
		     AVL Tree Objects for Python

This package implements a dual-personality builtin object for Python,
using AVL trees.  AVL trees (named after the inventors,
Adel'son-Vel'skii and Landis) are balanced binary search trees, where
the difference in height between any one node's left and right
branches is kept less than or equal to one.  Whenever a key is
inserted into a position in a way that would upset this balance, a
'rotation' is performed.  These rotations happen less than half the
time when inserting.  Deletion is similar, but may result in up to
log(n) rotations.

These trees can be used to implement dictionaries with fast lookup
and deletion of any single item, but can also act like sequential
lists. (In fact, the printed representation is just like that of a
list).

Each of the following operations can be done in O(log n) time:
i)	Find an item having a given key
ii)	Find the kth item, given k
iii)	Insert an item at a specified place
iv)	Delete a specified item

Other operations are also possible in comparable time, including
concatenation and splitting.

Objects are inserted and deleted based on their order with respect to
the builtin Python compare function, PyObject_Compare().
The __getslice__ method is provided, and returns a list rather than
a new tree object.

Each node uses four pointers (key, left, right, parent) and a long
(rank and balance), so the size overhead is probably comparable to
that of Python's dictionary object [which uses two pointers and a long,
and resizes whenever more than half full].  Another difference is that
the dictionary object holds (key, value) pairs, but avl objects hold
only single object (the key).

The algorithms are taken directly from Knuth's "Art of Computer
Programming, Volume 2: Searching and Sorting", and were prototyped
using Python (avl_tree.py).  They were then translated directly into
C (avl.c, avl.h), and a Python module wrapper was written (AVLmodule.c)

Several of the newer methods rely on an interesting order analysis
for the get_successor() and get_predecessor() functions in avl.c.
If you're an analysis of algorithms guru, please take a look at the
comments in avl_tree.py and check my work! 8^)

The avl 'library' (avl.c, avl.h) contains nothing Python-specific, so
it should be useful for other projects. (as if you'd want to after
becoming a Python convert!).  However it was necessary to implement
a few methods (getslice, concat) directly in AVLmodule.c.

===========================================================================
Building:

Unix:
First, cd $(AVL_LIB), then 'make libavl.a'

Then copy AVLmodule.c into your Modules directory.
add a few lines like this to your Python/Modules/Setup file.
---------------------------------------------------------------------------
# avl module
AVL_LIB=/usr/src/other-homes/rushing/python/avl
avl AVLmodule.o -DDEBUG_AVL -I$(AVL_LIB) -L$(AVL_LIB) -lavl
---------------------------------------------------------------------------

Win32:
First, cd $(AVL_LIB), then 'nmake -f makefile_nt.msc avl.lib'

add something like this to Python/Modules/makefile_nt.msc
(or a separate makefile like in Demo/dl.nt)
---------------------------------------------------------------------------
# avl module
AVL_LIB = \\squirl\rushing\python\avl

AVLmodule.obj: AVLmodule.c $(AVL_LIB)\avl.h
	$(cc) -DDEBUG_AVL -I$(AVL_LIB) $(cflags) $(ccustom) $(cdebug) $(cinclude) $(cpch) $(pythonopts) /DUSE_DL_IMPORT /Tp $*.c

avl.$(pythondllext): $(AVL_LIB)\avl.c $(AVL_LIB)\avl.h AVLmodule.obj
	link -dll -out:$*.$(pythondllext) AVLmodule.obj -implib:AVLmodule.lib -entry:_DllMainCRTStartup$(DLLENTRY) $(ldebug) /EXPORT:initavl $(lcustom) $(AVL_LIB)\avl.lib $(python_defs_lib)
---------------------------------------------------------------------------

Usage:

>>> t = avl.newavl()
>>> t
[]
>>> t.insert(50)
>>> t
[50]
>>> t.insert(45)
>>> t
[45, 50]
>>> t.remove(50)
>>> t
[45]

>>> t = avl.newavl()
>>> for x in range(20): t.insert(random.randint(0, 100))
...
>>> t
[3, 8, 12, 15, 15, 29, 32, 42, 44, 48, 50, 53, 55, 57, 69, 74, 75, 76, 79, 81, 88]
# if you've compiled in the debugging code (-DDEBUG_AVL)
#
# [Note: print_internal_structure() prints directly to stdout,
#   so it doesn't work in pythonwin, but will work with the
#   console-mode python.exe]

