This document explains how to use the avl_tree
template. Adelson-Velskii and
Landis Balanced Binary Search Trees (or AVL Trees) are described in many good
textbooks on fundamental data structures. The best web page I’ve been able to
find on the topic is “A Visual
Basic AVL Tree Container Class”.
This document, as well as the source code it describes, is in the public domain.
To avoid possible confusion about the terms that I use in this document (and in the source comments), here is a summary description of AVL Trees. An AVL Tree is a set of nodes (or elements). Each node is associated with a unique key value. The key values can be ordered from least to greatest. Each node in the tree may (or may not) have a less child node, and it may (or may not) have a greater child node. If node A is a child of node B, then B is the parent of A. If A is the less child of B, A’s key must be less than B’s key. Similarly, if A is the greater child of B, A’s key must be greater than B’s key. All nodes in a tree have exactly one parent, except for the root node, which has no parent. Node A is a descendant of node C if C is A’s parent, or if A’s parent is a descendant of C. If a node is not the root of the entire tree, it is the root of a subtree consisting of the node and all its descendants. The lesser subtree of a node is the subtree whose root is the less child of the node. The greater subtree of a node is the subtree whose root is the greater child of the node. The depth of a node is one more than the depth of its parent. The depth of the root node is 1. The depth of a tree is the maximum node depth. The balance factor of a node is the depth of its greater subtree minus the depth of its lesser subtree, with non-existent subtrees being considered to have a depth of 0. In an AVL tree, the balance factor of any node can only be -1, 0 or 1.
There are several open-source C and C++ implementations of AVL Trees
available (see “Hot Links”, then “Data Structures” at
C/C++ User’s Group Page ). But as far as
I know, this is the only one that manipulates the nodes of the tree using
abstract “handles” instead of concrete pointers. If all the nodes are in a
single array, you can use node indexes as handles instead of node pointers. This
approach makes it possible to compress the size of the nodes if memory is tight.
Index handles can make tree persistence as simple as writing the node array out
with a single disk write, and reading it back in with a single disk read. The
template also allows for a tree to be in secondary storage, with nodes being
“paged” in and out of memory.
To achieve the desired level of abstraction, the avl_tree
template uses lots of
short inline functions. Because of this, function inlining can significantly
improve performance when using the template. If the test suite (test_avl.cpp) is
compiled with GNU GCC using level 1 optimization (-O option), it executes twice
as fast as when the test suite is compiled without optimization (the default).
The template code makes no use of recursion. The implementation is
stack-friendly in general, except perhaps for the iter
class. Instances of
iter
contain an array of handles whose dimension is the maximum tree depth minus one.
Since key comparisons can potentially be complex, the code avoids repeated
comparisons of the same pair of node key values. To avoid clutter, default
destructor functions are not documented.
All of this code compiles with a contemporary version of GNU GCC, and with Visual C++ .NET.
To help describe the constraints on template class/typename parameters, or on member types of template class parameters, I like to use reference classes. This doesn’t necessary mean that the type being constrained has to use the reference class as its definition. It is only necessary that every possible usage of the reference class or one of its instances is also a possible usage of the constrained type or one of its instances. When an identifier with the prefix ANY_ is used, this means that all occurrences of that identifier should be substituted with the same type (or with types that implicitly convert to the substituted type). Take for example the function template:
template <class A> void foo(A &a) { a.x(a.y()); }
The reference class for the parameter A would be:
class A { public: void x(ANY_px p); ANY_px y(void); };
The following class could be passed as the class A parameter to the template:
struct someA { public: static double x(int aintp); signed char y(bool f = true) const; };
Since the return type of x()
is void
in the reference class, it can return
any type (or be void) in the actual parameter class. y()
can return
signed char
because signed char
implicitly converts to int
. Member functions can be made
static
or const
because these make the usage of a function more, not less,
flexible.
The avl_tree
template is in the abstract_container
namespace. The AVL Tree
header file also defines this enumerated type:
enum search_type { EQUAL = 1, LESS = 2, GREATER = 4, LESS_EQUAL = EQUAL | LESS, GREATER_EQUAL = EQUAL | GREATER };
in the abstract_container
namespace.
The avl_tree
template begins with:
template <class abstractor, unsigned max_depth = 32> class avl_tree . . .
All members of the reference class are public.
