mirror of
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✌️Build a red black tree successfully
This commit is contained in:
mbinary
parent
64c62a0ff8
commit
52c93f95ce
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@ -22,7 +22,7 @@ class node:
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def __lt__(self,nd):
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return self.val < nd.val
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def __eq__(self,nd):
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return nd is not None and self.val==nd.val
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return nd is not None and self.val == nd.val
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def setChild(self,nd,isLeft = True):
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if isLeft: self.left = nd
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else: self.right = nd
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@ -37,29 +37,11 @@ class node:
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def __repr__(self):
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return f'node({self.val},isBlack={self.isBlack})'
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class redBlackTree:
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def __init__(self):
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def __init__(self,unique=False):
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'''if unique is True, all node'vals are unique, else there may be equal vals'''
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self.root = None
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def getParent(self,val):
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if isinstance(val,node):val = val.val
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if self.root.val == val:return None
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nd = self.root
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while nd:
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if nd.val>val and nd.left is not None:
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if nd.left.val == val: return nd
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else: nd = nd.left
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elif nd.val<val and nd.right is not None:
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if nd.right.val == val: return nd
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else: nd = nd.right
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def find(self,val):
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if isinstance(val,node):val = val.val
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nd = self.root
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while nd:
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if nd.val ==val:
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return nd
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elif nd.val>val:
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nd = nd.left
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else:
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nd = nd.right
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self.unique = unique
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@staticmethod
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def checkBlack(nd):
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return nd is None or nd.isBlack
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@ -69,39 +51,85 @@ class redBlackTree:
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if isBlack is None or isBlack:
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nd.isBlack = True
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else:nd.isBlack = False
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def insert(self,val):
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if isinstance(val,node):val = val.val
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def _insert(root,nd):
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'''return parent'''
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while root:
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if root == nd:return None
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elif root>nd:
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if root.left :
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root=root.left
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else:
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root.left = nd
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return root
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else:
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if root.right:
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root = root.right
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else:
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root.right = nd
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return root
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# insert part
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nd = node(val)
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def sort(self,reverse = False):
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''' return a generator of sorted data'''
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def inOrder(root):
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if root is None:return
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if reverse:
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yield from inOrder(root.right)
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else:
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yield from inOrder(root.left)
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yield root
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if reverse:
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yield from inOrder(root.left)
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else:
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yield from inOrder(root.right)
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yield from inOrder(self.root)
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def getParent(self,chd):
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'''note that use is to find real node when different nodes have euqiv val'''
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if self.root is chd:return None
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nd = self.root
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while nd:
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if nd>chd and nd.left is not None:
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if nd.left is chd: return nd
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else: nd = nd.left
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elif nd<chd and nd.right is not None:
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if nd.right is chd: return nd
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else: nd = nd.right
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def find(self,val):
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nd = self.root
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while nd:
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if nd.val ==val:
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return nd
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elif nd.val>val:
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nd = nd.left
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else:
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nd = nd.right
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def getSuccessor(self,nd):
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if nd:
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if nd.right:
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nd = nd.right
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while nd.left:
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nd = nd.left
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return nd
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else:return self.getParent(nd)
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def transferParent(self,origin,new):
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if origin is self.root:
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self.root = new
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else:
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prt = self.getParent(origin)
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prt.setChild(new, prt.left is origin)
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def insert(self,nd):
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if not isinstance(nd,node):
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nd = node(nd)
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elif nd.isBlack: nd.isBlack = False
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if self.root is None:
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self.root = nd
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self.setBlack(self.root,True)
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return
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parent = _insert(self.root,nd)
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if parent is None: return
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if not parent.isBlack: self.fixUpInsert(parent,nd)
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self.root.isBlack = True
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else:
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parent = self.root
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while parent:
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if parent == nd : return None
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if parent>nd:
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if parent.left :
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parent = parent.left
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else:
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parent.left = nd
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break
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else:
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if parent.right:
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parent = parent.right
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else:
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parent.right = nd
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break
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self.fixUpInsert(parent,nd)
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def fixUpInsert(self,parent,nd):
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''' adjust color and level, there are two red nodes: the new one and its parent'''
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while not self.checkBlack(parent):
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grand = self.getParent(parent)
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isLeftPrt = grand.left == parent
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isLeftPrt = grand.left is parent
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uncle = grand.getChild(not isLeftPrt)
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if not self.checkBlack(uncle):
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# case 1: new node's uncle is red
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@ -112,7 +140,7 @@ class redBlackTree:
