Red-Black tree Rotation, Insertion, Deletion

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14.2-5 Show that any n-node tree can ve transformed to any other using O(n) rotations (hint: convert to a right going chain). I will start by showing weaker bounds - that O(n 2 ) and O(n log n) rotations suce - because that is how I proceeded when I rst saw the problem. First, observe that creating a right-going, for t2 path from t 1 and reversing the same construction gives a path from t 1 to t 2 . Note that it will take at most n rotations to make the lowest valued key the root. Once it is root, all keys are to the right of it, so no more rotations need go through it to create a right-going chain. Repeating with the second lowest key, third, etc. gives that O(n 2 ) rotations suce. Now that if we trytocreate a completely balanced tree instead. To getthe n=2 key to the root takes at most n rotations. Now eachsubtreehas half the nodes and we can recur... N N/2 N/2 N/4 N/4 N/4 N/4
To get a linear algorithm, wemust beware of trees like: 1 7 2 3 4 8 9 10 2 11 12 1 3 4 5 6 7 8 9 5 6 The correct answer is n ; 1 rotations suce to get to a rightmost chain. By picking the lowest node on the rightmost chain which has a left ancestor, we can add one node per rotation to the right most chain! zz y y x zz x z z Initially, the rightmost chain contained at least 1 node, so after n ; 1 rotations it contains all n. Slick!
Page 1 and 2: Rotations The basic restructuring s
Page 3: 7 4 11 x 3 6 9 18 y 2 14 19 12 17 2
Page 7 and 8: How can we x two reds in a row? It
Page 9 and 10: The Case of the Black Uncle If our
Page 11 and 12: Pseudocode and Figures
Page 13 and 14: Deletion Color Cases Suppose the no
Page 15: Conclusion Red-Black trees let us i

Red-Black tree Rotation, Insertion, Deletion

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