Each node has to be associated with a node handle, which is a unique value of the handle type. Here is the reference class for handle:
class handle { public: // No default value for handles is assumed by the template. handle(void); handle(handle &h); void operator = (handle &h); bool operator == (handle &h); };
Each node has to be associated with a key, which is a unique value of the key type. The difference between a key and a handle is that a node can be conveniently “looked up” using its handle, but it can’t be conveniently looked up using its key. In fact, the whole point of this template is to make it convenient to look up a node given its key. Here is the reference class for key:
classkey { public: //Only have to copy it key(key&k); };
The type size is char
, short
, int
or
long
, signed
or unsigned
. It must be
large enough the hold the maximum possible number of nodes in the tree.
handle get_less(handle h, bool access = true); handle get_greater(handle h, bool access = true);
Return the handle of the less/greater child of the node whose handle is h. If access is true, and the child node is in secondary storage, it has to be read into memory. If access is false, the child node does not have to be read into memory. Ignore the access parameter if your instantiation makes no use of secondary storage.
void set_less(handle h, handle lh); void set_greater(handle h, handle gh);
Given the handle h of a node, set the handle of the less/greater child of the node.
int get_balance_factor(handle h);
Return the balance factor of the node whose handle is h.
void set_balance_factor(handle h, int bf);
Set the balance factor of the node whose handle is h. The only possible balance factor values are -1, 0 and 1.
int compare_key_node(key k, handle h);
Compares a key with the key of a node. Returns a negative value if the key is less than the node’s key. Returns zero if the key is the same as the node’s key. Returns a positive value if the key is greater than the node’s key.
int compare_node_node(handle h1, handle h2);
Compares the keys of two nodes. Returns a negative value if the first node’s key is less than the second node’s key. Returns zero if the first node’s key is the same as the second node’s key. Returns a positive value if the first node’s key is greater than the second node’s key.
handle null(void);
Always returns the same, invalid handle value, which is called the null value.
bool read_error(void);
Returns true if there was an error reading secondary storage. If your instantiation of the template makes no use of secondary storage, use this definition:
bool read_error(void) { return(false); }
abstractor(void);
This is the maximum tree depth for an instance of the instantiated class. You almost certainly want to choose the maximum depth based on the maximum number of nodes that could possibly be in the tree instance at any given time. To do this, let the maximum depth be M such that:
where MN(d) means the minimum number of nodes in an AVL Tree of depth d. Here is a table of MN(d) values for d from 2 to 45.
D | MN(d) |
---|---|
2 | 2 |
3 | 4 |
4 | 7 |
5 | 12 |
6 | 20 |
7 | 33 |
8 | 54 |
9 | 88 |
10 | 143 |
11 | 232 |
12 | 376 |
13 | 609 |
14 | 986 |
15 | 1,596 |
16 | 2,583 |
17 | 4,180 |
18 | 6,764 |
19 | 10,945 |
20 | 17,710 |
21 | 28,656 |
22 | 46,367 |
23 | 75,024 |
24 | 121,392 |
25 | 196,417 |
26 | 317,810 |
27 | 514,228 |
28 | 832,039 |
29 | 1,346,268 |
30 | 2,178,308 |
31 | 3,524,577 |
32 | 5,702,886 |
33 | 9,227,464 |
34 | 14,930,351 |
35 | 24,157,816 |
36 | 39,088,168 |
37 | 63,245,985 |
38 | 102,334,154 |
39 | 165,580,140 |
40 | 267,914,295 |
41 | 433,494,436 |
42 | 701,408,732 |
43 | 1,134,903,169 |
44 | 1,836,311,902 |
45 | 2,971,215,072 |
If, in a particular instantiation, the maximum number of nodes in a tree instance is 1,000,000, the maximum depth should be 28. You pick 28 because MN(28) is 832,039, which is less than or equal to 1,000,000, and MN(29) is 1,346,268, which is strictly greater than 1,000,000.
If you insert a node that would cause the tree to grow to a depth greater than the maximum you gave, the results are undefined.
Each increase of 1 in the value of max_depth
increases the size of an
instance of the iter
class by sizeof(handle)
. The only other use of
max_depth
is
as the size of bit arrays used at various places in the code. Generally, the
number of bytes in a bit array is the size rounded up to a multiple of the
number of bits in an int
, and divided by the number of bits in a byte. All this
is a roundabout way of saying that, if you don’t use iter
instances, you can
guiltlessly add a big safety margin to the value of max_depth
.
Same as handle type member of the abstractor parameter class.
Same as key type member of the abstractor parameter class.
Same as size type member of the abstractor parameter class.
handle insert(handle h);
Insert the node with the given handle into the tree. The node must be
associated with a key value. The initial values of the node’s less/greater child
handles and its balance factor are don’t-cares. If successful, this function
returns the handle of the inserted node. If the node to insert has the same key
value as a node that’s already in the tree, the insertion is not performed, and
the handle of the node already in the tree is returned. Returns the null value
if there is an error reading secondary storage. Calling this function
invalidates all currently-existing instances of the iter
class (that are
iterating over this tree).
handle search(key k, search_type st = EQUAL);
Searches for a particular node in the tree, returning its handle if the node is found, and the null value if the node is not found. The node to search for depends on the value of the st parameter.