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parent = self.getParent(nd)
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else:
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# case 2: new node's uncle is black(including nil leaf)
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isLeftNode = parent.left==nd
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isLeftNode = parent.left is nd
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if isLeftNode ^ isLeftPrt:
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# case 2.1 the new node is inserted in left-right or right-left form
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# grand grand
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@ -130,105 +158,92 @@ class redBlackTree:
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parent.setChild(grand,not isLeftPrt)
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self.setBlack(grand, False)
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self.setBlack(parent, True)
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self.transferParent(grand,parent)
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self.setBlack(self.root,True)
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def sort(self,reverse = False):
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''' return a generator of sorted data'''
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def inOrder(root):
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if root is None:return
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if reverse:
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yield from inOrder(root.right)
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else:
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yield from inOrder(root.left)
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yield root
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if reverse:
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yield from inOrder(root.left)
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else:
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yield from inOrder(root.right)
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yield from inOrder(self.root)
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def getSuccessor(self,val):
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if isinstance(val,node):val = val.val
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def _inOrder(root):
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if root is None:return
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if root.val>= val:yield from _inOrder(root.left)
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yield root
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yield from _inOrder(root.right)
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gen = _inOrder(self.root)
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for i in gen:
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if i.val==val:
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try: return gen.__next__()
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except:return None
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def delete(self,val):
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# delete node in a binary search tree
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if isinstance(val,node):val = val.val
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nd = self.find(val)
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def copyNode(self,src,des):
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'''when deleting a node which has two kids,
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copy its succesor's data to his position
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data exclude left, right , isBlack
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'''
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des.val = src.val
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def delete(self,nd):
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'''delete node in a binary search tree'''
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if not isinstance(nd,node):
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nd = self.find(nd)
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if nd is None: return
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y = None
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if nd.left and nd.right:
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y= self.getSuccessor(val)
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y= self.getSuccessor(nd)
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else:
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y = nd
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py = self.getParent(y.val)
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py = self.getParent(y)
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x = y.left if y.left else y.right
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if py is None:
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self.root = x
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elif y==py.left:
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elif y is py.left:
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py.left = x
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else:
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py.right = x
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if y != nd:
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nd.val = y.val
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self.copyNode(y,nd)
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# adjust colors and rotate
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if self.checkBlack(y): self.fixUpDel(py,x)
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def fixUpDel(self,prt,chd):
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if self.root == chd or not self.checkBlack(chd):
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self.setBlack(chd, True)
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return
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isLeft = prt.left == chd
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brother = prt.getChild(not isLeft)
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if self.checkBlack(brother):
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# case 1: brother is black
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lb = self.checkBlack(brother.left)
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rb = self.checkBlack(brother.right)
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if lb and rb:
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# case 1.1: brother is black and two kids are black
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''' adjust colors and rotate '''
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while self.root != chd and self.checkBlack(chd):
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isLeft = prt.left is chd
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brother = prt.getChild(not isLeft)
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# brother is black
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lb = self.checkBlack(brother.getChild(isLeft))
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rb = self.checkBlack(brother.getChild(not isLeft))
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if not self.checkBlack(brother):
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# case 1: brother is red. converted to case 2,3,4
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# prt (isLeft) rotate
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prt.setChild(brother.getChild(isLeft), not isLeft)
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brother.setChild(prt, isLeft)
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self.setBlack(prt,False)
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self.setBlack(brother,True)
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self.transferParent(prt,brother)
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elif lb and rb:
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# case 2: brother is black and two kids are black.
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# conveted to the begin case
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self.setBlack(brother,False)
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chd = prt
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elif lb or rb:
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# case 1.2: brother is black and two kids's colors differ
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if self.checkBlack(brother.getChild(not isLeft)):
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prt = self.getParent(chd)
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else:
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if rb:
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# case 3: brother is black and left kid is red and right child is black
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# uncle's son is nephew, and niece for uncle's daughter
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nephew = brother.getChild(isLeft),
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print(nephew)
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nephew = brother.getChild(isLeft)
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self.setBlack(nephew,True)
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self.setBlack(brother,False)
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# brother right rotate
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# brother (not isLeft) rotate
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prt.setChild(nephew,not isLeft)
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brother.setChild(nephew.getChild(not isLeft),isLeft)
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nephew.setChild(brother, not isLeft)
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brother = prt.right
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brother = nephew
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# case 1.3: brother is black and two kids are red
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# case 4: brother is black and right child is red
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brother.isBlack = prt.isBlack
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self.setBlack(prt,True)
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self.setBlack(brother.right,True)
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self.setBlack(brother.getChild(not isLeft),True)
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# prt left rotate
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prt.setChild(brother.getChild(isLeft),not isLeft)
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brother.setChild(prt,isLeft)
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self.transferParent(prt,brother)
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chd = self.root
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else:
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# case 2: brother is red
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prt.setChild(brother.getChild(isLeft), not isLeft)
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brother.setChild(prt, isLeft)
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self.setBlack(prt,False)
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self.setBlack(brother,True)
