算法导论34--区间树（Python）

区间树：

闭区间是一个实数的有序对[t1, t2]，其中t1<=t2。区间便于表示占用一连续时间段的一些事件。

我们可以把一个区间[t1, t2]表示成一个对象i。其中属性i.low=t1为低端点，属性i.high=t2为高端点。

区间i和i'重叠：i.low<=i'high且i'.low<=i.high。

区间树是一种动态集合进行维护的红黑树，其中每个元素x都包含一个区间x.int。

每个结点x:包含一个区间属性x.int。且x的关键字为区间的低端点x.int.low、

附加信息：x.max：以x为根的子树中所有区间的端点的最大值。

操作：INTERVAL-SEARCH(T, i)它是用来找出树T中与区间i重叠的那个结点。

如果树中与i重叠的结点不存在，则下面过程返回指向哨兵T.ni的指针。

class Node:    def __init__(self,right,left,p,color,inter,maxx):        self.key=inter.low        self.right=right        self.left=left        self.p=p        self.color=color        #新增区间属性        self.inter=inter        #新增附加信息maxx        self.maxx=maxx#代表区间的类class Inter:    def __init__(self,low,high):        self.low=low        self.high=highclass tree:    def __init__(self,root,nil):        self.root=root        self.nil=nil    def tree_insert(self,z):        y=self.nil        x=self.root        while x!=self.nil:            y=x            if z.key<x.key:                x=x.left            else:                x=x.right        z.p=y        if y==self.nil:            self.root=z        elif z.key<y.key:            y.left=z        else:            y.right=z        z.left=self.nil        z.right=self.nil        z.color="RED"        z.maxx=max(z.inter.high,z.left.maxx,z.right.maxx)        #红黑树性质维护        self.rb_insert_fixup(z)        #更新父结点直到根结点的maxx        while z.p!=self.nil:            z.p.maxx=max(z.p.maxx,z.maxx)            z=z.p                def left_rotate(self,x):        y=x.right        x.right=y.left        if y.left!=self.nil:            y.left.p=x        y.p=x.p        if x.p==self.nil:            self.root=y        elif x==x.p.left:            x.p.left=y        else:            x.p.right=y        y.left=x        x.p=y        #左旋导致两个结点的max属性改变，更新如下        y.maxx=x.maxx        x.maxx=max(x.left.maxx,x.right.maxx,x.inter.high)            def right_rotate(self,y):        x=y.left        y.left=x.right        if x.right!=self.nil:            x.right.p=y        x.p=y.p          if y.p==self.nil:            self.root=x        elif y==y.p.left:            y.p.left=x        else:            y.p.right=x        x.right=y        y.p=x        #右旋导致两个结点的max属性改变，更新如下        x.maxx=y.maxx        y.maxx=max(y.right.maxx,y.left.maxx,y.inter.high)    def rb_insert_fixup(self,z):        while z.p.color=="RED":            if z.p==z.p.p.left:                y=z.p.p.right                if y.color=="RED":                    z.p.color="BLACK"                    y.color="BLACK"                    z.p.p.color="RED"                    z=z.p.p                else:                    if z==z.p.right:                        z=z.p                        self.left_rotate(z)                    z.p.color="BLACK"                    z.p.p.color="RED"                    self.right_rotate(z.p.p)            else:                y=z.p.p.left                if y.color=="RED":                    z.p.color="BLACK"                    y.color="BLACK"                    z.p.p.color="RED"                    z=z.p.p                else:                    if z==z.p.left:                        z=z.p                        self.right_rotate(z)                    z.p.color="BLACK"                    z.p.p.color="RED"                    self.left_rotate(z.p.p)        self.root.color="BLACK"    def inorder_tree_walk(self,x):        if x!=self.nil:            self.inorder_tree_walk(x.left)            print(x.key)            self.inorder_tree_walk(x.right)    def tree_search(self,x,k):        if x==self.nil or k==x.key:            return x        if k < x.key:            return self.tree_search(x.left,k)        else: return self.tree_search(x.right,k)    def rb_transplant(self,u,v):        if u.p==self.nil:            self.root=v        elif u==u.p.left:            u.p.left=v        else:            u.p.right=v        v.p=u.p    def tree_minimum(self,x):        while x.left!=self.nil:            x=x.left        return x        def rb_delete(self,z):        y=z        y_original_color=y.color        if z.left==self.nil:            x=z.right            self.rb_transplant(z,z.right)        elif z.right==self.nil:            x=z.left            self.rb_transplant(z,z.left)        else:            y=self.tree_minimum(z.right)            y_original_color=y.color            x=y.right            if y.p==z:                x.p=y            else:                self.rb_transplant(y,y.right)                y.right=z.right                y.right.p=y            self.rb_transplant(z,y)            y.left=z.left            y.left.p=y            y.color=z.color            if y_original_color=="BLACK":                self.rb_delete_fixup(x)                       def rb_delete_fixup(self,x):        while x!=self.root and x.color=="BLACK":            if x==x.p.left:                w=x.p.right                if w.color=="RED":                    w.color="BLACK"                    x.p.color="RED"                    self.left_rotate(x.p)                    w=x.p.right                if w.left.color=="BLACK" and w.right.color=="BLACK":                    w.color="RED"                    x=x.p                else:                    if w.right.color=="BLACK":                        w.left.color=="BLACK"                        w.color="RED"                        self.right_rotate(w)                        w=x.p.right                    w.color=x.p.color                    x.p.color="BLACK"                    w.right.color="BLACK"                    self.left_rotate(x.p)                    x=self.root            else:                w=x.p.left                if w.color=="RED":                    w.color="BLACK"                    x.p.color="RED"                    self.right_rotate(x.p)                    w=x.p.left                if w.right.color=="BLACK" and w.left.color=="BLACK":                    w.color="RED"                    x=x.p                else:                    if w.left.color=="BLACK":                        w.right.color=="BLACK"                        w.color="RED"                        self.left_rotate(w)                        w=x.p.left                    w.color=x.p.color                    x.p.color="BLACK"                    w.left.color="BLACK"                    self.right_rotate(x.p)                    x=self.root        x.color="BLACK"    def print_tree(self,z):        if z!=self.nil:            print(z.key,z.color,"[",z,inter.low,",",z.inter.high,"]",":",end='')            print("( ",end='')            print(z.left.key,z.left.color,"   ",end='')            print(z.right.key,z.right.color,end='')            print(" )",end='')    def interval_search(self,i):        x=self.root        while x!=self.nil and (i.high<x.inter.low or x.inter.high<i.low):            if x.left!=self.nil and x.left.maxx>=i.low:                x=x.left            else:                x=x.right        return xif __name__=="__main__":    inter=Inter(0,0)    nil=Node(None,None,None,"BLACK",inter,0)    inter=Inter(16,21)    root=Node(nil,nil,nil,"BLACK",inter,21)    t=tree(root,nil)    TT=[8,9,25,30,5,8,15,23,17,19,26,26,0,3,6,10,19,20]    for i in range(0,len(TT),2):        inter=Inter(TT[i],TT[i+1])        z=Node(nil,nil,nil,"RED",inter,0)        t.tree_insert(z)                

