package patricia-tree

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Patricia Tree data structure in OCaml for maps and sets. Supports generic key-value pairs


Dune Dependency






Published: 03 Jun 2024


Patricia Tree

This is an OCaml library that implements sets and maps as Patricia Trees, as described in Okasaki and Gill's 1998 paper Fast mergeable integer maps. It is a space-efficient prefix trie over the big-endian representation of the key's integer identifier.

The documentation for this library can be found online at

This library was written by Matthieu Lemerre then further improved by Dorian Lesbre, as part of the Codex semantics library, developed at CEA List.

Table of Contents:


This library can be installed with opam:

opam install patricia-tree

Alternatively, you can clone the source repository and install with dune:

git clone
cd patricia-tree
opan install . --deps-only
dune build -p patricia-tree
dune install
# To build documentation
opam install . --deps-only --with-doc
dune build @doc


  • Similar to OCaml's Map and Set, using the same function names when possible and the same convention for order of arguments. This should allow switching to and from Patricia Tree with minimal effort.

  • The functor parameters (KEY module) requires an injective to_int : t -> int function instead of a compare function. to_int should be fast and injective. This works well with hash-consed types.

  • The Patricia Tree representation is stable, contrary to maps, inserting nodes in any order will return the same shape. This allows different versions of a map to share more subtrees in memory, and the operations over two maps to benefit from this sharing. The functions in this library attempt to maximally preserve sharing and benefit from sharing, allowing very important improvements in complexity and running time when combining maps or sets is a frequent operation.

    To do so, these functions often have extra requirements on their argument (e.g. inter f m1 m2 can be optimized by not inspecting common subtrees when f is idempotent). To avoid accidental errors, they are renamed (e.g. to idempotent_inter for the efficient version and nonidempotent_inter_no_share for the general one)

  • Since our Patricia Tree use big-endian order on keys, the maps and sets are sorted in increasing unsigned order of keys. This means negative keys are sorted above positive keys, with -1 being the largest possible key, and 0 the smallest. This also avoids a bug in Okasaki's paper discussed in QuickChecking Patricia Trees by Jan Mitgaard.

    It also affects functions like unsigned_min_binding and pop_unsigned_minimum. They will return the smallest positive integer of both positive and negative keys are present; and not the smallest negative, as one might expect.

  • Supports generic maps and sets: a 'm map that maps 'k key to ('k, 'm) value. This is especially useful when using GADTs for the type of keys. This is also sometimes called a dependent map.

  • Allows easy and fast operations across different types of maps and set which have the same type of keys (e.g. an intersection between a map and a set).

  • Multiple choices for internal representation (NODE), which allows for efficient storage (no need to store a value for sets), or using weak nodes only (values removed from the tree if no other pointer to it exists). This system can also be extended to store size information in nodes if needed.

  • Exposes a common interface (view) to allow users to write their own pattern matching on the tree structure without depending on the NODE being used.

  • hash-consed versions of heterogeneous/homogeneous maps/sets are available. These provide constant time equality and comparison, and ensure maps/set with the same constants are always physically equal. It comes at the cost of a constant overhead in memory usage (at worst, as hash-consing may allow memory gains) and constant time overhead when calling constructors.

Quick overview


This library contains a single module, PatriciaTree. The main functors used to build our maps and sets are the following:

(** {2 Homogeneous maps and sets} *)

module MakeMap(Key: KEY) : MAP with type key = Key.t
module MakeSet(Key: KEY) : SET with type elt = Key.t

(** {2 Heterogeneous maps and sets} *)

  with type 'a elt = 'a Key.t
module MakeHeterogeneousMap(Key: HETEROGENEOUS_KEY)(Value: HETEROGENEOUS_VALUE) :
  with type 'a key = 'a Key.t
   and type ('k,'m) value = ('k,'m) Value.t

There are also hash-consed versions of these four functors: MakeHashconsedMap, MakeHashconsedSet, MakeHashconsedHeterogeneousMap and MakeHashconsedHeterogeneousSet. These uniquely number their nodes, which means:

  • equal and compare become constant time operations;

  • two maps with the same bindings (where keys are compared by KEY.to_int and values by HASHED_VALUE.polyeq) will always be physically equal;

  • functions that benefit from sharing will see improved performance;

  • constructors are slightly slower, as they now require a hash-table lookup;

  • memory usage is increased: nodes store their tags inside themselves, and a global hash-table of all built nodes must be maintained;

  • hash-consed maps assume their values are immutable;

  • WARNING: when using physical equality as HASHED_VALUE.polyeq, some maps of different types may be given the same identifier. See the end of the documentation of HASHED_VALUE.polyeq for details. Note that this is the case in the default implementations HashedValue and HeterogeneousHashedValue.

  • All hash-consing functors are generative, since each functor call will create a new hash-table to store the created nodes. Calling a functor twice with same arguments will lead to two numbering systems for identifiers, and thus the types should not be considered compatible.


