SWI-Prolog 9.1.21
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PublicShow sourcerbtrees.pl -- Red black trees

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.

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
Source rb_new(-Tree) is det
Create a new Red-Black tree Tree.
deprecated
- Use rb_empty/1.
Source rb_empty(?Tree) is semidet
Succeeds if Tree is an empty Red-Black tree.
Source rb_lookup(+Key, -Value, +Tree) is semidet
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.
Source rb_min(+Tree, -Key, -Value) is semidet
Key is the minimum key in Tree, and is associated with Val.
Source rb_max(+Tree, -Key, -Value) is semidet
Key is the maximal key in Tree, and is associated with Val.
Source rb_next(+Tree, +Key, -Next, -Value) is semidet
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.
Source rb_previous(+Tree, +Key, -Previous, -Value) is semidet
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.
Source rb_update(+Tree, +Key, ?NewVal, -NewTree) is semidet
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.
Source rb_update(+Tree, +Key, -OldVal, ?NewVal, -NewTree) is semidet
Same as rb_update(Tree, Key, NewVal, NewTree) but also unifies OldVal with the value associated with Key in Tree.
Source rb_apply(+Tree, +Key, :G, -NewTree) is semidet
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.
Source rb_in(?Key, ?Value, +Tree) is nondet
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.

Source rb_insert(+Tree, +Key, ?Value, -NewTree) is det
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.
Source rb_insert_new(+Tree, +Key, ?Value, -NewTree) is semidet
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.
Source 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.
Source 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.
Source 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.
Source 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.
Source rb_visit(+Tree, -Pairs) is det
Pairs is an infix visit of tree Tree, where each element of Pairs is of the form Key-Value.
Source rb_map(+T, :Goal) is semidet
True if call(Goal, Value) is true for all nodes in T.
Source rb_map(+Tree, :G, -NewTree) is semidet
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).
Source 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.

Source rb_clone(+TreeIn, -TreeOut, -Pairs) is det
`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.
Source 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).
Source rb_keys(+Tree, -Keys) is det
Keys is unified with an ordered list of all keys in the Red-Black tree Tree.
Source list_to_rbtree(+List, -Tree) is det
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.
Source ord_list_to_rbtree(+List, -Tree) is det
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.
Source rb_size(+Tree, -Size) is det
Size is the number of elements in Tree.
Source is_rbtree(@Term) is semidet
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.