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''' updated 01/06/11 to simplify empty tree definition Module for balanced binary search trees. Usage: from bbst import * defines two classes, bbstree and bbsnode. tree = bbstree() creates a new, empty tree. tnode = tree.lookup(k) searches tree for a node with key k. Returns the bbsnode with key k, or None if k is not found. tnode.key, tnode.value are usable attributes when found. tree.forward() is a generator that returns each tree node in ascending key order. tree.reverse() is a generator that returns each tree node in descending key order. for tnode in tree.forward/reverse(): print(tnode.key, tnode.value) tree.height() returns the height of the tree, 0 for the empty tree. The tree's population is between 2**height and 2**(height+1), that is, if height is 7, the tree has from 128 to 255 nodes. The expected search length is height, the maximum is height+2. tree.insert(k, v=None) creates a node with key=k and value=v and inserts it into tree. If a node with key=k already exists, its value is updated to v. Nothing is returned. tree.instest(k, v=None) looks for a node k in the tree, and inserts it only if it does not exist. Returns the node. Use this for example if you are using the tree to keep a tally of items k: tree.instest(k,0) installs node k only if it didn't exist, but returns it, so tree.instest(k,0).value += 1 keeps a tally. tree.delete(k) deletes the tree node with key k if it exists, and rebalances the tree as necessary. Any data type may be used for a key (not just integers). However, during lookup, insert, and delete, the key k may be compared to any node's key. Thus all keys must support ==, >, and < comparisons and be mutually comparable by Python type rules. The value of a node may be any Python type. To associate a value with a key, present it as v in insert(k,v) or instest(k,v). To retrieve a value, access tree.lookup(k).value or tree.instest(k,v).value. To change the value of an existing key, assign to tree.lookup(k).value or to tree.instest(k,v).value, or simply call tree.insert(k,newvalue) The tree is balanced; adding and deleting keys in any sequence will not produce an unbalanced tree. The algorithms are based on the discussion by Julienne Walker found at http://eternallyconfuzzled.com/tuts/datastructures/jsw_tut_andersson.aspx ''' # # A "node" is an object containing a key, a value, a level, and # links to at most two other nodes. # # Walker's algorithms expect the subtrees of any leaf node to be filled # with pointers to a nil node that acts as a sentinel. It has level=0 # and left and right subs that point to itself. Here we create that one # node in the bbst module namespace. # class bbsnode_0: def __init__(self): self.level = 0 self.key = None self.value = None self.subs = [None,None] # NIL_NODE = bbsnode_0() NIL_NODE.subs = [NIL_NODE,NIL_NODE] # # Here we create an official object of type node, with subtrees initialized # to NIL_NODE: # class bbsnode: def __init__(self,k,v=None): self.level = 1 self.key = k self.value = v self.subs = [NIL_NODE,NIL_NODE] # # End of class bbsnode. # class bbstree: # # A "tree" is an object that contains zero or more "nodes." The tree, # which is the "self" argument to all these member functions, is not # itself a node, but rather contains nodes, or actually, links to the # one node which is currently the root of the tree. That in turn links # to sub-nodes and they to sub-sub-nodes etc. # # Here we initialize a new tree. Its only node is NIL_NODE which has # a level of 0, hence the height of an empty tree is 0. # def __init__(self): self.root = NIL_NODE # the tree is empty self.lastinsert = None # save last-inserted node # # Height # def height(self): return self.root.level # # Look for key k in the tree; if found, return its node, else return None. # tree.lookup() is the "public" method; it handles the special case of # the empty tree. We use __lookup() to recurse into the tree. # def lookup(self, k): if self.root != NIL_NODE: # we are not an empty tree return self.__lookup(self.root, k) else: # this tree has no nodes return None # not found # # Recursive lookup # def __lookup(self, tnode, k): if tnode.key == k: return tnode else: sub = tnode.subs[k > tnode.key] # k is in left or right? if sub != NIL_NODE: # we have a subtree that side return self.__lookup(sub, k) else: return None # # Generator functions, forward() for stepping through the tree in order, and # reverse() backwards. Self is a tree object, and the generator really needs to # operate on the tree nodes, so the real generator is __scan. To scan forward, # one goes first down the left (0) side then down the right (1) side. To scan # backward, the reverse. Since the code is the same except for the subscripts # of tnode.subs[], those numbers are passed in as arguments to __scan(). # The scanning function is not recursive as this would be awkward for a python # generator (yes you can have recursive generators but a generator that calls # itself only returns another generator object!). It emulates recursion with a # stack. Stack size is not an issue with a balanced tree; if we stack 31 # tnodes we have a tree of 2 giga-nodes and stack size is not our main issue! # An empty tree has self.root = NIL_NODE, terminating the generator immediately. # def forward(self): return self.