Value of st | Node to search for |
---|---|
EQUAL | Node whose key is equal to the key k. |
LESS | Node whose key is the maximum of the keys of all the nodes with keys less than the key k. |
GREATER | Node whose key is the minimum of the keys of all the nodes with keys greater than the key k. |
LESS_EQUAL | Node whose key is the maximum of the keys of all the nodes with keys less than or equal to the key k. |
GREATER_EQUAL | Node whose key is the minimum of the keys of all the nodes with keys greater than or equal to the key k. |
handle search_least(void);
Returns the handle of the node whose key is the minimum of the keys of all the nodes in the tree. Returns the null value if the tree is empty or an error occurs reading from secondary storage.
handle search_greatest(void);
Returns the handle of the node whose key is the maximum of the keys of all the nodes in the tree. Returns the null value if the tree is empty or an error occurs reading from secondary storage.
handle remove(key k);
Removes the node with the given k from the tree. Returns the handle of the
node removed. Returns the null value if there is no node in the tree with the
given key, or an error occurs reading from secondary storage. Calling this
function invalidates all currently-existing instances of the iter
class (that
are iterating over this tree).
void purge(void);
Removes all nodes from the tree, making it empty.
bool is_empty(void);
Returns true if the tree is empty.
void read_error(void);
Returns true if an error occurred while reading a node of the tree from secondary storage. When a read error has occurred, the tree is in an undefined state.
avl_tree(void);
Initializes the tree to the empty state.
template<typename fwd_iter>
bool build(fwd_iter p, size num_nodes);
Builds a tree from an sequence of nodes that are sorted in ascending order by their
key values. The number of nodes in the sequence is given by num_nodes
. p is a
forward iterator that initially refers to the first node in the sequence. Here
is the reference class for the fwd_iter
:
class fwd_iter { public: fwd_iter(fwd_iter &); handle operator * (void); void operator ++ (int); };
Any nodes in the tree (prior to calling this function) are purged. The iterator will be incremented one last time when it refers to the last node in the sequence. build() returns false if a read error occurs while trying to build the tree. The time complexity of this function is O(n x log n), but it is more efficient than inserting the nodes in the sequence one at a time, and the resulting tree will generally have better balance.
If the abstractor class has a copy constructor and assignment operator, the
avl_tree
instantiation will have a (default) copy constructor and assignment
operator.
Instances of this member class are bi-directional iterators over the
ascendingly sorted (by key) sequence of nodes in a tree. The subsections of this
section describe the public members of iter
iter(void);
Initializes the iterator to the null state
void start_iter(avl_tree &tree, key k, search_type st = EQUAL);
Causes the iterator to refer to a particular node in the tree that is specified as the first parameter. If the particular node cannot be found in the tree, or if a read error occurs, the iterator is put into the null state. The particular node to refer to is determined by the st parameter.
Value of st | Node to search for |
---|---|
EQUAL | Node whose key is equal to the key k. |
LESS | Node whose key is the maximum of the keys of all the nodes with keys less than the key k. |
GREATER | Node whose key is the minimum of the keys of all the nodes with keys greater than the key k. |
LESS_EQUAL | Node whose key is the maximum of the keys of all the nodes with keys less than or equal to the key k. |
GREATER_EQUAL | Node whose key is the minimum of the keys of all the nodes with keys greater than or equal to the key k. |
void start_iter_least(avl_tree &tree);
Cause the iterator to refer to the node with the minimum key in the given tree. Puts the iterator into the null state if the tree is empty or a read error occurs.
void start_iter_greatest(avl_tree &tree);
Cause the iterator to refer to the node with the maximum key in the given tree. Puts the iterator into the null state if the tree is empty or a read error occurs.
handle operator * (void);
Returns the handle of the node that the iterator refers to. Returns the null value if the iterator is in the null state
void operator ++ (void); void operator ++ (int);
Causes the iterator to refer to the node whose key is the next highest after the key of the node the iterator currently refers to. Puts the iterator into the null state if the key of the node currently referred to is the maximum of the keys of all the nodes in the tree, or if a read error occurs. Has no effect if the iterator is already in the null state.
void operator -- (void); void operator -- (int);
Causes the iterator to refer to the node whose key is the next lowest after the key of the node the iterator currently refers to. Puts the iterator into the null state if the key of the node currently referred to is the minimum of the keys of all the nodes in the tree, or if a read error occurs. Has no effect if the iterator is already in the null state.
Returns true if a read error occurred.
These member functions exist and can be safely used
abstractor abs;
handle root;
Contains the handle of the root node of the AVL Tree. Contains null value if the tree is empty.
The other protected members are most easily understood by reading the source code.
12 Sep 2002 - fixed some problems in the code for handling errors if tree nodes are in secondary strorage.