Red-Black Trees Goodrich, Tamassia. Red-Black Trees 1

Transcription

1 Red-Black Trees Goodrich, Tamassia Red-Black Trees 1

2 From (,) to Red-Black Trees A red-black tree is a representation of a (,) tree by means of a binary tree hose nodes are colored red or black In comparison ith its associated (,) tree, a red-black tree has same logarithmic time performance simpler implementation ith a single node type OR Goodrich, Tamassia Red-Black Trees

3 Red-Black Trees A red-black tree can also be defined as a binary search tree that satisfies the folloing properties: Root Property: the root is black External Property: eery leaf is black Internal Property: the children of a red node are black Depth Property: all the leaes hae the same black depth Goodrich, Tamassia Red-Black Trees 3

4 Height of a Red-Black Tree Theorem: A red-black tree storing n entries has height O(log n) Proof: The height of a red-black tree is at most tice the height of its associated (,) tree, hich is O(log n) The search algorithm for a binary search tree is the same as that for a binary search tree By the aboe theorem, searching in a red-black tree takes O(log n) time 00 Goodrich, Tamassia Red-Black Trees

5 Insertion To perform operation insert(k, o), e execute the insertion algorithm for binary search trees and color red the nely inserted node unless it is the root We presere the root, external, and depth properties If the parent of is black, e also presere the internal property and e are done Else ( is red ) e hae a double red (i.e., a iolation of the internal property), hich requires a reorganiation of the tree Example here the insertion of causes a double red: Goodrich, Tamassia Red-Black Trees 5

6 Remedying a Double Red Consider a double red ith child and parent, and let be the sibling of Case 1: is black The double red is an incorrect replacement of a -node Restructuring: e change the -node replacement Case : is red The double red corresponds to an oerflo Recoloring: e perform the equialent of a split Goodrich, Tamassia Red-Black Trees

7 Restructuring A restructuring remedies a child-parent double red hen the parent red node has a black sibling It is equialent to restoring the correct replacement of a -node The internal property is restored and the other properties are presered Goodrich, Tamassia.... Red-Black Trees 7

8 Restructuring (cont.) There are four restructuring configurations depending on hether the double red nodes are left or right children 00 Goodrich, Tamassia Red-Black Trees 8

9 Recoloring A recoloring remedies a child-parent double red hen the parent red node has a red sibling The parent and its sibling become black and the grandparent u becomes red, unless it is the root It is equialent to performing a split on a 5-node The double red iolation may propagate to the grandparent u Goodrich, Tamassia Red-Black Trees 9

10 Analysis of Insertion Algorithm insert(k, o) 1. We search for key k to locate the insertion node. We add the ne entry (k, o) at node and color red 3. hile doublered() if isblack(sibling(parent())) restructure() return else { sibling(parent() is red } recolor() Recall that a red-black tree has O(log n) height Step 1 takes O(log n) time because e isit O(log n) nodes Step takes O(1) time Step 3 takes O(log n) time because e perform O(log n) recolorings, each taking O(1) time, and at most one restructuring taking O(1) time Thus, an insertion in a redblack tree takes O(log n) time 00 Goodrich, Tamassia Red-Black Trees 10

11 Deletion To perform operation remoe(k), e first execute the deletion algorithm for binary search trees Let be the internal node remoed, the external node remoed, and r the sibling of If either of r as red, e color r black and e are done Else ( and r ere both black) e color r double black, hich is a iolation of the internal property requiring a reorganiation of the tree Example here the deletion of 8 causes a double black: 3 8 r 3 r 00 Goodrich, Tamassia Red-Black Trees 11

12 Remedying a Double Black The algorithm for remedying a double black node ith sibling y considers three cases Case 1: y is black and has a red child We perform a restructuring, equialent to a transfer, and e are done Case : y is black and its children are both black We perform a recoloring, equialent to a fusion, hich may propagate up the double black iolation Case 3: y is red We perform an adjustment, equialent to choosing a different representation of a 3-node, after hich either Case 1 or Case applies Deletion in a red-black tree takes O(log n) time 00 Goodrich, Tamassia Red-Black Trees 1

13 Red-Black Tree Reorganiation Insertion Red-black tree action restructuring recoloring Deletion Red-black tree action restructuring recoloring adjustment remedy double red (,) tree action change of -node representation split remedy double black (,) tree action transfer fusion change of 3-node representation result double red remoed double red remoed or propagated up result double black remoed double black remoed or propagated up restructuring or recoloring follos 00 Goodrich, Tamassia Red-Black Trees 13