Union-Find Lattice

This package centers around the union-find-lattice library, whose root module is Union_Find_Lattice. It extends the classic union-find structures with:

• A persistent union: t -> node -> node -> t which returns a new copy instead of modifying the source

• Lattice operations:

  • join: t -> t -> t, which is the union-find containing only the equalities that are true in both arguments;
  • meet: t -> t -> t, which is the union-find containing the equalities that are true in either argument;
  • incl: t -> t -> bool, which checks if all equalities from the left argument hold in the right argument;

These lattice operations and the algorithms implementing the are described by Lesbre and Lemerre, A Lattice of Union-Finds, SAS 2026, specifically, the difference-based lattice operations using union-by-rank.

This library was originally written by Dorian Lesbre and Matthieu Lemerre. Copyright (C) 2026 CEA (Commissariat à l'énergie atomique et aux énergies alternatives). It is provided here under a LGPL v2.1 license.

Data-structures

Each variant proposes four different data-structures to represent the union-find lattice:

• Patricia trees use functional maps and thus pay logarithmic cost for find and union, but are truly immutable and are thus the most flexible and easy to use.

• Persistent arrays are faster when repeatedly accessing one version, but pay a rerooting cost when switching (i.e. calling any function) on another version.

  • They are also limited to lattice operations between versions deriving from the same common ancestor.
  • As they are not truly persistent, they do not support unsynchronized access.
• PersistentArrayNCA is a variant of persistent array that adds a version tag to the arrays. It allows join to build the result on the nearest common ancestor, rather than one of the arguments. This takes a bit more memory but should result in shorter version chains when stacking joins on top of joins.
• ArrayWithCopy uses a standard union-find implementation based on a single mutable array. It is NOT persistent, but provides an explicit copy function. It uses linear lattice operations rather than the difference based one, which can be faster when the difference (i.e. the number of changes between both versions) is comparable to the number of nodes.

Variants

In addition to the Union_Find_Lattice.Classic implementations, we provide several variants

• Union_Find_Lattice.Valued extend union-find by attaching a Value to each equivalence class. These values also have a lattice structure. As an example, if union-find is representing a set of equalities between terms, these values can represent the set of possible values for each term equivalence class.
• Union_Find_Lattice.Labeled extend union-find by attaching a relation to each link between a node and its parents. These relations have a group structure. They allow representing more complex relations than equality, for instance, equality up to a constant. See Lesbre et al, Relational Abstractions based on Labeled Union-Find, PLDI 2024 for an in-depth description of labeled union-find.
• Union_Find_Lattice.Polymorphic is the same as Union_Find_Lattice.Labeled, but replaces the monomorphic types node and relation by polymorphic ones ('a node and ('a, 'b) relation
• Union_Find_Lattice.LabeledValued and Union_Find_Lattice.PolymorphicValued combine values and labels.

Example usage

Installation

To use the library, download the package with opam:

opam install union-find-lattice

Alternatively, clone the repository on github, install dependencies and build locally:

git clone git@github.com:codex-semantics-library/union-find-lattice.git
cd union-find-lattice
opan install . --deps-only
dune build -p union-find-lattice
dune install -p union-find-lattice
# To build tests and benchmarks
opan install . --deps-only --with-test --with-dev-setup
dune build
# To build documentation
opam install . --deps-only --with-doc
dune build @doc

Next add the library as a dependency in your dune files:

(executable ; or library
  ...
  (libraries union-find-lattice ...)
)

Example

Here is a minimal example of a union-find join with integer nodes

module UF = Union_Find_Lattice.Classic.PatriciaTree
  (Union_Find_Lattice.DefaultConfig)
  (struct
    include Int
    let to_int x = x
    let pretty = Format.pp_print_int
  end)

let root = UF.make 0 (* the number passed to make only matters for arrays *)

let left = UF.union (UF.union root 0 1) 1 3 (* one class: {0,1,3} *)
let right = UF.union (UF.union root 0 2) 2 3 (* one class: {0,2,3} *)

let join = UF.join left right (* one class: {0,3} *)
let meet = UF.meet left right (* one class: {0,1,2,3} *)

# UF.check_related join 0 3;;
- : bool = true

# UF.check_related join 0 1;;
- : bool = false

# UF.check_related meet 1 2 (* meet creates new equalities by transitivity *);;
- : bool = true

# UF.incl meet left && UF.incl meet right && UF.incl left join && UF.incl right join
  (* join is the least upper bound, meet the greatest lower bound *);;
- : bool = true

Other included libraries

This package also includes the following libraries:

• union-find-lattice.persistent-array, with root module PersistentArray, the persistent array structure used. It is very close to the one described by Conchon and Filliâtre, A Persistent Union-Find Data Structure, ML 2007, with the following differences:

  • we added our difference operator: PersistentArray.S.diff and PersistentArray.S.diff_key;
  • we added a variant, PersistentArray.Versioned, which add a version tag to arrays, allowing diff to return the nearest common ancestor;
  • we made our arrays extendable, similar to dynamic arrays/vectors. See PersistentArray.S.append and PersistentArray.S.extend
• union-find-lattice.utils, with root module Utils. Undocumented internal utilities