From (2,4) to Red-Black Trees
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1 Red-Black Trees 3/0/1 Presentation for use ith the textbook Data Structures and Algorithms in Jaa, th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldasser, Wiley, 01 Red-Black Trees Goodrich, Tamassia, Goldasser Red-Black Trees 1 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 n same logarithmic time performance n simpler implementation ith a single node type OR 5 01 Goodrich, Tamassia, Goldasser Red-Black Trees 1
2 Red-Black Trees 3/0/1 Red-Black Trees! A red-black tree can also be defined as a binary search tree that satisfies the folloing properties: n Root Property: the root is black n External Property: eery leaf is black n Internal Property: the children of a red node are black n Depth Property: all the leaes hae the same black depth Goodrich, Tamassia, Goldasser Red-Black Trees 3 Height of a Red-Black Tree! Theorem: A red-black tree storing n items has height O(log n) Proof: n 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 01 Goodrich, Tamassia, Goldasser Red-Black Trees
3 Red-Black Trees 3/0/1 Insertion! To insert (k, o), e execute the insertion algorithm for binary search trees and color red the nely inserted node unless it is the root n We presere the root, external, and depth properties n If the parent of is black, e also presere the internal property and e are done n 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, Goldasser Red-Black Trees 5 Remedying a Double Red! Consider a double red ith child and parent, and let be the sibling of Case 1: is black n The double red is an incorrect replacement of a -node n Restructuring: e change the -node replacement Case : is red n The double red corresponds to an oerflo n Recoloring: e perform the equialent of a split Goodrich, Tamassia, Goldasser Red-Black Trees 3
4 Red-Black Trees 3/0/1 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, Goldasser Red-Black Trees Restructuring (cont.)! There are four restructuring configurations depending on hether the double red nodes are left or right children 01 Goodrich, Tamassia, Goldasser Red-Black Trees 8
5 Red-Black Trees 3/0/1 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 01 Goodrich, Tamassia, Goldasser Red-Black Trees 9 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 n O(log n) recolorings, each taking O(1) time, and n at most one restructuring taking O(1) time! Thus, an insertion in a redblack tree takes O(log n) time 01 Goodrich, Tamassia, Goldasser Red-Black Trees 10 5
6 Red-Black Trees 3/0/1 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 n If either of r as red, e color r black and e are done n 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 01 Goodrich, Tamassia, Goldasser Red-Black Trees 11 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 n We perform a restructuring, equialent to a transfer, and e are done Case : y is black and its children are both black n We perform a recoloring, equialent to a fusion, hich may propagate up the double black iolation Case 3: y is red n 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 01 Goodrich, Tamassia, Goldasser Red-Black Trees 1
7 Red-Black Trees 3/0/1 Red-Black Tree Reorganiation Insertion remedy double red Red-black tree action (,) tree action result restructuring recoloring Deletion change of -node representation split remedy double black Red-black tree action (,) tree action result double red remoed double red remoed or propagated up restructuring transfer double black remoed recoloring adjustment fusion change of 3-node representation double black remoed or propagated up restructuring or recoloring follos 01 Goodrich, Tamassia, Goldasser Red-Black Trees 13 Jaa Implementation 01 Goodrich, Tamassia, Goldasser Red-Black Trees 1
8 Red-Black Trees 3/0/1 Jaa Implementation, 01 Goodrich, Tamassia, Goldasser Red-Black Trees 15 Jaa Implementation, 3 01 Goodrich, Tamassia, Goldasser Red-Black Trees 1 8
9 Red-Black Trees 3/0/1 Jaa Implementation, 01 Goodrich, Tamassia, Goldasser Red-Black Trees 1 9
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