Here is a brief overview of the various module types of our library:

  • BASE_MAP: the underlying module type of all our trees (maps end sets). It represents a 'b map binding 'a key to ('a,'b) value, as well as all functions needed to manipulate them.

    It can be accessed from any of the more specific maps types, thus providing a unified representation, useful for cross map operations. However, for practical purposes, it is often best to use the more specific interfaces:

    • HETEROGENEOUS_MAP for heterogeneous maps (this is just BASE_MAP with a WithForeign functor).

    • MAP for homogeneous maps, this interface is close to Stdlib.Map.S.

    • HETEROGENEOUS_SET for heterogeneous sets (sets of 'a elt). These are just maps to unit, but with a custom node representation to avoid storing unit in nodes.

    • SET for homogeneous sets, this interface is close to Stdlib.Set.S.

  • The parameter of our functor are either KEY or HETEROGENEOUS_KEY. These just consist of a type, a (polymorphic) equality function, and an injective to_int coercion.

    The heterogeneous map functor also has a HETEROGENEOUS_VALUE parameter to specify the ('a, 'b) value type

  • The internal representations of our tree can be customized to use different internal NODE. Each node come with its own private constructors and destructors, as well as a cast to a uniform view type used for pattern matching.

    A number of implementations are provided:

    • SimpleNode: exactly the NODE.view type;

    • WeakNode: only store weak pointer to its elements;

    • NodeWithId: node which contains a unique identifier;

    • SetNode: optimized for sets, doesn't store the [unit] value;

    • WeakSetNode: both a WeakNode and as SetNode

    • HashconsedNode: performs hash-consing (it also stores a unique identifier, but checks when building a new node whether a node with similar content already exists);

    • HashconsedSetNode: both a HashconsedNode and a SetNode.

    Use the functors MakeCustomMap and MakeCustomSet (or their heterogeneous versions MakeCustomHeterogeneousMap and MakeCustomHeterogeneousSet) to build maps using these nodes, or any other custom nodes.


Homogeneous map

Here is a small example of a non-generic map:

(** Create a key struct *)
module Int (*: PatriciaTree.KEY*) = struct
  type t = int
  let to_int x = x

(** Call the map and/or set functors *)
module IMap = PatriciaTree.MakeMap(Int)
module ISet = PatriciaTree.MakeSet(Int)

(** Use all the usual map operations *)
let map =
  IMap.empty |>
  IMap.add 1 "hello" |>
  IMap.add 2 "world" |>
  IMap.add 3 "how do you do?"
  (* Also has an [of_list] and [of_seq] operation for initialization *)

let _ = IMap.find 1 map (* "hello" *)
let _ = IMap.cardinal map (* 3 *)

(** The strength of Patricia Tree is the speedup of operation on multiple maps
    with common subtrees. *)
let map2 =
  IMap.idempotent_inter_filter (fun _key _l _r -> None)
  (IMap.add 4 "something" map) (IMap.add 5 "something else" map)
let _ = map == map2 (* true *)
(* physical equality is preserved as much as possible, although some intersections
   may need to build new nodes and won't be fully physically equal, they will
   still share subtrees if possible. *)

(** Many operations preserve physical equality whenever possible *)
let _ = (IMap.add 1 "hello" map) == map (* true: already present *)

(** Example of cross map/set operation: only keep the bindings of [map]
    whose keys are in a given set *)
let set = ISet.of_list [1; 3]
module CrossOperations = IMap.WithForeign(ISet.BaseMap)
let restricted_map = CrossOperations.nonidempotent_inter
  { f = fun _key value () -> value } map set

Heterogeneous map

(** Very small typed expression language *)
type 'a expr =
  | G_Const_Int : int -> int expr
  | G_Const_Bool : bool -> bool expr
  | G_Addition : int expr * int expr -> int expr
  | G_Equal : 'a expr * 'a expr -> bool expr

module Expr : PatriciaTree.HETEROGENEOUS_KEY with type 'a t = 'a expr = struct
  type 'a t = 'a expr

  (** Injective, so long as expression are small enough
      (encodes the constructor discriminant in two lowest bits).
      Ideally, use a hash-consed type, to_int needs to be fast *)
  let rec to_int : type a. a expr -> int = function
    | G_Const_Int i ->   0 + 4*i
    | G_Const_Bool b ->  1 + 4*(if b then 1 else 0)
    | G_Addition(l,r) -> 2 + 4*(to_int l mod 10000 + 10000*(to_int r))
    | G_Equal(l,r) ->    3 + 4*(to_int l mod 10000 + 10000*(to_int r))

