## A.45 library(rbtrees): Red black trees

author
  Vitor Santos Costa, Jan Wielemaker, Samer Abdallah, Peter Ludemann.
See also
  - `library(pairs)`, `library(assoc)`
  - "Introduction to Algorithms", Second Edition Cormen, Leiserson, Rivest, and Stein, MIT Press

Red-Black trees are balanced search binary trees. They are named because nodes can be classified as either red or black. The code we include is based on "Introduction to Algorithms", second edition, by Cormen, Leiserson, Rivest and Stein. The library includes routines to insert, lookup and delete elements in the tree.

A Red black tree is represented as a term `t(Nil, Tree)`, where Nil is the Nil-node, a node shared for each nil-node in the tree. Any node has the form `colour(Left, Key, Value, Right)`, where colour is one of `red` or `black`.

Warning: instantiation of keys

Red-Black trees depend on the Prolog standard order of terms to organize the keys as a (balanced) binary tree. This implies that any term may be used as a key. The tree may produce wrong results, such as not being able to find a key, if the ordering of keys changes after the key has been inserted into the tree. The user is responsible to ensure that variables used as keys or appearing in a term used as key that may affect ordering are not unified, with the exception of unification against new fresh variables. For this reason, ground terms are safe keys. When using non-ground terms, either make sure the variables appear in places that do not affect the standard order relative to other keys in the tree or make sure to not unify against these variables as long as the tree is being used.

[det]rb_new(-Tree)
  Create a new Red-Black tree Tree.
  deprecated
    Use rb_empty/1.
[semidet]rb_empty(?Tree)
  Succeeds if Tree is an empty Red-Black tree.
[semidet]rb_lookup(+Key, -Value, +Tree)
  True when Value is associated with Key in the Red-Black tree Tree. The given Key may include variables, in which case the RB tree is searched for a key with equivalent variables (using (`==`)/2). Time complexity is O(log N) in the number of elements in the tree.
  See also
    rb_in/3 for backtracking over keys.
[semidet]rb_min(+Tree, -Key, -Value)
  Key is the minimum key in Tree, and is associated with Val.
[semidet]rb_max(+Tree, -Key, -Value)
  Key is the maximal key in Tree, and is associated with Val.
[semidet]rb_next(+Tree, +Key, -Next, -Value)
  Next is the next element after Key in Tree, and is associated with Val. Fails if Key isn't in Tree or if Key is the maximum key.
[semidet]rb_previous(+Tree, +Key, -Previous, -Value)
  Previous is the previous element after Key in Tree, and is associated with Val. Fails if Key isn't in Tree or if Key is the minimum key.
[semidet]rb_update(+Tree, +Key, ?NewVal, -NewTree)
  Tree NewTree is tree Tree, but with value for Key associated with NewVal. Fails if Key is not in Tree (using (`==`)/2). This predicate may fail or give unexpected results if Key is not sufficiently instantiated.
  See also
    rb_in/3 for backtracking over keys.
[semidet]rb_update(+Tree, +Key, -OldVal, ?NewVal, -NewTree)
  Same as `rb_update(Tree, Key, NewVal, NewTree)` but also unifies OldVal with the value associated with Key in Tree.
[semidet]rb_apply(+Tree, +Key, :G, -NewTree)
  If the value associated with key Key is Val0 in Tree, and if `call(G,Val0,ValF)` holds, then NewTree differs from Tree only in that Key is associated with value ValF in tree NewTree. Fails if it cannot find Key in Tree, or if `call(G,Val0,ValF)` is not satisfiable.
[nondet]rb_in(?Key, ?Value, +Tree)
  True when Key-Value is a key-value pair in red-black tree Tree. Same as below, but does not materialize the pairs.

      rb_visit(Tree, Pairs), member(Key-Value, Pairs)

  Leaves a choicepoint even if Key is instantiated; to avoid a choicepoint, use rb_lookup/3.

