Red Black Trees

1. Red Black Trees CSC 172 SPRING 2002 LECTURE 18

2. Red Black Tree BST with a coloring convention for each element Nodes are colored according to rules One rule involves paths Specifically, paths from nodes with no children or one child

3. Example 30 5 45 40 50 9 2 41

4. Two Rules Red Rule If an element is red, all of it’s children are black Path Rule The number of black elements must be the same in all paths from the root element to element with no children or with one child

5. Example 30 5 45 40 50 9 2 41

6. Height of a Red-Black Tree Claim: Let y be the root of a subtree of a red-black tree. The number of black elements is the same in a path from y to any one of its descendants with no child or one child.

7. b0 b1 b2 General case x In general, b0+b1=b0+b2 y So, b1=b2 We define blackHeight(z) = bh(z) = The number of black elements in any path from z to any descendant with 0 or 1 child z1 z2

8. Height of a red-black tree For any nonempty subtree of a red-black tree

9. Basis height(t) = 0 n(t) = 0 if the root is red n(t) = 1 if the root is black Either way 1 >= 2 bh(root(t)) - 1

10. Induction case 1 Let k be any nonnegative integer BTIH n(t) >= 2 bh(root(t)) - 1 for height(t) <= k If the root of t has one child, we must have bh(root) = 1

11. Induction case 2 The root of t has two children, (v1,v2) If the root is red bh(root) = bh(v1) = bh(v2) If the root is black bh(root) = bh(v1) + 1 = bh(v2) + 1 Either way bh(v1) >= bh(root(t)) – 1 bh(v2) >= bh(root(t)) – 1

12. Induction case 2 BTIH: n(leftTree(t)) >= 2 bh(v1) – 1 n(rightTree(t)) >= 2 bh(v2) – 1 The number of elements in t is one more than the number of elements in leftTree(t) + rightTree(t)

13. Ergo

14. Finally For any red-black tree with n elements height(t) is O(log n) By the red rule, at most half of the elements in a path from the root to the farthest leaf can be red. So, at least half those elements must be black bh(root(t)) >= height(t)/2 n(t) >= 2bh(root(t)) – 1 n(t) >= 2height(t)/2 – 1 So, height(t) <= 2 log2(n(t)+1)

15. AVL vs RB AVL : height < 1.75 log2(n) Red-Black height <= 2 log2(n+1) So, AVLs are “bushier” that red-black. However, maintaining a red-black is simpler Which is why Java’s TreeSet class uses red-black

16. Helpers private static final boolean RED = false; private static final boolean BLACK= true; private static boolean colorOf(Entry p) { return (p==null?BLACK:p.color); } private static void setColor(Entry p,boolean c) { if(p!=null) p.color – c; } private static Entry parentOf(Entry p) { return (p==null?null:p.parent); }

17. Insertions on red-black trees FixAfterInsertion • Let t be (ref) the parent of the new insertion • Create a new Entry object pointer to by t.left or t.right • Set new Entry’s fields • Recolor and restructure • Set the root to BLACK

18. Insertions on red-black trees Suppose we insert Enty x We set x’s color to red Do we need to recolor & rotate? If x is root, “no” If x’s parent is BLACK, “no” So, loop while(x!= root && x.parent.color == RED)

19. Aunt’s & Uncles Because of rotation we need to consider the color of the sibling of x’s parent. When x is parent’s left child: y = x.parent.parent.right; // could be null

20. 45 40 50 x y x 41 Case 1:colorOf(y) == RED setColor(parentOf(x),BLACK); setColor(y,BLACK); setColor(parentOf(parentOf(x)),RED) x=parentOf(x); //keep looping 45 40 50 41

21. 45 Y is null 41 40 x x Case 2:colorOf(y) = BLACK && x is RC x=parentOf(x); rotateLeft(x); 45 40 41

22. Y is null 41 45 x x 40 Case 3:colorOf(y) = BLACK && x is LC setColor(parentOf(x),BLACK); setColor(parentOf(parentOf(x)),RED); if(parentOf(parentOf(x) )!= null) rotateRight(parentOf(parentOf(x)); 45 41 40