# Question

There are four basic operations on red-black trees that perform structural modifications: node insertions, node deletions, rotations, and color modifications. We have seen that RB-INSERT and RB-DELETE use only O (1) rotations, node insertions, and node deletions to maintain the red-black properties, but they may make many more color modifications.

a. Describe a legal red-black tree with n nodes such that calling RB-INSERT to add the (n + 1) st node causes Ω (lg n) color modifications. Then describe a legal red-black tree with n nodes for which calling RB-DELETE on a particular node causes Ω (lg n) color modifications.

Although the worst-case number of color modifications per operation can be logarithmic, we shall prove that any sequence of m RB-INSERT and RB-DELETE operations on an initially empty red-black tree causes O (m) structural modifications in the worst case.

b. Some of the cases handled by the main loop of the code of both RB-INSERT-FIXUP and RB-DELETE-FIXUP are terminating: once encountered, they cause the loop to terminate after a constant number of additional operations.

For each of the cases of RB-INSERT-FIXUP and RB-DELETE-FIXUP, specify which are terminating and which are not. We shall first analyze the structural modifications when only insertions are performed. Let T be a red-black tree, and define Φ (T) to be the number of red nodes in T . Assume that 1 unit of potential can pay for the structural modifications performed by any of the three cases of RBINSERT- FIXUP.

c. Let T′ be the result of applying Case 1 of RB-INSERT-FIXUP to T. Argue that Φ (T′) = Φ(T) - 1.

d. Node insertion into a red-black tree using RB-INSERT can be broken down into three parts. List the structural modifications and potential changes resulting from lines 1–16 of RB-INSERT, from non-terminating cases of RB-INSERT-FIXUP, and from terminating cases of RB-INSERT-FIXUP.

e. Using part (d), argue that the amortized number of structural modifications performed by any call of RB-INSERT is O (1).

We now wish to prove that there is O (m) structural modifications when there are both insertions and deletions. Let us define, for each node x, Now we redefine the potential of a red-black tree T as and let

T′ be the tree that results from applying any non-terminating case of RB-INSERTFIXUP or RB-DELETE-FIXUP to T.

f. Show that Φ (T′) ≤ Φ (T) - 1 for all non-terminating cases of RB-INSERT-FIXUP. Argue that the amortized number of structural modifications performed by any call of RB-INSERT-FIXUP is O (1).

g. Show that Φ (T′) ≤ Φ (T) - 1 for all non-terminating cases of RB-DELETE-FIXUP. Argue that the amortized number of structural modifications performed by any call of RB-DELETE-FIXUP is O (1).

h. Complete the proof that in the worst case, any sequence of m RB-INSERT and RBDELETE operations performs O (m) structural modifications.

a. Describe a legal red-black tree with n nodes such that calling RB-INSERT to add the (n + 1) st node causes Ω (lg n) color modifications. Then describe a legal red-black tree with n nodes for which calling RB-DELETE on a particular node causes Ω (lg n) color modifications.

Although the worst-case number of color modifications per operation can be logarithmic, we shall prove that any sequence of m RB-INSERT and RB-DELETE operations on an initially empty red-black tree causes O (m) structural modifications in the worst case.

b. Some of the cases handled by the main loop of the code of both RB-INSERT-FIXUP and RB-DELETE-FIXUP are terminating: once encountered, they cause the loop to terminate after a constant number of additional operations.

For each of the cases of RB-INSERT-FIXUP and RB-DELETE-FIXUP, specify which are terminating and which are not. We shall first analyze the structural modifications when only insertions are performed. Let T be a red-black tree, and define Φ (T) to be the number of red nodes in T . Assume that 1 unit of potential can pay for the structural modifications performed by any of the three cases of RBINSERT- FIXUP.

c. Let T′ be the result of applying Case 1 of RB-INSERT-FIXUP to T. Argue that Φ (T′) = Φ(T) - 1.

d. Node insertion into a red-black tree using RB-INSERT can be broken down into three parts. List the structural modifications and potential changes resulting from lines 1–16 of RB-INSERT, from non-terminating cases of RB-INSERT-FIXUP, and from terminating cases of RB-INSERT-FIXUP.

e. Using part (d), argue that the amortized number of structural modifications performed by any call of RB-INSERT is O (1).

We now wish to prove that there is O (m) structural modifications when there are both insertions and deletions. Let us define, for each node x, Now we redefine the potential of a red-black tree T as and let

T′ be the tree that results from applying any non-terminating case of RB-INSERTFIXUP or RB-DELETE-FIXUP to T.

f. Show that Φ (T′) ≤ Φ (T) - 1 for all non-terminating cases of RB-INSERT-FIXUP. Argue that the amortized number of structural modifications performed by any call of RB-INSERT-FIXUP is O (1).

g. Show that Φ (T′) ≤ Φ (T) - 1 for all non-terminating cases of RB-DELETE-FIXUP. Argue that the amortized number of structural modifications performed by any call of RB-DELETE-FIXUP is O (1).

h. Complete the proof that in the worst case, any sequence of m RB-INSERT and RBDELETE operations performs O (m) structural modifications.

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