>>> t.print_internal_structure()
                                       +-[- 88 001]
                          +-[/ 81 001]-|
             +-[\ 79 008]-|
             |            |                         +-[- 76 001]
             |            |            +-[- 75 002]-|
             |            |            |            +-[- 74 001]
             |            +-[- 69 004]-|
             |                         |            +-[- 57 001]
             |                         +-[- 55 002]-|
             |                                      +-[- 53 001]
+-[- 50 011]-|
             |                                      +-[- 48 001]
             |                         +-[/ 44 001]-|
             |            +-[/ 42 002]-|
             |            |            +-[- 32 001]
             +-[- 29 006]-|
                          |                        +-[- 15 001]
                          |           +-[- 15 002]-|
                          |           |            +-[- 12 001]
                          +-[/ 8 002]-|
                                      +-[- 3 001]

# [now isn't that pretty?.]
# FYI, the leftmost char in each node represents the 'balance factor'
# for that node, whether the tree is heavy on the left ('\'), right
# ('/'), or not at all ('-').  the middle item is 'repr(key)', and the
# final number is the 'rank' of the node, used internally to quickly
# locate a node of a certain index - it's the number of nodes in the
# left subtree, plus one.

# create a new tree from a list.
# Note: this sorts the list as a side-effect, if you don't
# want that, use a copy (i.e., 'l = avl.from_list(my_list[:])')

>>> t = avl.newavl (range(30))
>>> len(t)
30
>>>

# You can address individual elements in the sequence:
# Note: you cannot assign to the items in the tree, as
# this might change the relative order of the keys.

>>> t[25]
25
>>> t[-1]
30

# You can take slices of the list. The default slice
# operation returns a new avl tree, not a list.
# [If you want a list, use the slice_as_list() method]

>>> random_tree[3:8]
[45, 87, 29, 73, 12]
>>> type(_)
<type 'AVL tree'>

# You can insert any python object:

>>> t = avl.newavl(range(10))
>>> t.insert('Hey, where will this go?')
>>> t
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 'Hey, where will this go?']
>>> 

# is a certain element in the tree?

>>> t.lookup (5)
5
>>> t.lookup (34)
Traceback (innermost last):
  File "<stdin>", line 1, in ?
KeyError: 34
>>> t.has_key (5)
1
>>> t.has_key (34)
0
>>>

lookup is a little tricky, because it returns the first object in the
tree that compares equal to the key you give.  For builtin objects,
the builtin compare is used to order the tree, but by using your own
objects with your own __cmp__ functions, you can do all sorts of
interesting things.

*** -------------------- Note --------------------***
It is important that you not directly change an object stored in an
avl tree in a way that would change its ordered position.  If you need
to do this, first remove() the item, change it, then reinsert it!
This is still very fast, as both operations are O(log n)

The results of such an illegal modification are unspecified, and are
likely to crash the program.  The only way to completely avoid this
problem is to use only immutable types.  This restriction seemed to
severe to enforce.  So be careful out there!
*** -------------------- Note --------------------***

# to copy a tree:
>>> t2 = avl.newavl(t)

# to concatenate two trees
>>> t3 = t1 + t2
# Note: this is the equivalent of creating a
#  copy of t1, and then inserting all the elements
#  of t2 in turn.

# the 'repeat' operation (t1 * 5) is currently undefined
# drop me a line if you need it. (and describe what you think
# it should do)

# ===========================================================================
# release 2.0 (february 1997)
# ===========================================================================
Major changes to the C library have been applied for this version.
The comparison function is now a member of the avl_tree structure,
instead of being passed in with all of the API functions.  An extra
'compare_arg' pointer is included to allow attaching 'state' to the
compare function - this is a bit ugly, but was necessary to cleanly
interface to python.

Several new functions are visible from python, including two from
David Ascher <david_ascher@brown.edu> (at_least() and at_most()).

# 't.span()' returns a pair of indices into the three that 'span' the
# given key or key pair.  For example:

>>> t
[1,2,3,4,5]
>>> t.span(4)
(3,4)
>>> t[3:4]
[4]
>>> t = avl.newavl(map (lambda x: random.randint(0,1000), range(10)))
>>> t
[15, 34, 371, 536, 659, 691, 714, 754, 847, 936]
>>> t.span(200,500)
(2, 3)
>>> t[2:3]
[371]
>>> t.span(300,700)
(2, 6)
>>> t[2:6]
[371, 536, 659, 691]
>>> 

# at_least() returns the first object comparing greater to or equal to the <key> argument.

>>> t.at_least (400)
536

# at_most() returns the first object comparing less than or equal to the <key> argument.
>>> t.at_most (800)
754
