Red Black Trees Bottom Up Deletion Recall ordinary BST Delete 1 2 3 If node to be deleted is a leaf just delete it If node to be deleted has just one child replace it with that child splice If node to be deleted has two children replace the value in the node by its in order predecessor successor s value then delete the in order predecessor successor a recursive step 8 3 2007 UMBC CSMC 341 Red Black Trees 2 2 Bottom Up Deletion 1 2 8 3 2007 Do ordinary BST deletion Eventually a case 1 or case 2 deletion will be done leaf or just one child If deleted node U is a leaf think of deletion as replacing U with the NULL pointer V If U had one child V think of deletion as replacing U with V What can go wrong UMBC CSMC 341 Red Black Trees 2 3 Which RB Property may be violated after deletion 1 If U is Red Not a problem no RB properties violated 2 If U is Black If U is not the root deleting it will change the black height along some path 8 3 2007 UMBC CSMC 341 Red Black Trees 2 4 Fixing the problem Think of V as having an extra unit of blackness This extra blackness must be absorbed into the tree by a red node or propagated up to the root and out of the tree There are four cases our examples and rules assume that V is a left child There are symmetric cases for V as a right child 8 3 2007 UMBC CSMC 341 Red Black Trees 2 5 Terminology The node just deleted was U The node that replaces it is V which has an extra unit of blackness The parent of V is P The sibling of V is S Black Node Red or Black and don t care Red Node 8 3 2007 UMBC CSMC 341 Red Black Trees 2 6 Bottom Up Deletion Case 1 V s sibling S is Red Rotate S around P and recolor S P NOT a terminal case One of the other cases will now apply All other cases apply when S is Black 8 3 2007 UMBC CSMC 341 Red Black Trees 2 7 Case 1 Diagram P V S Rotate S around P P S V S P Recolor S P V 8 3 2007 UMBC CSMC 341 Red Black Trees 2 8 Bottom Up Deletion Case 2 V s sibling S is Black and has two Black children Recolor S to be Red P absorbs V s extra blackness 8 3 2007 If P is Red we re done it absorbed the blackness If P is Black it now has extra blackness and problem has been propagated up the tree UMBC CSMC 341 Red Black Trees 2 9 Case 2 diagram Recolor S P absorbs blackness P S V V P S Either extra Black absorbed by P or P now has extra blackness 8 3 2007 UMBC CSMC 341 Red Black Trees 2 10 Bottom Up Deletion Case 3 S is Black S s right child is RED Left child either color Rotate S around P Swap colors of S and P and color S s right child Black This is the terminal case we re done 8 3 2007 UMBC CSMC 341 Red Black Trees 2 11 Case 3 diagrams S P Rotate S around P S V P V S P V 8 3 2007 Swap colors of S P Color S s right child Black UMBC CSMC 341 Red Black Trees 2 12 Bottom Up Deletion Case 4 S is Black S s right child is Black and S s left child is Red Rotate S s left child around S Swap color of S and S s left child Now in case 3 8 3 2007 UMBC CSMC 341 Red Black Trees 2 13 Case 4 Diagrams P V P S V P Rotate S s left around S S V S 8 3 2007 UMBC CSMC 341 Red Black Trees 2 Swap colors of S and S s original left child 14 Top Down Deletion An alternative to the recursive bottom up deletion is top down deletion This method is iterative It moves down the tree only fixing things as it goes What is the goal of top down deletion 8 3 2007 UMBC CSMC 341 Red Black Trees 2 15 65 50 10 80 70 60 90 62 Perform the following deletions in the order specified Delete 90 Delete 80 Delete 70 8 3 2007 UMBC CSMC 341 Red Black Trees 2 16
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