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self.setBlack(chd,True)
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def display(self):
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def getHeight(nd):
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if nd is None:return 0
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@ -238,25 +253,39 @@ class redBlackTree:
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lst = deque([root])
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level = []
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h = getHeight(root)
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lv = 0
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ct = 0
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while lv<=h:
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ct = lv = 0
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while 1:
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ct+=1
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nd = lst.popleft()
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if ct >= 2**lv:
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lv+=1
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if lv>h:break
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level.append([])
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level[-1].append(str(nd))
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if nd is not None:
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lst.append(nd.left)
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lst.append(nd.right)
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lst += [nd.left,nd.right]
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else:
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lst.append(None)
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lst.append(None)
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lst +=[None,None]
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return level
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def addBlank(lines):
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width = 5
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sep = ' '*width
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n = len(lines)
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for i,oneline in enumerate(lines):
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k = 2**(n-i) -1
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new = [sep*((k-1)//2)]
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for s in oneline:
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new.append(s.ljust(width))
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new.append(sep*k)
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lines[i] = new
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return lines
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lines = levelVisit(self.root)
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print('-'*5+ 'level visit' + '-'*5)
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return '\n'.join([' '.join(line) for line in lines])
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lines = addBlank(lines)
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li = [''.join(line) for line in lines]
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li.insert(0,'red-black-tree'.rjust(48,'-') + '-'*33)
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li.append('end'.rjust(42,'-')+'-'*39+'\n')
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return '\n'.join(li)
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def __str__(self):
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return self.display()
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@ -270,44 +299,41 @@ def genNum(n =10):
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if d not in nums:
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nums.append(d)
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break
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#nums = [3,4,2,0,1,6]
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return nums
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def buildTree(n=10,nums=None,visitor=None):
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if nums is None: nums = genNum(n)
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if nums is None or nums ==[]: nums = genNum(n)
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rbtree = redBlackTree()
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print(f'build a red-black tree using {nums}')
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for i in nums:
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rbtree.insert(i)
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if visitor:
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visitor(rbtree)
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rbtree.insert(i)
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return rbtree
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return rbtree,nums
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def testInsert():
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def visitor(t):
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print(t)
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nums = [66, 14, 7, 2, 52, 96, 63, 51, 16, 53]
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rbtree = buildTree(nums = nums,visitor = visitor)
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rbtree,nums = buildTree(visitor = visitor)
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print('-'*5+ 'in-order visit' + '-'*5)
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for i,j in enumerate(rbtree.sort()):
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print(f'{i+1}: {j}')
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def testSuc():
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rbtree = buildTree()
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rbtree,nums = buildTree()
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for i in rbtree.sort():
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print(f'{i}\'s suc is {rbtree.getSuccessor(i)}')
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def testDelete():
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#nums = [56, 89, 31, 29, 24, 8, 62, 96, 20, 75] #tuple
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nums = [66, 14, 7, 2, 52, 96, 63, 51, 16, 53]
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rbtree = buildTree(nums = nums)
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#nums = [2,3,3,2,6,7,2,1]
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nums = None
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rbtree,nums = buildTree(nums = nums)
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print(rbtree)
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shuffle(nums)
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for i in nums:
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print(f'deleting {i}')
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rbtree.delete(i)
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print(rbtree)
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if __name__=='__main__':
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testInsert()
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#testDelete()
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#testSuc()
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#testInsert()
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testDelete()
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721
notes/red-black-tree.md
Normal file
721
notes/red-black-tree.md
Normal file
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@ -0,0 +1,721 @@
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---
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title: 『数据结构』红黑树(red-black tree)
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date: 2018-07-12 19:58
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categories: 数据结构与算法
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tags: [数据结构,红黑树]
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keywords:
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mathjax: true
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description:
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<!-- TOC -->
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- [1. 定义与性质](#1-定义与性质)
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- [1.1. 数据域](#11-数据域)
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- [1.2. 红黑性质](#12-红黑性质)
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- [1.3. 黑高度](#13-黑高度)
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- [2. 旋转](#2-旋转)
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- [3. 插入](#3-插入)
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- [3.1. 二叉查找树的插入](#31-二叉查找树的插入)
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- [3.2. 颜色调整与旋转](#32-颜色调整与旋转)
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- [3.2.1. 问题](#321-问题)
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- [3.2.2. 情况](#322-情况)
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- [3.2.2.1. case1: x 的叔叔是红色的](#3221-case1--x-的叔叔是红色的)
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- [3.2.2.2. case2: x 的叔叔是黑色, x,p(x), p(p(x)),方向为 left-right 或者 right-left](#3222-case2-x-的叔叔是黑色-xpx-ppx方向为-left-right-或者-right-left)
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- [3.2.2.3. case3: x 的叔叔是黑色, x,p(x), p(p(x)),方向为 left-left 或者 right-right](#3223-case3-x-的叔叔是黑色-xpx-ppx方向为-left-left-或者-right-right)
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- [3.2.3. 总体解决方案](#323-总体解决方案)
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- [4. 删除](#4-删除)
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- [4.1. 二叉查找树删除结点](#41-二叉查找树删除结点)
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- [4.2. 调整颜色与旋转](#42-调整颜色与旋转)
|
||||
- [5. 数据结构的扩张](#5-数据结构的扩张)
|
||||
- [5.1. 平衡树的扩张](#51-平衡树的扩张)
|
||||
- [6. python 代码](#6-python-代码)
|
||||
- [7. 参考](#7-参考)
|
||||
|
||||
<!-- /TOC -->
|
||||
|
||||
|
||||
<a id="markdown-1-定义与性质" name="1-定义与性质"></a>
|
||||
# 1. 定义与性质
|
||||
红黑树是一种平衡的二叉查找树
|
||||
<a id="markdown-11-数据域" name="11-数据域"></a>
|
||||
## 1.1. 数据域
|
||||
每个结点有 5 个数据域
|
||||
* color: red or black
|
||||
* key: keyword
|
||||
* left: pointer to left child
|
||||
* right:pointer to right child
|
||||
* p: pointer to nil leaf
|
||||
|
||||
<a id="markdown-12-红黑性质" name="12-红黑性质"></a>
|
||||
## 1.2. 红黑性质
|
||||
满足下面的 `红黑性质` 的二叉查找树就是红黑树:
|
||||
* 每个结点或是红色或是黑色
|
||||
* 根是黑
|
||||
* nil leaf 是 黑
|
||||
* 红结点的孩子是黑
|
||||
* 从每个结点出发,通过子孙到达叶子结点的各条路径上 黑结点数相等
|
||||
|
||||
如,叶子结点 是 nil, 即不存储任何东西, 为了编程方便,相对的,存有数据的结点称为内结点
|
||||
|
||||
|
||||
为了节省空间, 可以如下实现, 只需要一个 nil 结点
|
||||
|
||||
|
||||
<a id="markdown-13-黑高度" name="13-黑高度"></a>
|
||||
## 1.3. 黑高度
|
||||
从某个结点 x 到叶结点的黑色结点数,称为此结点的黑高度, 记为 $h_b(x)$
|
||||
树的黑高度是根的黑高度
|
||||
|
||||
>1. 以 x 为 根的子树至少包含 $2^{h_b(x)}-1$个结点
|
||||
>2. 一颗有 n 个内结点的红黑树高度至多为$2lg(n+1)$
|
||||
|
||||
可用归纳法证明1
|
||||
证明 2:
|
||||
设树高 h
|
||||
由红黑性质4, 根结点到叶子路径上的黑结点数至少 $\frac{h}{2}$,即 $h_b(root)\geqslant \frac{h}{2}$
|
||||
再由1,
|
||||
$$n \geqslant 2^{h_b(x)} -1 \geqslant 2^{\frac{h}{2}} -1$$
|
||||
|
||||
即 $ h\leqslant 2lg(n+1)$
|
||||
|
||||
<a id="markdown-2-旋转" name="2-旋转"></a>
|
||||
# 2. 旋转
|
||||
由于上面证明的红黑树高为 $O(logn)$,红黑树的 insert, delete, search 等操作都是, $O(logn)$.
|
||||
进行了 insert, delete 后可能破坏红黑性质, 可以通过旋转来保持.
|
||||
|
||||
|
||||
下面是对结点 x 进行 左旋与右旋.
|
||||
注意进行左旋时, 右孩子不是 nil(要用来作为旋转后 x 的双亲), 同理 右旋的结点的左孩子不是nil
|
||||
|
||||
总结起来就是: 父亲旋转,顺时针就是右旋,逆时针就是左旋, 旋转的结果是儿子成为原来父亲的新父亲, 即旋转的结点下降一层, 它的一个儿子上升一层.
