Introduction to Algorithms Learning Notes--13th chapter red and black Trees

Red and black Trees

A red-black tree is a binary lookup tree, but adds a storage node color to each node, which can be red or black. By limiting the color of the nodes on any path from root to leaf, the red-black tree ensures that no path is twice times longer than the other paths, and thus is nearly balanced.

Each node contains 5 domains, color,key,left,right,p

Meets the following red-black properties:

1. Each knot is red or black

2. The root node is black

3. Each leaf node (nil) is black

4, if a node is red, then its two sub-nodes are black

5. For each node, the number of black nodes is the same from the node to the path of each descendant node.

Left rotation

1left-ROTATE (t,x)2 Y←right[x]3 Right[x]←left[y]4 p[left[y]]←x5 P[y]←p[x]6 ifp[x]==Nil[t]7 Then root[t]←y8     Else ifx=Left[p[x]]9 Then left[p[x]]←yTen         ElseRight[p[x]]←y One left[y]←x AP[x]←y

Insert

1rb-INSERT (t,z)2 Y←nil[t]3 X←root[t]4  whilex!=Nil[t]5      Doy←x6         ifkey[z]<Key[x]7 Then X←left[x]8             ElseX←right[x]9 P[z]←yTen ify=Nil[t] One Then root[t]←z A     Else ifkey[z]<Key[y] - Then left[y]←z -         Elseright[y]←z the Left[z]←nil[t] - Right[z]←nil[t] - color[z]←red -rb-insert-fixup (t,z) +  - //keep the red and black nature +rb-insert-fixup (t,z) A  whilecolor[p[x]]=RED at      Do ifp[z]=Left[p[p[z] ] - Then Y←right[p[p[z] ] -             ifcolor[y]=RED - Then color[p[z]]←black - Color[y]←black - color[p[p[z]]]←red in Z←p[p[z]] -                 Else ifz=Right[p[x]] to Then Z←p[x] +left-ROTATE (t,z) - Color[p[z]]←black the color[p[p[z]]]←red *right-ROTATE (t,p[p[z]) $             Else(Same asThen clause with" Right"and" Left"exchanged)Panax NotoginsengColor[root[t]]←black

Delete

1rb-DELETE (t,z)2 ifLeft[z]=nil[t] or right[z]=Nil[t]3 Then y←z4     Elsey←tree-successor (z)5 ifleft[y]!=Nil[t]6 Then X←left[y]7     ElseX←right[y]8 P[x]←p[y]9 ifp[y]=Nil[t]Ten Then root[t]←x One     Else ify=Left[p[y]] A Then left[p[y]]←x -         Elseright[p[y]]←x - ify!=Z the Then Key[x]←key[y] -Copy y's satellite data into Z - ifcolor[y]=BLACK -Then rb-delete-fixup (t,x) + returny -  + //keep the red and black nature Arb-delete-fixup (t,x) at  whileX!=root[t] and color[x]=BLACK -      Do ifx=Left[p[x]] - Thenω←right[p[x]] -             ifcolor[ω]=RED -Then color[ω]=BLACK - color[p[x]]←red inleft-ROTATE (t,p[x]) - Ω←right[p[x]] to             ifColor[left[ω]]=black and color[right[ω]]=BLACK + Then color[ω]←red - X←p[x] the                 Else ifcolor[right[ω]]=BLACK * Then color[left[ω]]←black $ color[ω]←redPanax Notoginsengright-ROTATE (t,ω) - Ω←right[p[x]] the Color[ω]←color[p[x]] + Color[p[x]]←black A Color[right[ω]]←black theleft-ROTATE (t,p[x]) + X←root[t] -         Else(Same asThen clause with" Right"and" Left"exchanged) $Color[x]←black

Introduction to Algorithms Learning Notes--13th chapter red and black Trees