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Red-Black Trees

Learn about Red-Black Trees: properties, insertion, restructuring, recoloring, conversion from 2-3-4 trees, and analysis of insertion algorithm.

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Red-Black Trees

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  1. Red-Black Trees 6 v 8 3 z 4 Red-Black Trees

  2. A red-black (RB) tree is a representation of a 2-3-4 tree by means of a binary tree whose nodes are colored red or black In comparison with its associated 2-3-4 tree, a red-black tree has Same logarithmic time performance Easier to implement (with a single node type) Harder to understand 4 2 6 7 3 5 4 From 2-3-4 to Red-Black Trees 5 3 6 OR 3 5 2 7

  3. Red-Black Trees • An RB tree is a BST with the following properties: • Root Property: the root is black • External Property: every leaf is black • Internal Property: the children of a red node are black  No path from the root to the leaf has two consecutive red nodes. • Depth Property: all the leaves have the same black depth  Each of the paths from the root to the leaves have the same number of black nodes. 9 4 15 21 2 6 12 7 Red-Black Trees

  4. More about RB Trees • Insertion • Nodes are inserted in the same way as a BST • Tree is updated by restructuring and recoloring Red-Black Trees

  5. Height of a Red-Black Tree • Theorem: A red-black tree storing nentries has height O(log n) Proof: • The height of a red-black tree is at most twice the height of its associated 2-3-4 tree, which is O(log n) • The search algorithm for a binary search tree is the same as that for a binary search tree • By the above theorem, searching in a red-black tree takes O(log n) time How about an alternative proof without using 2-3-4 trees? Red-Black Trees

  6. Conversion between 2-3-4 and RB Trees 2-3-4 tree Red-black trees Red-Black Trees

  7. To perform operation put(k, o), we execute the insertion algorithm for binary search trees and color red the newly inserted node z unless it is the root We preserve the root, external, and depth properties If the parent v of z is black, we also preserve the internal property and we are done Else (v is red ) we have a double red (i.e., a violation of the internal property), which requires a reorganization of the tree Example where the insertion of 4 causes a double red: Insertion 6 6 v v 8 8 3 3 z z 4 The inserted node is always red, why?

  8. Remedying a Double Red • Consider a double red with child z and parent v, and let w be the sibling of v Case 1: w is black • The double red is an incorrect replacement of a 4-node • Restructuring: we change the 4-node replacement Case 2: w is red • The double red corresponds to an overflow • Recoloring: we perform the equivalent of a split 4 4 w v v w 7 7 2 2 z z 6 6 Uncle of z 4 6 7 2 4 6 7 .. 2 .. Red-Black Trees

  9. Restructuring (1/2) • A restructuring remedies a child-parent double red when the uncle of the inserted node is black • It is equivalent to restoring the correct replacement of a 4-node • The internal property is restored and the other properties are preserved z 6 4 v v w 7 7 2 4 z w 2 6 Uncle of z 4 6 7 4 6 7 .. 2 .. .. 2 .. Red-Black Trees

  10. 6 2 6 4 4 2 2 6 4 Restructuring (2/2) • There are four restructuring configurations depending on whether the double red nodes are left or right children 2 6 4 4 2 6 Red-Black Trees

  11. Recoloring • A recoloring remedies a child-parent double red when the uncle of the inserted node is red • The parent v and its sibling w become black and the grandparent u becomes red, unless it is the root  To keep the depth property • It is equivalent to performing a split on a 5-node • The double red violation may propagate to the grandparent u May propagate… 4 4 v v w w 7 7 2 2 z z 6 6 Uncle of z … 4 … 2 4 6 7 2 6 7 Red-Black Trees

  12. Summary of Insertion • Two cases of insertions • Uncle is black  Restructure • Takes O(1) • Could be a single or double rotations • Uncle is red  Recolor • Takes O(log n) since it may propagate • Corresponds to splitting a node in a 2-3-4 tree Red-Black Trees

  13. Example of Insertion (1/3) Insert 7 Insert 12 Restructure Insert 15 Recolor Insert 3 Insert 5 Red-Black Trees

  14. Example of Insertion (2/3) Insert 14 Restructure Insert 18 Recolor Red-Black Trees

  15. Example of Insertion (3/3) Insert 16 Restructure Double red! Insert 17 Cascaded restructure Recolor Red-Black Trees

  16. Resources of RB Trees • Insertion animation • Animation • Pay attention to the propagation… • More interactive animation • Insertion rule • Rationale behind insertion • Proof of logarithmic height (w/o using 2-3-4 trees) Red-Black Trees

  17. Analysis of Insertion Algorithmput(k, o) 1. We search for key k to locate the insertion node z 2. We add the new entry (k, o) at node z and color z red 3. whiledoubleRed(z) if isBlack(sibling(parent(z))) z  restructure(z) return else {sibling(parent(z) is red } z  recolor(z) • Recall that a red-black tree has O(log n) height • Step 1 takes O(log n) time because we visit O(log n) nodes • Step 2 takes O(1) time • Step 3 takes O(log n) time because we perform • O(log n) recolorings, each taking O(1) time, and • at most one restructuring taking O(1) time • Thus, an insertion in a red-black tree takes O(log n) time Red-Black Trees

  18. Deletion • To do erase(k), we first execute the deletion algorithm for BST • Let v be the internal node removed, w the external node removed, and r the sibling of w • If either v or r was red, we color r black and we are done • Else (v and r were both black) we color rdouble black, which is a violation of the depth property requiring a update of the tree • Example where the deletion of 8 causes a double black: 6 6 v r 8 3 3 r w 4 4 Red-Black Trees

  19. Remedying a Double Black • To remedy a double black node w with sibling y: Case 1: y is black and has a red child • We perform a restructuring, equivalent to a transfer , and we are done Case 2: y is black and its children are both black • We perform a recoloring, equivalent to a fusion, which may propagate up the double black violation Case 3: y is red • We perform an adjustment, equivalent to choosing a different representation of a 3-node, after which either Case 1 or Case 2 applies • Deletion in a red-black tree takes O(log n) time Red-Black Trees

  20. Either R or B Case 1:Restructuring Either R or B Either R or B

  21. Case 2: Recoloring

  22. Case 3: Adjustment

  23. Example of Deletion (1/3) Delete 3 Case 1 of restructuring (Transfer in 2-3-4 trees) Restructure Delete 12

  24. Example of Deletion (2/3) Delete 17 Delete 18 Delete 15 Recolor Case 2 of recoloring (Fusion in 2-3-4 trees)

  25. Example of Deletion (3/3) Case 3 of adjustment (A different rep. of 3-node ) Case 3 of adjustment (Fusion in 2-3-4 trees ) Delete 16 Recolor Adjust

  26. Case 1: Restructuring Case 1 of restructuring (Transfer in 2-3-4 trees) Restructure 7, 14, 16 5, 14, 16 Transfer 17, 18 17, 18 4, 5 x 4 7 15 15

  27. Case 2: Recoloring Case 2 of recoloring (Fusion in 2-3-4 trees) Recolor 5, 14, 16 5, 14 Fusion 15, 16 x 4 7 4 7 15 Red-Black Trees

  28. Case 3: Adjustment Case 3 of adjustment (A different rep. of 3-node ) Case 3 of adjustment (Fusion in 2-3-4 trees ) Recolor Adjust 5 5, 14 Fusion 7, 14 4 x 4 7

  29. Red-Black Tree Update

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