<Section Label="avlidea">
<Heading>The idea of AVL trees</Heading>
AVL trees are balanced binary trees called <Q>AVL trees</Q> in honour
of their inventors G.M. Adelson-Velskii and E.M. Landis (see <Cite
Key="AVL"/>). A description in English can be found in
<Cite Key="ACP3"/> in Section 6.2.3 about balanced trees.
<P/>
The general idea is to store data in a binary tree such that all
entries in the left subtree of a node are smaller than the entry at
the node and all entries in the right subtree are bigger. The tree is
kept <Q>balanced</Q> which means that for each node the depth of the
left and right subtrees differs by at most 1. In this way, finding
something in the tree, adding a new entry, deleting an entry all have
complexity <M>log(n)</M> where <M>n</M> is the number of entries in
the tree. If one additionally stores the number of entries in the left
subtree of each node, then finding the <M>k</M>-th entry, removing the
<M>k</M>-th entry and inserting an entry in position <M>k</M> also
have complexity <M>log(n)</M>. The <Package>orb</Package> contains an
implementation of such tree objects providing all these operations.
<P/>
<Q>Entries</Q> in AVL tree objects are key-value pairs and the sorting
is done by the key. If all values as <K>true</K> then no memory is
needed to store the values (see the corresponding behaviour for hash
tables). The only requirement on the type of the keys is that two
arbitrary keys must be comparable in the sense that one can decide
which of them is smaller. If &GAP;s standard comparison operations
<M><</M> and <M>=</M> work for your keys, no further action is
required, if not, then you must provide your own three-way comparison
function (see below).
<P/>
Note that the AVL trees implemented here can be used in basically two
different ways, which can sometimes be mixed: The usual way is by
accessing entries by their key, the tree is then automatically kept
sorted. The alternative way is by accessing entries by their index in
the tree! Since the nodes of the trees remember how many elements are
stored in their left subtree, it is in fact possible to access the
<M>k</M>-th entry in the tree or delete it. It is even possible to
insert something in position <M>k</M>. However, note that if you do
this latter operation, you are yourself responsible to keep the
entries in the tree sorted. You can ignore this responsibility, but
then you can no longer access the entries in the tree by their key
and the corresponding functions might fail or even run into errors.
<P/>
This usage can be useful, since in this way AVL trees provide an
implementation of a list data structure where the operation list
access (by index), adding an element (in an arbitrary position) and
deleting an element (by its index) all have complexity
<M>log(n)</M> where <M>n</M> is the number of entries in the list.
</Section>
An AVL tree is created using the following function:
<ManSection>
<Func Name="AVLTree" Arg="[ opt ]"/>
<Returns> A new AVL tree object </Returns>
<Description>
This function creates a new AVL tree object. The optional argument
<A>opt</A> is an options record, in which you can bind the
following components:
<P/>
<C>cmpfunc</C> is a three-way comparison function taking two
arguments <A>a</A> and <A>b</A> and returning <M>-1</M> if
<M><A>a</A> < <A>b</A></M>, <M>+1</M> if
<M><A>a</A> > <A>b</A></M> and <M>0</M> if
<M><A>a</A> = <A>b</A></M>. If no function is given then
the generic function <Ref Func="AVLCmp"/> is taken. This three-way
comparison function is stored with the tree and is used for all
comparisons in tree operations.
<C>allocsize</C> is the number of nodes which are allocated for
the tree initially. It can be useful to specify this if you know
that your tree will eventually contain a lot of entries, since
then the tree object does not have to grow that many times.
</Description>
</ManSection>
For every AVL tree a three-way comparison function is needed, usually
you can get away with using the following default one:
<ManSection>
<Func Name="AVLCmp" Arg="a, b"/>
<Returns> -1, 0 or 1 </Returns>
<Description>
This function calls the <K><</K> operation and the <K>=</K>
operation to provide a generic three-way comparison function
to be used in AVL tree operations. See <Ref Func="AVLTree"/>
for a description of the return value. This function is implemented in
the kernel and should be particularly fast.
</Description>
</ManSection>
The following functions are used to access entries by key:
<ManSection>
<Func Name="AVLAdd" Arg="t, key, val"/>
<Returns> <K>true</K> or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function stores the
key <A>key</A> with value <A>value</A> in the tree assuming that the keys
in it are sorted according to the three-way comparison function stored
with the tree. If <A>value</A> is <K>true</K> then no additional memory
is needed. It is an error if there is already a key equal to <A>key</A>
in the tree, in this case the function returns <K>fail</K>. Otherwise it
returns <K>true</K>.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLLookup" Arg="t, key"/>
<Returns> an value or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function looks up the
key <A>key</A> in the tree and returns the value which is associated to it.
