartitions n = map (`partitionFromBits` [0..n-1]) [0 .. 2^(n-1) - 1]allPartitions 4Ø[[[0,1,2,3]],[[0],[1,2,3]],[[0],[1],[2,3]],[[0,1],[2,3]],[[0,1],[2],[3]],[[0],[1],[2],[3!]],[[0],[1,2],[3]],[[0,1,2],[3]]] random-cycleëPrimarily a testing utility, to compute directly the lengths of each partition element for a list of size n, using . Note this uses . random-cycle1Internal. Partition a list as determined by bits bs+, but shortcircuit if the local condition r% is false for some partition element. random-cycleInternal. Inner logic of å that carries around the input list length to avoid recomputation. It is the callers job to ensure n == length xs. random-cycle*Draw a random partition of the input list xs² 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.4This function preserves the order of the input list.Examplesimport System.Random.StatefulpureGen = mkStdGen 03runStateGen_ pureGen $ uniformPartition [1..5::Int][[1,2,3],[4],[5]]5runStateGen_ pureGen $ uniformPartition ([] :: [Int])[] random-cycle,Generate a partition with a local condition rû on each partition element. Construction of a partition shortcircuits to failure as soon as the local condition is false.îSince 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 rÜ 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.ÌSome cases can perhaps be deceptively expensive: For example, the condition r = ((>= 2) . length)ñ leads to huge runtimes, since the number of partitions with at least one element of length 1 is exponential in n.Examplesimport System.Random.Stateful maxit = 1000pureGen = mkStdGen 0r = (>= 2) . length?runStateGen_ pureGen $ uniformPartitionThin maxit r [1..5::Int]Just [[1,2],[3, 4, 5]]ÍrunStateGen_ pureGen $ uniformPartitionThin maxit (const False) ([] :: [Int])Just []