运行：

>>> TT=[0,5,6,8,15,16,17,19,25,26]>>> for i in TT:z=t.tree_search(t.root,i)print(z.key,z.color,"[",z.inter.low,",",z.inter.high,"]",z.maxx,":",end='')print("( ",end='')print(z.left.key,z.left.color,"   ",end='')print(z.right.key,z.right.color,end='')print(" )",end='')print()0 RED [ 0 , 3 ] 3 :( 0 BLACK    0 BLACK )5 BLACK [ 5 , 8 ] 10 :( 0 RED    6 RED )6 RED [ 6 , 10 ] 10 :( 0 BLACK    0 BLACK )8 RED [ 8 , 9 ] 23 :( 5 BLACK    15 BLACK )15 BLACK [ 15 , 23 ] 23 :( 0 BLACK    0 BLACK )16 BLACK [ 16 , 21 ] 30 :( 8 RED    25 RED )17 BLACK [ 17 , 19 ] 20 :( 0 BLACK    19 RED )19 RED [ 19 , 20 ] 20 :( 0 BLACK    0 BLACK )25 RED [ 25 , 30 ] 30 :( 17 BLACK    26 BLACK )26 BLACK [ 26 , 26 ] 26 :( 0 BLACK    0 BLACK )>>> t.interval_search(Inter(22,25))<__main__.Node object at 0x038B1F10>>>> t.interval_search(Inter(22,25)).key15>>> t.interval_search(Inter(11,14)).key0