|
||||
|
||||
<a id="markdown-3-插入" name="3-插入"></a>
|
||||
# 3. 插入
|
||||
插入的过程:
|
||||
* 先同二叉查找树那样插入, 做为叶子(不为空)
|
||||
* 然后将新结点的 左右孩子设为 nil , 颜色设为红色
|
||||
* 最后再进行颜色调整以及旋转(维持红黑性质)
|
||||
|
||||
这是算法导论[^1]上的算法
|
||||
```python
|
||||
RB-INSERT(T, z)
|
||||
y ← nil[T] // 新建节点“y”，将y设为空节点。
|
||||
x ← root[T] // 设“红黑树T”的根节点为“x”
|
||||
while x ≠ nil[T] // 找出要插入的节点“z”在二叉树T中的位置“y”
|
||||
do y ← x
|
||||
if key[z] < key[x]
|
||||
then x ← left[x]
|
||||
else x ← right[x]
|
||||
p[z] ← y // 设置 “z的父亲” 为 “y”
|
||||
if y = nil[T]
|
||||
then root[T] ← z // 情况1：若y是空节点，则将z设为根
|
||||
else if key[z] < key[y]
|
||||
then left[y] ← z // 情况2：若“z所包含的值” < “y所包含的值”，则将z设为“y的左孩子”
|
||||
else right[y] ← z // 情况3：(“z所包含的值” >= “y所包含的值”)将z设为“y的右孩子”
|
||||
left[z] ← nil[T] // z的左孩子设为空
|
||||
right[z] ← nil[T] // z的右孩子设为空。至此，已经完成将“节点z插入到二叉树”中了。
|
||||
color[z] ← RED // 将z着色为“红色”
|
||||
RB-INSERT-FIXUP(T, z) // 通过RB-INSERT-FIXUP对红黑树的节点进行颜色修改以及旋转，让树T仍然是一颗红黑树
|
||||
```
|
||||
<a id="markdown-31-二叉查找树的插入" name="31-二叉查找树的插入"></a>
|
||||
## 3.1. 二叉查找树的插入
|
||||
可以用python 实现如下
|
||||
```python
|
||||
def insert(self,nd):
|
||||
if not isinstance(nd,node):
|
||||
nd = node(nd)
|
||||
elif nd.isBlack: nd.isBlack = False
|
||||
|
||||
if self.root is None:
|
||||
self.root = nd
|
||||
self.root.isBlack = True
|
||||
else:
|
||||
parent = self.root
|
||||
while parent:
|
||||
if parent == nd : return None
|
||||
if parent>nd:
|
||||
if parent.left :
|
||||
parent = parent.left
|
||||
else:
|
||||
parent.left = nd
|
||||
break
|
||||
else:
|
||||
if parent.right:
|
||||
parent = parent.right
|
||||
else:
|
||||
parent.right = nd
|
||||
break
|
||||
self.fixUpInsert(parent,nd)
|
||||
```
|
||||
<a id="markdown-32-颜色调整与旋转" name="32-颜色调整与旋转"></a>
|
||||
## 3.2. 颜色调整与旋转
|
||||
<a id="markdown-321-问题" name="321-问题"></a>
|
||||
### 3.2.1. 问题
|
||||
在插入后,可以发现后破坏的红黑性质只有以下两条(且互斥)
|
||||
|
||||
1. root 是红 (这可以直接将root 颜色设为黑调整)
|
||||
2. 红结点的孩子是黑
|
||||
|
||||
所以下面介绍如何保持 红结点的孩子是黑 , 即插入结点的双亲结点是红的情况.
|
||||
|
||||
下面记 结点 x 的 双亲为 p(x), 新插入的结点为 x, 记 uncle 结点 为 u(x)
|
||||
|
||||
由于 p(x) 是红色, 而根结点是黑色, 所以 p(x)不是根, p(p(x))存在
|
||||
<a id="markdown-322-情况" name="322-情况"></a>
|
||||
### 3.2.2. 情况
|
||||
|
||||
有如下三种情况
|
||||
|
||||
|
||||
|
||||
每种情况的解决方案如下
|
||||
|
||||
<a id="markdown-3221-case1--x-的叔叔是红色的" name="3221-case1--x-的叔叔是红色的"></a>
|
||||
#### 3.2.2.1. case1: x 的叔叔是红色的
|
||||
这里只需改变颜色, 将 p(x)变为 黑, p(p(x))变为红, u(x) 变为黑色 (x为右孩子同样)
|
||||
|
||||
|
||||
|
||||
<a id="markdown-3222-case2-x-的叔叔是黑色-xpx-ppx方向为-left-right-或者-right-left" name="3222-case2-x-的叔叔是黑色-xpx-ppx方向为-left-right-或者-right-left"></a>
|
||||
#### 3.2.2.2. case2: x 的叔叔是黑色, x,p(x), p(p(x)),方向为 left-right 或者 right-left
|
||||
即 x,p(x), p(p(x)) 成折线状
|
||||
<a id="markdown-3223-case3-x-的叔叔是黑色-xpx-ppx方向为-left-left-或者-right-right" name="3223-case3-x-的叔叔是黑色-xpx-ppx方向为-left-left-或者-right-right"></a>
|
||||
#### 3.2.2.3. case3: x 的叔叔是黑色, x,p(x), p(p(x)),方向为 left-left 或者 right-right
|
||||
即 x,p(x), p(p(x)) 成直线状
|
||||
|
||||
|
||||
|
||||
|
||||
当 x 为右孩子时, 通过旋转变成p(x) 的双亲, 然后相当于 新插入 p(x)作为左孩子, 再进行转换.
|
||||
|
||||
即将新结点的双亲向上一层旋转,颜色变为黑色, 而新节点的祖父向下一层, 颜色变为红色
|
||||
|
||||
<a id="markdown-323-总体解决方案" name="323-总体解决方案"></a>
|
||||
### 3.2.3. 总体解决方案
|
||||
我最开始也没有弄清楚, 有点绕晕的感觉, 后来仔细读了书上伪代码, 然后才发现就是一个状态机, 画出来就一目了然了.
