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
- rb_new(-Tree) is det
- Create a new Red-Black tree Tree.
- deprecated
- - Use rb_empty/1.
- rb_empty(?Tree) is semidet
- Succeeds if Tree is an empty Red-Black tree.
- 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.
- rb_min(+Tree, -Key, -Value) is semidet
- Key is the minimum key in Tree, and is associated with Val.
- rb_max(+Tree, -Key, -Value) is semidet
- Key is the maximal key in Tree, and is associated with Val.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- rb_map(+T, :Goal) is semidet
- True if
call(Goal, Value)
is true for all nodes in T.
- 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).
- 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.
- 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.
- 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).
- rb_keys(+Tree, -Keys) is det
- Keys is unified with an ordered list of all keys in the Red-Black
tree Tree.
- 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.
- 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.
- rb_size(+Tree, -Size) is det
- Size is the number of elements in Tree.
- 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.