If the key is not in the tree, the value <K>fail</K> is returned. This
function assumes that the keys in the tree are sorted according to the
three-way comparison function stored with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLDelete" Arg="t, key"/>
<Returns> an value or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function looks up the
key <A>key</A> in the tree, deletes it and returns the value which was
associated with it. If <A>key</A> is not contained in the tree then
<K>fail</K> is returned. This function assumes that the keys in the
tree are sorted according to the three-way comparison function stored
with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLFindIndex" Arg="t, key"/>
<Returns> an integer or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function looks up the
key <A>key</A> in the tree and returns the index, under which it is stored
in the tree. This index is one-based, that is, it takes values from 1 to
the number of entries in the tree. If <A>key</A> is not contained in the
tree then <K>fail</K> is returned. This function assumes that the keys in
the tree are sorted according to the three-way comparison function
stored with the tree.
</Description>
</ManSection>
The following functions are used to access entries in trees by their
index:
<ManSection>
<Func Name="AVLIndex" Arg="t, index"/>
<Returns> a key or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function returns the
key at index <A>index</A> in the tree, so <A>index</A> must be an integer
in the range 1 to the number of elements in the tree. If the value is out
of these bounds, <K>fail</K> is returned. Note that to use this function it
is not necessary that the keys in the tree are sorted according to the
three-way comparison function stored with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLIndexLookup" Arg="t, index"/>
<Returns> a value or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function returns the
value associated to the key at index <A>index</A> in the tree, so
<A>index</A> must be an integer in the range 1 to the number of elements
in the tree. If the value is out of these bounds, <K>fail</K> is
returned. Note that to use this function it is not necessary that the keys
in the tree are sorted according to the three-way comparison function
stored with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLIndexAdd" Arg="t, key, value, index"/>
<Returns> a key or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function inserts
the key <A>key</A> at index <A>index</A> in the tree and associates the
value <A>value</A> with it. If <A>value</A> is <K>true</K> then no
additional memory is needed to store the value. The index <A>index</A>
must be an integer in the range 1 to <M>n+1</M> where <M>n</M> is the
number of entries in the tree. The new key is inserted before the key which
currently is stored at index <A>index</A>, so calling with <A>index</A>
equal to <M>n+1</M> puts the new key at the end. If <A>index</A> is not in
the corrent range, this function returns <K>fail</K> and the tree remains
unchanged.
<P/>
<B>Caution:</B> With this function it is possible to put a key into the tree
at a position such that the keys in the tree are no longer sorted according
to the three-way comparison function stored with the tree! If you do this,
the functions <Ref Func="AVLAdd"/>, <Ref Func="AVLLookup"/>, <Ref
Func="AVLDelete"/> and <Ref Func="AVLFindIndex"/> will no longer work since
they assume that the keys are sorted!
</Description>
</ManSection>
<ManSection>
<Func Name="AVLIndexDelete" Arg="t, index"/>
<Returns> a key or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function deletes
the key at index <A>index</A> in the tree and returns the value which was
associated with it.
</Description>
</ManSection>
The following functions allow low level access to the AVL tree object:
<ManSection>
<Func Name="AVLFind" Arg="t, key"/>
<Returns> an integer or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function locates the
key <A>key</A> in the tree and returns the position in the positional
object, at which the node which contains the key is stored. This position
will always be divisible by 4. Use the functions <Ref Func="AVLData"/> and
<Ref Func="AVLValue"/> to access the key and value of the node
respectively. The function returns <K>fail</K> if the key is not found in
the tree. This function assumes that the keys in
the tree are sorted according to the three-way comparison function
stored with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLIndexFind" Arg="t, index"/>
<Returns> an integer or <K>fail</K> </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree. This function locates the
index <A>index</A> in the tree and returns the position in the positional
object, at which the node which hash this index is stored. This position
will always be divisible by 4. Use the functions <Ref Func="AVLData"/> and
<Ref Func="AVLValue"/> to access the key and value of the node
respectively. The function returns <K>fail</K> if the key is not found in
the tree. This function does not assume that the keys in
the tree are sorted according to the three-way comparison function
stored with the tree.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLData" Arg="t, pos"/>
<Returns> an key </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree and the second a position
in the positional object corresponding to a node as returned by <Ref
Func="AVLFind"/>. The function returns the key associated with this node.
</Description>
</ManSection>
<ManSection>
<Func Name="AVLValue" Arg="t, pos"/>
<Returns> a value </Returns>
<Description>
The first argument <A>t</A> must be an AVL tree and the second a position
in the positional object corresponding to a node as returned by <Ref
Func="AVLFind"/>. The function returns the value associated with this node.
</Description>
</ManSection>
The following convenience methods for standard list methods are
implemented for AVL tree objects:
<ManSection>
<Meth Name="Display" Arg="t"/>
<Returns> nothing </Returns>
<Description>
This function displays the tree in a user-friendly way. Do not try this
with trees containing many nodes!
</Description>
</ManSection>
<ManSection>
<Meth Name="ELM_LIST" Arg="t, index"/>
<Returns> A key or <K>fail</K> </Returns>
<Description>
This method allows for easy access to the key at index <A>index</A> in the
tree using the square bracket notation <C><A>t</A>[<A>index</A>]</C>.
It does exactly the same as <Ref Func="AVLIndex"/>.
