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Lecture 22: The Rolling Stones, Masters of the Balanced Tree

CSC 213 – Large Scale Programming. Lecture 22: The Rolling Stones, Masters of the Balanced Tree. Red-Black Tree Properties. Root Property: Root node painted black External Property: Leaves are painted black Internal Property: Red node s’ children are black

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Lecture 22: The Rolling Stones, Masters of the Balanced Tree

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  1. CSC 213 – Large Scale Programming Lecture 22:The Rolling Stones, Masters of the Balanced Tree

  2. Red-Black Tree Properties • Root Property:Root node painted black • External Property: Leaves are painted black • Internal Property: Red nodes’ children are black • Depth Property: Leaves have identical blackdepth • Number of blackancestorsfor the node 9 4 15 21 2 6 12 7

  3. Insertion • Begins as BST insertion (just like splay & AVL) • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes

  4. Insertion • Begins as BST insertion (just like splay & AVL) • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes • Example: insert(3) 6 8

  5. Insertion • Begins as BST insertion (just like splay & AVL) • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes • Example: insert(3) 6 3 8

  6. Insertion • Begins as BST insertion (just like splay & AVL) • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes • Example: insert(3) 6 3 8

  7. Insertion • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes • If node’s parent isred, violates internal property • Must reorganize tree to remove double red • Example: insert(4) 6 3 8 4

  8. Insertion • New node’s initial color set by where it is • If node is root, paint it black • Other nodes colored red when insertion completes • If node’s parent isred, violates internal property • Must reorganize tree to remove double red • Example: insert(4)makes tree unbalanced 6 3 8 4

  9. Double Red With Red Aunt • Double red represents creation of 5-node • Perform recoloringto find if balance really off 6 8 3 4

  10. Double Red With Red Aunt • Parent & uncle painted black, grandparent red • When grandparent is root, must paint it black • When easier, promote 2nd Entryto parent • Must continue check, double red can propagate 4 7 2 6

  11. Double Red With Red Aunt • Parent & uncle painted black, grandparent red • When grandparent is root, must paint it black • When easier, promote 2nd Entryto parent • Must continue check, double red can propagate 4 4 7 7 2 2 6 6

  12. Double Red With Red Aunt • Parent & uncle painted black, grandparent red • When grandparent is root, must paint it black • When easier, promote 2nd Entryto parent • Must continue check, double red can propagate 4 4 7 7 2 2 6 6

  13. Double Red With BlackAunt • Poorly balanced tree causes this double red • Restore tree balance to use AVL tree restructuring • Preserves overall balance of the tree 6 8 3 4

  14. Double Red With BlackAunt • Rebalance tree using AVL tree restructuring • Recolors nodes also, but changes are very small 4 7 2 6

  15. Double Red With BlackAunt • Rebalance tree using AVL tree restructuring • Recolors nodes also, but changes are very small 6 4 7 7 2 4 2 6

  16. Double Red Restructuring • 4 different restructuresneeded to remedy • Differ in how node, parent, & grandparent related • Identical result no matter where we start 7 4 7 4 7 4 6 6 6 6 4 7

  17. Double Red Restructuring • 4 different restructuresneeded to remedy • Differ in how node, parent, & grandparent related • Identical result no matter where we start 7 4 7 4 7 4 6 6 6 6 4 7 6 4 7

  18. Deletion • Start with normal BST deletion • If Entry inred node or leafs’ssibling red • Leaf’s sibling is paintedblack • Example: remove(1) 6 8 3 1 4

  19. Deletion • Start with normal BST deletion • If Entry inred node or leafs’ssibling red • Leaf’s sibling is paintedblack • Example: remove(1) 6 8 3 1 4

  20. Deletion • Start with normal BST deletion • If Entry inred node or leafs’ssibling red • Leaf’s sibling is paintedblack • Example: remove(1) 6 8 3 4

  21. Deletion • Start with normal BST deletion • If Entry inred node or leafs’ssibling red • Leaf’s sibling is paintedblack • Example: remove(1) 6 8 3 4

  22. What’s Blacker Than Black? • If removed Entry & leaf’s siblingalready black • Paint sibling double black • This is an illegal state – violates internal property • Example: remove(8) 6 8 3 4

  23. What’s Blacker Than Black? • If removed Entry & leaf’s siblingalready black • Paint sibling double black • This is an illegal state – violates internal property • Example: remove(8) 6 8 3 4

  24. What’s Blacker Than Black? • If removed Entry & leaf’s siblingalready black • Paint sibling double black • This is an illegal state – violates internal property • Example: remove(8) 6 3 4

  25. What’s Blacker Than Black? • If removed Entry & leaf’s siblingalready black • Paint sibling double black • This is an illegal state – violates internal property • Example: remove(8) causesdouble black 6 3 4

  26. Remedying Double Black • Case 1: sibling is blackwithredchild • Reorder nodes using AVL tree restructure • Case 2: sibling and its children are black • Equal to (2,4) tree underflow, so recolor nodes

  27. Remedying Double Black • Case 1: sibling is blackwithredchild • Reorder nodes using AVL tree restructure • Case 2: sibling and its children are black • Just a bad balance, so just recolor nodes • Case 3: sibling is red • Adjust subtreeto better balance tree • Once complete apply case 1 or case 2

  28. Black Sibling With Red Niece • Solve double black using AVL restructuring 9 10 6 8

  29. Black Sibling With Red Niece • Solve double black using AVL restructuring 9 6 8

  30. Black Sibling With Red Niece • Solve double black using AVL restructuring 8 9 9 6 6 8

  31. Sibling & Children are Black • Solve double black recoloring parent & sibling 5 9 … 6 10

  32. Sibling & Children are Black • Solve double black recoloring parent & sibling 5 9 … 6

  33. Sibling & Children are Black • Solve double black recoloring parent & sibling 5 5 9 9 … … 6 6

  34. Sibling & Children are Black • Solve double black recoloring parent & sibling 5 5 9 9 … … 6 6

  35. Sibling is Red • Adjustingdouble blackstalls for time • Transforms situation into something we can fix 9 10 5 4

  36. Sibling is Red • Adjustingdouble blackstalls for time • Transforms situation into something we can fix 9 10 5 4

  37. Sibling is Red • Adjustingdouble blackstalls for time • Transforms situation into something we can fix 9 5 4

  38. Sibling is Red • Adjustingdouble blackstalls for time • Transforms situation into something we can fix 9 5 4

  39. Sibling is Red • Adjustingdouble blackstalls for time • Transforms situation into something we can fix • Once completed, re-examine double blacknode 9 5 5 4 9 4

  40. Do the Activity

  41. For Next Lecture • Weekly assignment available to test skills • Due at regular time Tuesday after break • Talk to me if struggling on problems • Last idea from Map/Dictionary after break • What are Sets? Why are we getting to them now? • It may not appear related, but how are they used? • Reminder: lab phase #2 due Monday after break

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