  (** Full polymorphic equality *)
  let rec polyeq : type a b. a expr -> b expr -> (a, b) PatriciaTree.cmp =
    fun l r -> match l, r with
    | G_Const_Int l, G_Const_Int r -> if l = r then Eq else Diff
    | G_Const_Bool l, G_Const_Bool r -> if l = r then Eq else Diff
    | G_Addition(ll, lr), G_Addition(rl, rr) -> (
        match polyeq ll rl with
        | Eq -> polyeq lr rr
        | Diff -> Diff)
    | G_Equal(ll, lr), G_Equal(rl, rr) ->    (
        match polyeq ll rl with
        | Eq -> (match polyeq lr rr with Eq -> Eq | Diff -> Diff) (* Match required by typechecker *)
        | Diff -> Diff)
    | _ -> Diff

(** Map from expression to their values: here the value only depends on the type
    of the key, not that of the map *)
module EMap = PatriciaTree.MakeHeterogeneousMap(Expr)(struct type ('a, _) t = 'a end)

(** You can use all the usual map operations *)
let map : unit EMap.t =
  EMap.empty |>
  EMap.add (G_Const_Bool false) false |>
  EMap.add (G_Const_Int 5) 5 |>
  EMap.add (G_Addition (G_Const_Int 3, G_Const_Int 6)) 9 |>
  EMap.add (G_Equal (G_Const_Bool false, G_Equal (G_Const_Int 5, G_Const_Int 7))) true

let _ = EMap.find (G_Const_Bool false) map (* false *)
let _ = EMap.cardinal map (* 4 *)

(** Fast operations on multiple maps with common subtrees. *)
let map2 =
    { f = fun _key _l _r -> None } (* polymorphic 1rst order functions are wrapped in records *)
    (EMap.add (G_Const_Int 0) 8 map)
    (EMap.add (G_Const_Int 0) 9 map)

Release status

This should be close to a stable release. It is already being used as part of a larger project successfully, and this usage as helped us mature the interface. As is, we believe the project is usable, and we don't anticipate any major change before 1.0.0. We didn't commit to a stable release straight away as we would like a bit more time using this library before doing so.

Known issues

There is a bug in the OCaml typechecker which prevents us from directly defining non-generic maps as instances of generic maps. To avoid this, non-generic maps use a separate value type (instead of just using 'b)

type (_, 'b) snd = Snd of 'b [@@unboxed]

It should not incur any extra performance cost as it is unboxed, but can appear when manipulating non-generic maps.

For more details about this issue, see the OCaml discourse discussion.

Comparison to other OCaml libraries

ptmap and ptset

There are other implementations of Patricia Tree in OCaml, namely ptmap and ptset, both by J.C. Filliatre. These are smaller and closer to OCaml's built-in Map and Set, however:

  • Our library allows using any type key that comes with an injective to_int function, instead of requiring key = int.

  • We support generic (heterogeneous) types for keys/elements.

  • We support operations between sets and maps of different types.

  • We use a big-endian representation, allowing easy access to min/max elements of maps and trees.

  • Our interface and implementation tries to maximize the sharing between different versions of the tree, and to benefit from this memory sharing. Theirs do not.

  • These libraries work with older version of OCaml (>= 4.05 I believe), whereas ours requires OCaml >= 4.14 (for the new interface of Ephemeron used in WeakNode).


Additionally, there is a dependent map library: dmap, which gave us the idea of making our PatriciaTree dependent. It allows creating type safe dependent maps similar to our heterogeneous maps. However, its maps aren't Patricia trees. They are binary trees build using a (polymorphic) comparison function, similarly to the maps of the standard library.

Another difference is that the type of values in the map is independent of the type of the keys, allowing keys to be associated with different values in different maps. i.e. we map 'a key to any ('a, 'b) value type, whereas dmap only maps 'a key to 'a or 'a value.

dmap also works with OCaml >= 4.12, whereas we require OCaml >= 4.14.

Contributions and bug reports

Any contributions are welcome!

You can report any bug, issues, or desired features using the Github issue tracker. Please include OCaml, dune, and library version information in you bug reports.

If you want to contribute code, feel free to fork the repository on Github and open a pull request. By doing so you agree to release your code under this project's license (LGPL-2.1).

There is no imposed coding style for this repository, here are just a few guidelines and conventions:

  • Module type names should use SCREAMING_SNAKE_CASE.

  • Module and functor names use PascalCase, functors names start with Make.

  • Even though the library implements homogeneous maps as a specialization of heterogeneous ones, the naming convention is that no prefix means homogeneous, and all heterogeneous objects are prefixed with heterogeneous.

  • Please document any new functions in the interface, using ocamldoc style comments.

  • Please consider adding test for new features/fixed bugs if at all possible. This library uses a QuickCheck framework for tests.

Dependencies (2)

  1. dune >= "3.0"
  2. ocaml >= "4.14"

Dev Dependencies (5)

  1. sherlodoc with-doc
  2. odoc >= "2.4.0" & with-doc
  3. mdx >= "2.4.1" & with-test
  4. ppx_inline_test >= "v0.16.0" & with-test
  5. qcheck-core >= "0.21.2" & with-test

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