|
||||
|
||||
|
||||
|
||||
|
||||
现在算是知其然了, 那么怎样知其所以然呢? 即 为什么要分类这三个 case, 不重不漏了吗?
|
||||
|
||||
其实也简单, 只是太繁琐.
|
||||
就是将各种情况枚举出来, 一一分析即可. 我最开始试过, 但是太多,写在代码里很容易写着写着就混了.
|
||||
而算法导论上分成这三个case , 很简洁, 只是归纳了一下而已. 如果想看看枚举情况的图与说明,可以参考[^2] .
|
||||
|
||||
|
||||
|
||||
算法导论上的伪代码
|
||||
```python
|
||||
RB-INSERT-FIXUP(T, z)
|
||||
while color[p[z]] = RED // 若“当前节点(z)的父节点是红色”，则进行以下处理。
|
||||
do if p[z] = left[p[p[z]]] // 若“z的父节点”是“z的祖父节点的左孩子”，则进行以下处理。
|
||||
then y ← right[p[p[z]]] // 将y设置为“z的叔叔节点(z的祖父节点的右孩子)”
|
||||
if color[y] = RED // Case 1条件：叔叔是红色
|
||||
then color[p[z]] ← BLACK ▹ Case 1 // (01) 将“父节点”设为黑色。
|
||||
color[y] ← BLACK ▹ Case 1 // (02) 将“叔叔节点”设为黑色。
|
||||
color[p[p[z]]] ← RED ▹ Case 1 // (03) 将“祖父节点”设为“红色”。
|
||||
z ← p[p[z]] ▹ Case 1 // (04) 将“祖父节点”设为“当前节点”(红色节点)
|
||||
else if z = right[p[z]] // Case 2条件：叔叔是黑色，且当前节点是右孩子
|
||||
then z ← p[z] ▹ Case 2 // (01) 将“父节点”作为“新的当前节点”。
|
||||
LEFT-ROTATE(T, z) ▹ Case 2 // (02) 以“新的当前节点”为支点进行左旋。
|
||||
color[p[z]] ← BLACK ▹ Case 3 // Case 3条件：叔叔是黑色，且当前节点是左孩子。(01) 将“父节点”设为“黑色”。
|
||||
color[p[p[z]]] ← RED ▹ Case 3 // (02) 将“祖父节点”设为“红色”。
|
||||
RIGHT-ROTATE(T, p[p[z]]) ▹ Case 3 // (03) 以“祖父节点”为支点进行右旋。
|
||||
else (same as then clause with "right" and "left" exchanged) // 若“z的父节点”是“z的祖父节点的右孩子”，将上面的操作中“right”和“left”交换位置，然后依次执行。
|
||||
color[root[T]] ← BLACK
|
||||
```
|
||||
我用python 实现如下. 由于左右方向不同, 如果向上面伪代码那样实现, fixup 代码就会有两份类似的(即 right left 互换), 为了减少代码冗余, 我就定义了 `setChild`, `getChild` 函数, 传递左或是右孩子这个方向的数据(代码中是isLeft), 所以下面的就是完整功能的 fixup, 可以减少一般的代码量, haha😄,
|
||||
(下文 删除结点同理)
|
||||
|
||||
其实阅读代码也简单, 可以直接当成 isLeft 取真值.
|
||||
```python
|
||||
def fixUpInsert(self,parent,nd):
|
||||
''' adjust color and level, there are two red nodes: the new one and its parent'''
|
||||
while not self.checkBlack(parent):
|
||||
grand = self.getParent(parent)
|
||||
isLeftPrt = grand.left is parent
|
||||
uncle = grand.getChild(not isLeftPrt)
|
||||
if not self.checkBlack(uncle):
|
||||
# case 1: new node's uncle is red
|
||||
self.setBlack(grand, False)
|
||||
self.setBlack(grand.left, True)
|
||||
self.setBlack(grand.right, True)
|
||||
nd = grand
|
||||
parent = self.getParent(nd)
|
||||
else:
|
||||
# case 2: new node's uncle is black(including nil leaf)
|
||||
isLeftNode = parent.left is nd
|
||||
if isLeftNode ^ isLeftPrt:
|
||||
# case 2.1 the new node is inserted in left-right or right-left form
|
||||
# grand grand
|
||||
# parent or parent
|
||||
# nd nd
|
||||
parent.setChild(nd.getChild(isLeftPrt),not isLeftPrt)
|
||||
nd.setChild(parent,isLeftPrt)
|
||||
grand.setChild(nd,isLeftPrt)
|
||||
nd,parent = parent,nd
|
||||
# case 2.2 the new node is inserted in left-left or right-right form
|
||||
# grand grand
|
||||
# parent or parent
|
||||
# nd nd
|
||||
grand.setChild(parent.getChild(not isLeftPrt),isLeftPrt)
|
||||
parent.setChild(grand,not isLeftPrt)
|
||||
self.setBlack(grand, False)
|
||||
self.setBlack(parent, True)
|
||||
self.transferParent(grand,parent)
|
||||
self.setBlack(self.root,True)
|
||||
```
|
||||
|
||||
|
||||
<a id="markdown-4-删除" name="4-删除"></a>
|
||||
# 4. 删除
|
||||
|
||||
|
||||
算法导论上的算法
|
||||
|
||||
写的很简练👍
|
||||
|
||||
|
||||
<a id="markdown-41-二叉查找树删除结点" name="41-二叉查找树删除结点"></a>
|
||||
## 4.1. 二叉查找树删除结点
|
||||
下面 z 是要删除的结点, y 是 其后继或者是它自己, x 是 y 的一个孩子(如果 y 的孩子为 nil,则为 nli, 否则 y 只有一个非 nil 孩子, 为 x)
|
||||
|
||||
* 当 z 孩子全是 nil (y==z): 直接让其双亲对应的孩子为 nil
|
||||
* 当 z 只有一个非 nil 孩子 x (y==z):
|
||||
1. 如果 z 为根, 则让 x 为根.