This is to make AVL trees behave more like lists.
</Description>
</ManSection>
<ManSection>
<Meth Name="Position" Arg="t, key"/>
<Returns> an integer or <K>fail</K> </Returns>
<Description>
This method allows to use the <C>Position</C> operation to locate the index
at which the key <A>key</A> is stored in the tree. It does exactly the same
as <Ref Func="AVLFindIndex"/>. This is to make AVL trees behave more like
lists.
</Description>
</ManSection>
<ManSection>
<Meth Name="Add" Arg="t, key [, index]"/>
<Returns> nothing </Returns>
<Description>
This method allows to use the <C>Add</C> operation to add a key (with
associated value <K>true</K>) to the tree at index <A>index</A>. It does
exactly the same as <Ref Func="AVLIndexAdd"/>, so the same warning about
sortedness as there applies! If <A>index</A> is omitted, the key is
added at the end. This is to make AVL trees behave more like lists.
</Description>
</ManSection>
<ManSection>
<Meth Name="Remove" Arg="t, index"/>
<Returns> a key </Returns>
<Description>
This method allows to use the <C>Remove</C> operation to remove a key from
the tree at index <A>index</A>. If <A>index</A> is omitted, the last key in
the tree is remove. This method returns the deleted key or <K>fail</K> if
the tree was empty. This is to make AVL trees behave more like lists.
</Description>
</ManSection>
<ManSection>
<Meth Name="Length" Arg="t"/>
<Returns> a key </Returns>
<Description>
This method returns the number of entries stored in the tree <A>t</A>. This
is to make AVL trees behave more like lists.
</Description>
</ManSection>
<ManSection>
<Meth Name="\in" Arg="key, t"/>
<Returns> <K>true</K> or <K>false</K> </Returns>
<Description>
This method tests whether or not the key <A>key</A> is stored in the AVL
tree <A>t</A>. This is to make AVL trees behave more like lists.
</Description>
</ManSection>
</Section>
<Section>
<Heading>The internal data structures</Heading>
An AVL tree is a positional object in which the first 7 positions are
used for administrative data (see table below) and then from position
8 on follow the nodes of the tree. Each node uses 4 positions such
that all nodes begin at positions divisible by 4. The system allocates
the positional object larger than actually needed such that not every new
node leads to the object being copied. Nodes which become free are
collected in a free list.
The following table contains the information what is stored in each of
the first 7 entries:
<Table Align="l|l">
<Row>
<Item>1</Item>
<Item>last actually used position, is always congruent 3 mod 4
</Item>
</Row>
<Row>
<Item>2</Item>
<Item>position of first node in free list</Item>
</Row>
<Row>
<Item>3</Item>
<Item>number of currently used nodes in the tree</Item>
</Row>
<Row>
<Item>4</Item>
<Item>position of largest allocated position is always congruent 3
mod 4</Item>
</Row>
<Row>
<Item>5</Item>
<Item>three-way comparison function</Item>
</Row>
<Row>
<Item>6</Item>
<Item>position of the top node</Item>
</Row>
<Row>
<Item>7</Item>
<Item>a plain list holding the values stored under the keys</Item>
</Row>
</Table>
The four positions used for a node contain the following information,
recall that each node starts at a position divisible by 4:
<Table Align="l|l">
<Row>
<Item>0 mod 4</Item>
<Item>reference to the key</Item>
</Row>
<Row>
<Item>1 mod 4</Item>
<Item>position of left node or 0 if empty, balance factor (see below)</Item>
</Row>
<Row>
<Item>2 mod 4</Item>
<Item>position of right node or 0 if empty</Item>
</Row>
<Row>
<Item>3 mod 4</Item>
<Item>index: number of nodes in left subtree plus one</Item>
</Row>
</Table>
Since all positions of nodes are divisible by 4, we can use the least
significant two bits of the left node reference for the so called balance
factor. Balance factor 0 (both bits 0) indicates that the depth of the
left subtree is equal to the depth of the right subtree. Balance factor 1
(bits 01) indicates that the depth of the right subtree is one greater than
the depth of the left subtree. Balance factor 2 (or -1 in <Cite
Key="ACP3"/>, here bits 10) indicates that the depth of the left subtree is
one greater than the depth of the right subtree.
<P/>
For freed nodes the position of the next free node in the free list is held
in the 0 mod 4 position and 0 means the end of the free list.
<P/>
Position 7 in the positional object can contain the value <K>fail</K>, in
this case all stored values are <K>true</K>. This is a measure to limit the
memory usage in the case that the only relevant information in the tree is
the key and no values are stored there. This is in particular interesting
if the tree structure is just used as a list implementation.
<P/>
Note that all functions dealing with AVL trees are both implemented on the
&GAP; level and on the kernel level. Both implementations do exactly the
same thing, the kernel version is only much faster and tuned for efficiency
whereas the &GAP; version documents the functionality better and is used
as a fallback if the C-part of the <Package>orb</Package> is not compiled.
</Section>
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