[det]rb_insert(+Tree, +Key, ?Value, -NewTree)
  Add an element with key Key and Value to the tree Tree creating a new red-black tree NewTree. If Key is a key in Tree, the associated value is replaced by Value. See also rb_insert_new/4. Does not validate that Key is sufficiently instantiated to ensure the tree remains valid if a key is further instantiated.
[semidet]rb_insert_new(+Tree, +Key, ?Value, -NewTree)
  Add a new element with key Key and Value to the tree Tree creating a new red-black tree NewTree. Fails if Key is a key in Tree. Does not validate that Key is sufficiently instantiated to ensure the tree remains valid if a key is further instantiated.
rb_delete(+Tree, +Key, -NewTree)
  Delete element with key Key from the tree Tree, returning the value Val associated with the key and a new tree NewTree. Fails if Key is not in Tree (using (`==`)/2).
  See also
    rb_in/3 for backtracking over keys.
rb_delete(+Tree, +Key, -Val, -NewTree)
  Same as `rb_delete(Tree, Key, NewTree)`, but also unifies Val with the value associated with Key in Tree.
rb_del_min(+Tree, -Key, -Val, -NewTree)
  Delete the least element from the tree Tree, returning the key Key, the value Val associated with the key and a new tree NewTree. Fails if Tree is empty.
rb_del_max(+Tree, -Key, -Val, -NewTree)
  Delete the largest element from the tree Tree, returning the key Key, the value Val associated with the key and a new tree NewTree. Fails if Tree is empty.
[det]rb_visit(+Tree, -Pairs)
  Pairs is an infix visit of tree Tree, where each element of Pairs is of the form Key-Value.
[semidet]rb_map(+T, :Goal)
  True if `call(Goal, Value)` is true for all nodes in T.
[semidet]rb_map(+Tree, :G, -NewTree)
  For all nodes Key in the tree Tree, if the value associated with key Key is Val0 in tree Tree, and if `call(G,Val0,ValF)` holds, then the value associated with Key in NewTree is ValF. Fails if `call(G,Val0,ValF)` is not satisfiable for all Val0. If G is non-deterministic, rb_map/3 will backtrack over all possible values from `call(G,Val0,ValF)`. You should not depend on the order of tree traversal (currently: key order).
rb_fold(:Goal, +Tree, +State0, -State)
  Fold the given predicate over all the key-value pairs in Tree, starting with initial state State0 and returning the final state State. Pred is called as

      call(Pred, Key-Value, State1, State2)

  Determinism depends on Goal.

[det]rb_clone(+TreeIn, -TreeOut, -Pairs)
  ‘Clone' the red-back tree TreeIn into a new tree TreeOut with the same keys as the original but with all values set to unbound values. Pairs is a list containing all new nodes as pairs K-V.
rb_partial_map(+Tree, +Keys, :G, -NewTree)
  For all nodes Key in Keys, if the value associated with key Key is Val0 in tree Tree, and if `call(G,Val0,ValF)` holds, then the value associated with Key in NewTree is ValF, otherwise it is the value associated with the key in Tree. Fails if Key isn't in Tree or if `call(G,Val0,ValF)` is not satisfiable for all Val0 in Keys. Assumes keys are sorted and not repeated (fails if this is not true).
[det]rb_keys(+Tree, -Keys)
  Keys is unified with an ordered list of all keys in the Red-Black tree Tree.
[det]list_to_rbtree(+List, -Tree)
  Tree is the red-black tree corresponding to the mapping in List, which should be a list of Key-Value pairs. List should not contain more than one entry for each distinct key, but this is not validated by list_to_rbtree/2.
[det]ord_list_to_rbtree(+List, -Tree)
  Tree is the red-black tree corresponding to the mapping in list List, which should be a list of Key-Value pairs. List should not contain more than one entry for each distinct key, but this is not validated by ord_list_to_rbtree/2. List is assumed to be sorted according to the standard order of terms.
[det]rb_size(+Tree, -Size)
  Size is the number of elements in Tree.
[semidet]is_rbtree(@Term)
  True if Term is a valid Red-Black tree. Processes the entire tree, checking the coloring of the nodes, the balance and the ordering of keys. Does not validate that keys are sufficiently instantiated to ensure the tree remains valid if a key is further instantiated.