Uniform selection of partitions and cycle partitions

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.

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.

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). Therefore, sampling a uniform cycle partition without conditions is as simple as sampling a uniform permutation.

Note self-loops are allowed in the possible configurations.

`uniformCyclePartitionThin`

samples uniformly from the set of cycle partition
graphs satisfying a predicate, in O(n/p) time on average, where p is the
proportion of cycle partition graphs satisfying the predicate. It works by
rejection sampling, so the user is asked to provide a maximum number of
sampling attempts to guarantee termination.

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.

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.

At the moment, the `uniformPartitionThin`

is implemented only for lists.

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`

.

Similarly, if the predicate itself is short-circuiting, the sampler will short-circuit.

Send by email, without need for an account, to ~brendanrbrown/random-cycle@todo.sr.ht

Man pages for tickets on SourceHut, particularly the "Email access" section.

Man pages for sending patches upstream.