|
||||
2. 让 y 的双亲连接到 x
|
||||
* 当 z 有两个非nil孩子(y!=z): 复制其后继 y 的内容到 z (除了指针,颜色) , 将其后继 y 的孩子(最多只有一个 非 nil ,不然就不是后继了)连接到其后继的双亲, 删除 其后继y,
|
||||
|
||||
即[^3] 如果要删除有两个孩子的结点 z , 则找到它的后继y(前趋同理), 可以推断 y 一定没有左孩子, 右孩子可能有,可能没有. 也就是最多一个孩子.
|
||||
所以将 y 的值复制到 x 位置, 现在相当于删除 y 处的结点.
|
||||
这样就化为 删除的结点最多一个孩子的情况.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<a id="markdown-42-调整颜色与旋转" name="42-调整颜色与旋转"></a>
|
||||
## 4.2. 调整颜色与旋转
|
||||
可以发现只有当 y 是黑色,才进行颜色调整以及旋转(维持红黑性质), 因为如果删除的是红色, 不会影响黑高度, 所有红黑性质都不会破坏
|
||||
伪代码如下, (我的python代码见文末)
|
||||
|
||||
|
||||
如果被删除的结点 y 是黑色的, 有三种破坏红黑性质的情况
|
||||
1. y是根, 则 y 的一个红色孩子成为新根
|
||||
2. 进行删除结点过程中, p(y) 的孩子有 x, 两者都是红色
|
||||
3. 删除 y 导致包含y 的路径上的黑结点 少 1个
|
||||
|
||||
修复3的思路:
|
||||
如果可能,在兄弟一支,通过旋转,改变颜色修复
|
||||
否则, 将红结点一直向上推(因为当前路径上少了一个黑结点,向上推的过程中使红结点所在的子树都少一个黑结点), 直到到达树根, 那么全部路径都少一个黑结点, 3就修复了, 这时只需将根设为黑就修复了 1
|
||||
|
||||
代码中的 while 循环的目的是将额外的黑色沿树上移,直到
|
||||
* x 指向一个红黑结点
|
||||
* x 指向根,这时可以简单地消除额外的黑色
|
||||
* 颜色修改与旋转
|
||||
|
||||
在 while 中, x 总是指向具有双重黑色的那个非根结点, 在第 2 行中要判断 x 是其双亲的左右孩子
|
||||
w 表示 x 的相抵. w 不能为 nil(因为 x 是双重黑色)
|
||||
|
||||
算法中的四种情况如图所示
|
||||
|
||||
|
||||
即
|
||||
* x 的兄弟 w 是红色的
|
||||
|
||||
* x 的兄弟 w 是黑色的, w的两个孩子都是黑色的
|
||||
|
||||
* x 的兄弟 w 是黑色的, w 的左孩子是红,右孩子是黑
|
||||
* x 的兄弟 w 是黑色的, w 的孩子是红色的
|
||||
|
||||
>>注意上面都是先考虑的左边, 右边可以对称地处理.
|
||||
|
||||
同插入一样, 为了便于理解, 可以作出状态机.
|
||||
而且这些情形都是归纳化简了的, 你也可以枚举列出基本的全部情形.
|
||||
|
||||
|
||||
<a id="markdown-5-数据结构的扩张" name="5-数据结构的扩张"></a>
|
||||
# 5. 数据结构的扩张
|
||||
<a id="markdown-51-平衡树的扩张" name="51-平衡树的扩张"></a>
|
||||
## 5.1. 平衡树的扩张
|
||||
通过在平衡树(如红黑树上的每个结点 加上 一个数据域 size (表示以此结点为根的子树的结点数.) 可以使`获得第 i 大的数` 的时间复杂度为 $O(logn)$
|
||||
|
||||
在 $O(n)$ 时间内建立, python代码如下
|
||||
```python
|
||||
def setSize(root):
|
||||
if root is None:return 0
|
||||
root.size = setSize(root.left) + setSize(root.right)+1
|
||||
```
|
||||
在$O(logn)$时间查找,
|
||||
```python
|
||||
def find(root,i):
|
||||
r = root.left.size +1
|
||||
if r==i:
|
||||
return root
|
||||
if r > i:
|
||||
return find(root.left,i)
|
||||
else:
|
||||
return find(root.right,i-r)
|
||||
```
|
||||
|
||||
|
||||
|
||||
<a id="markdown-6-python-代码" name="6-python-代码"></a>
|
||||
# 6. python 代码
|
||||
|
||||
**[github地址](https://github.com/mbinary/algorithm-and-data-structure.git)**
|
||||
|
||||
我用了 setChild, getChild 来简化代码量, 其他的基本上是按照算法导论上的伪代码提到的case 来实现的. 然后display 只是测试的时候,为了方便调试而层序遍历打印出来
|
||||
|
||||
效果如下
|
||||
|
||||
|
||||
```python
|
||||
'''
|
||||
#########################################################################
|
||||
# File : redBlackTree.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.coding.me
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2018-07-12 20:34
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
from functools import total_ordering
|
||||
from random import randint, shuffle
|
||||
|
||||
@total_ordering
|
||||
class node:
|
||||
def __init__(self,val,left=None,right=None,isBlack=False):
|
||||
self.val =val
|
||||
self.left = left
|
||||
self.right = right
|
||||
self.isBlack = isBlack
|
||||
def __lt__(self,nd):
|
||||
return self.val < nd.val
|
||||
def __eq__(self,nd):
|
||||
return nd is not None and self.val == nd.val
|
||||
def setChild(self,nd,isLeft = True):
|
||||
if isLeft: self.left = nd
|
||||
else: self.right = nd
|
||||
def getChild(self,isLeft):
|
||||
if isLeft: return self.left
|
||||
else: return self.right
|
||||
def __bool__(self):
|
||||
return self.val is not None
|
||||
def __str__(self):
|
||||
color = 'B' if self.isBlack else 'R'
|
||||
return f'{color}-{self.val:}'
|
||||
def __repr__(self):
|
||||
return f'node({self.val},isBlack={self.isBlack})'
|
||||
class redBlackTree:
|
||||
def __init__(self,unique=False):
|
||||
'''if unique is True, all node'vals are unique, else there may be equal vals'''
|
||||
self.root = None
|
||||
self.unique = unique
|
||||
|
||||
@staticmethod
|
||||
def checkBlack(nd):
|
||||
return nd is None or nd.isBlack
|
||||
@staticmethod
|
||||
def setBlack(nd,isBlack):
|
||||
if nd is not None:
|
||||
if isBlack is None or isBlack:
|
||||
nd.isBlack = True
|
||||
else:nd.isBlack = False
|
||||
def sort(self,reverse = False):
|
||||
''' return a generator of sorted data'''
|
||||
def inOrder(root):
|
||||
if root is None:return
|
||||
if reverse:
|
||||
yield from inOrder(root.right)
|
||||
else:
|
||||
yield from inOrder(root.left)
|
||||
yield root
|
||||
if reverse:
|
||||
yield from inOrder(root.left)
|
||||
else:
|
||||
yield from inOrder(root.right)
|
||||