__scan(self.root,0,1) def reverse(self): return self.__scan(self.root,1,0) def __scan(self,tnode,first,second): direction = 0 stack = [NIL_NODE] # when popped, ends the loop while tnode != NIL_NODE: if direction == 0: while tnode.subs[first] != NIL_NODE: stack.append(tnode) tnode = tnode.subs[first] yield tnode if tnode.subs[second] != NIL_NODE: tnode = tnode.subs[second] direction = 0 else: tnode = stack.pop() direction = 1 # # The following are sub-functions of __insert, used to balance the tree. # One basic operation is the skew: "A skew removes left horizontal links by # rotating right at the parent." In the example, a/b/c/d/e are key values # and the digits are level values. Skew(node-with-d): # # d,2 b,2 # / \ / \ # b,2 e,1 --> a,1 d,2 # / \ / \ # a,1 c,1 c,1 e,1 # # The following is Walker's non-recursive skew. Although it gets a self # arg by virtue of being a class attribute of tree, "self" (the tree object) # is ignored. The function operates only on the argument (a node) t. # def __skew(self, t): if t.level != 0: if t.subs[0].level == t.level: tmp = t.subs[0] t.subs[0] = tmp.subs[1] tmp.subs[1] = t t = tmp return t # # The second basic operation is the split: "A split removes consecutive # horizontal links by rotating left and increasing the level of the parent." # Continuing the example following the above skew, split(node-with-b) # # b,2 d,3 # / \ / \ # a,1 d,2 --> b,2 e,2 # / \ / \ # c,1 e,2 a,1 c,1 # # The following is Walker's non-recursive split. Again self (a tree) is ignored. # def __split(self, t): if t.level != 0: if t.level == t.subs[1].subs[1].level: tmp = t.subs[1] t.subs[1] = tmp.subs[0] tmp.subs[0] = t t = tmp t.level += 1 return t # # With that, we can insert key k into the tree (or simply find it, if # it already exists). Nothing is returned. As with tree.lookup(), # tree.insert() is the public method and handles the special case of # the empty tree; then __insert() is called to do the recursion. # def insert(self, k, v=None): if self.root != NIL_NODE: # we are not an empty tree self.root = self.__split( self.__skew( self.__insert(self.root, k, v) ) ) else: # empty tree getting its first node self.root = bbsnode(k,v) self.lastinsert = self.root # # Recursive insertion # def __insert(self, tnode, k, v): if k == tnode.key: # key exists, update value tnode.value = v else: # it goes in the left or the right side = k > tnode.key sub = tnode.subs[ side ] if sub != NIL_NODE: # there is a subtree that side tnode.subs[side] = self.__split( self.__skew( self.__insert(sub, k, v) ) ) else: # no subtree on that side, make new leaf self.lastinsert = bbsnode(k,v) tnode.subs[side] = self.lastinsert return tnode # # Because of balancing, it is impractical for insert() to return the inserted # node: __insert() may return a different node as a result of balancing, and # insert() doesn't know if self.lastinsert was updated or not. This makes it # awkward to use the tree e.g. to keep a running tally of input symbols, where # each symbol is stored as a key and the tally is the value. Here we provide # instest() for this purpose: it can know that self.lastinsert was updated # because of the failure of the preceding lookup. # def instest(self, k, v): tnode = self.lookup(k) if tnode == None: self.insert(k, v) tnode = self.lastinsert return tnode # # Deletion, still following the lead of Julienne Walker although, you know, # rewriting pretty freely. Again the public method is part of a tree and # handles the special case of the empty tree. __delete() does the recursion. # def delete(self, k): if self.root != NIL_NODE: # we are not an empty tree self.root = self.__delete(self.root, k) # # Recursive deletion with nonrecursive skew and split. This is the second # version Walker gives, described as "simple, but unconventional enough # to be confusing." Yup. See her discussion for why it is always enough # to do three skews and two splits. Also recall that the skew and split # routines are conditional and do nothing but a test when not needed. # def __delete(self, tnode, k): if tnode != NIL_NODE: if k == tnode.key: if (tnode.subs[0] != NIL_NODE) & (tnode.subs[1] != NIL_NODE) : heir = tnode.subs[0] while heir.subs[1] != NIL_NODE: heir = heir.subs[1] tnode.key = heir.key tnode.value = heir.value tnode.subs[0] = self.__delete(tnode.subs[0], tnode.key) else: tnode = tnode.subs[tnode.subs[0]==NIL_NODE] else: side = tnode.key < k # side==0 on key > OR EQUAL tnode.subs[side] = self.__delete(tnode.subs[side], k) lv = tnode.level-1 if (tnode.subs[0].level < lv) | (tnode.subs[1].level < lv) : tnode.level = lv if tnode.subs[1].level > lv: tnode.subs[1].level = lv tnode = self.__skew(tnode) tnode.subs[1] = self.__skew(tnode.subs[1]) tnode.subs[1].subs[1] = self.__skew(tnode.subs[1].subs[1]) tnode = self.__split(tnode) tnode.subs[1] = self.__split(tnode.subs[1]) return tnode