yield from inOrder(self.root)
|
||||
def getParent(self,chd):
|
||||
'''note that use is to find real node when different nodes have euqiv val'''
|
||||
if self.root is chd:return None
|
||||
nd = self.root
|
||||
while nd:
|
||||
if nd>chd and nd.left is not None:
|
||||
if nd.left is chd: return nd
|
||||
else: nd = nd.left
|
||||
elif nd<chd and nd.right is not None:
|
||||
if nd.right is chd: return nd
|
||||
else: nd = nd.right
|
||||
def find(self,val):
|
||||
nd = self.root
|
||||
while nd:
|
||||
if nd.val ==val:
|
||||
return nd
|
||||
elif nd.val>val:
|
||||
nd = nd.left
|
||||
else:
|
||||
nd = nd.right
|
||||
def getSuccessor(self,nd):
|
||||
if nd:
|
||||
if nd.right:
|
||||
nd = nd.right
|
||||
while nd.left:
|
||||
nd = nd.left
|
||||
return nd
|
||||
else:return self.getParent(nd)
|
||||
def transferParent(self,origin,new):
|
||||
if origin is self.root:
|
||||
self.root = new
|
||||
else:
|
||||
prt = self.getParent(origin)
|
||||
prt.setChild(new, prt.left is origin)
|
||||
|
||||
def insert(self,nd):
|
||||
if not isinstance(nd,node):
|
||||
nd = node(nd)
|
||||
elif nd.isBlack: nd.isBlack = False
|
||||
|
||||
if self.root is None:
|
||||
self.root = nd
|
||||
self.root.isBlack = True
|
||||
else:
|
||||
parent = self.root
|
||||
while parent:
|
||||
if parent == nd : return None
|
||||
if parent>nd:
|
||||
if parent.left :
|
||||
parent = parent.left
|
||||
else:
|
||||
parent.left = nd
|
||||
break
|
||||
else:
|
||||
if parent.right:
|
||||
parent = parent.right
|
||||
else:
|
||||
parent.right = nd
|
||||
break
|
||||
self.fixUpInsert(parent,nd)
|
||||
def fixUpInsert(self,parent,nd):
|
||||
''' adjust color and level, there are two red nodes: the new one and its parent'''
|
||||
while not self.checkBlack(parent):
|
||||
grand = self.getParent(parent)
|
||||
isLeftPrt = grand.left is parent
|
||||
uncle = grand.getChild(not isLeftPrt)
|
||||
if not self.checkBlack(uncle):
|
||||
# case 1: new node's uncle is red
|
||||
self.setBlack(grand, False)
|
||||
self.setBlack(grand.left, True)
|
||||
self.setBlack(grand.right, True)
|
||||
nd = grand
|
||||
parent = self.getParent(nd)
|
||||
else:
|
||||
# case 2: new node's uncle is black(including nil leaf)
|
||||
isLeftNode = parent.left is nd
|
||||
if isLeftNode ^ isLeftPrt:
|
||||
# case 2.1 the new node is inserted in left-right or right-left form
|
||||
# grand grand
|
||||
# parent or parent
|
||||
# nd nd
|
||||
parent.setChild(nd.getChild(isLeftPrt),not isLeftPrt)
|
||||
nd.setChild(parent,isLeftPrt)
|
||||
grand.setChild(nd,isLeftPrt)
|
||||
nd,parent = parent,nd
|
||||
# case 2.2 the new node is inserted in left-left or right-right form
|
||||
# grand grand
|
||||
# parent or parent
|
||||
# nd nd
|
||||
grand.setChild(parent.getChild(not isLeftPrt),isLeftPrt)
|
||||
parent.setChild(grand,not isLeftPrt)
|
||||
self.setBlack(grand, False)
|
||||
self.setBlack(parent, True)
|
||||
self.transferParent(grand,parent)
|
||||
self.setBlack(self.root,True)
|
||||
|
||||
def copyNode(self,src,des):
|
||||
'''when deleting a node which has two kids,
|
||||
copy its succesor's data to his position
|
||||
data exclude left, right , isBlack
|
||||
'''
|
||||
des.val = src.val
|
||||
def delete(self,nd):
|
||||
'''delete node in a binary search tree'''
|
||||
if not isinstance(nd,node):
|
||||
nd = self.find(nd)
|
||||
if nd is None: return
|
||||
y = None
|
||||
if nd.left and nd.right:
|
||||
y= self.getSuccessor(nd)
|
||||
else:
|
||||
y = nd
|
||||
py = self.getParent(y)
|
||||
x = y.left if y.left else y.right
|
||||
if py is None:
|
||||
self.root = x
|
||||
elif y is py.left:
|
||||
py.left = x
|
||||
else:
|
||||
py.right = x
|
||||
if y != nd:
|
||||
self.copyNode(y,nd)
|
||||
|
||||
if self.checkBlack(y): self.fixUpDel(py,x)
|
||||
|
||||
|
||||
def fixUpDel(self,prt,chd):
|
||||
''' adjust colors and rotate '''
|
||||
while self.root != chd and self.checkBlack(chd):
|
||||
isLeft = prt.left is chd
|
||||
brother = prt.getChild(not isLeft)
|
||||
# brother is black
|
||||
lb = self.checkBlack(brother.getChild(isLeft))
|
||||
rb = self.checkBlack(brother.getChild(not isLeft))
|
||||
if not self.checkBlack(brother):
|
||||
# case 1: brother is red. converted to case 2,3,4
|
||||
# prt (isLeft) rotate
|
||||
prt.setChild(brother.getChild(isLeft), not isLeft)
|
||||
brother.setChild(prt, isLeft)
|
||||
|
||||
self.setBlack(prt,False)
|
||||
self.setBlack(brother,True)
|
||||
|
||||
self.transferParent(prt,brother)
|
||||
elif lb and rb:
|
||||
# case 2: brother is black and two kids are black.
|
||||
# conveted to the begin case
|
||||
self.setBlack(brother,False)
|
||||
chd = prt
|
||||
prt = self.getParent(chd)
|
||||
else:
|
||||
if rb:
|
||||
# case 3: brother is black and left kid is red and right child is black
|
||||
# uncle's son is nephew, and niece for uncle's daughter
|
||||
nephew = brother.getChild(isLeft)
|
||||
self.setBlack(nephew,True)
|
||||
self.setBlack(brother,False)
|
||||
|
||||
# brother (not isLeft) rotate
|
||||
prt.setChild(nephew,not isLeft)
|
||||
brother.setChild(nephew.getChild(not isLeft),isLeft)
|
||||
nephew.setChild(brother, not isLeft)
|
||||
brother = nephew
|
||||
|
||||
# case 4: brother is black and right child is red
|
||||
brother.isBlack = prt.isBlack
|
||||
self.setBlack(prt,True)
|
||||
self.setBlack(brother.getChild(not isLeft),True)
|
||||
|
||||
# prt left rotate
|
||||
prt.setChild(brother.getChild(isLeft),not isLeft)
|
||||
brother.setChild(prt,isLeft)
|
||||
|
||||
self.transferParent(prt,brother)
|
||||
|
||||
chd = self.root
|
||||
self.setBlack(chd,True)
|
||||
|
||||
|
||||
def display(self):
|
||||
def getHeight(nd):
|
||||
if nd is None:return 0
|
||||
return max(getHeight(nd.left),getHeight(nd.right)) +1
|
||||
def levelVisit(root):
|
||||
from collections import deque
|
||||
lst = deque([root])
|
||||
level = []
|
||||
h = getHeight(root)
|
||||
ct = lv = 0
|
||||
while 1:
|
||||
ct+=1
|
||||
nd = lst.popleft()
|
||||
if ct >= 2**lv:
|
||||
lv+=1
|
||||
if lv>h:break
|
||||
level.append([])
|
||||
level[-1].append(str(nd))
|
||||
if nd is not None:
|
||||
lst += [nd.left,nd.right]
|
||||
else:
|
||||
lst +=[None,None]
|
||||
return level
|
||||
def addBlank(lines):
|
||||
width = 5
|
||||
sep = ' '*width
|
||||
n = len(lines)
|
||||
for i,oneline in enumerate(lines):
|
||||
k = 2**(n-i) -1
|
||||
new = [sep*((k-1)//2)]
|
||||
for s in oneline:
|
||||
new.append(s.ljust(width))
|
||||
new.append(sep*k)
|
||||
lines[i] = new
|
||||
return lines
|
||||
|
||||
lines = levelVisit(self.root)
|
||||
lines = addBlank(lines)
|
||||
li = [''.join(line) for line in lines]
|
||||
li.insert(0,'red-black-tree'.rjust(48,'-') + '-'*33)
|
||||
li.append('end'.rjust(42,'-')+'-'*39+'\n')
|
||||
return '\n'.join(li)
|
||||
|
||||
def __str__(self):
|
||||
return self.display()
|
||||
```
|
||||
测试代码
|
||||
```python
|
||||
def genNum(n =10):
|
||||
nums =[]
|
||||
for i in range(n):
|
||||
while 1:
|
||||
d = randint(0,100)
|
||||
if d not in nums:
|
||||
nums.append(d)
|
||||
break
|
||||
return nums
|
||||
|
||||
def buildTree(n=10,nums=None,visitor=None):
|
||||
if nums is None or nums ==[]: nums = genNum(n)
|
||||
rbtree = redBlackTree()
|
||||
print(f'build a red-black tree using {nums}')
|
||||
for i in nums:
|
||||
rbtree.insert(i)
|
||||
if visitor:
|
||||
visitor(rbtree)
|
||||
return rbtree,nums
|
||||
def testInsert():
|
||||
def visitor(t):
|
||||
print(t)
|
||||
rbtree,nums = buildTree(visitor = visitor)
|
||||
print('-'*5+ 'in-order visit' + '-'*5)
|
||||
for i,j in enumerate(rbtree.sort()):
|
||||
print(f'{i+1}: {j}')
|
||||
|
||||
def testSuc():
|
||||
rbtree,nums = buildTree()
|
||||
for i in rbtree.sort():
|
||||
print(f'{i}\'s suc is {rbtree.getSuccessor(i)}')
|
||||
|
||||
def testDelete():
|
||||
#nums = [2,3,3,2,6,7,2,1]
|
||||
nums = None
|
||||
rbtree,nums = buildTree(nums = nums)
|
||||
print(rbtree)
|
||||
for i in nums:
|
||||
print(f'deleting {i}')
|
||||
rbtree.delete(i)
|
||||
print(rbtree)
|
||||
|
||||
if __name__=='__main__':
|
||||
#testSuc()
|
||||
#testInsert()
|
||||
testDelete()
|
||||
```
|
||||
|
||||
<a id="markdown-7-参考" name="7-参考"></a>
|
||||
# 7. 参考
|
||||
[^1]: 算法导论
|
||||
|
||||
[^2]: https://www.jianshu.com/p/a5514510f5b9?utm_campaign=maleskine&utm_content=note&utm_medium=seo_notes&utm_source=recommendation
|
||||
[^3]: https://www.jianshu.com/p/0b68b992f688?utm_campaign=maleskine&utm_content=note&utm_medium=seo_notes&utm_source=recommendation
|
||||
Loading…
Reference in New Issue
Block a user