Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

C-10.8
Let D be an ordered dictionary with n entries. Show how to modify the AVL
tree to implement the following method for D in time O(logn):
countAllInRange(k 1 ,k 2 ): Compute and return the number of entries in D
with key k such that k 1 ≤ k ≤ k 2 .
C-10.9
Show that the nodes that become unbalanced in an AVL tree after operation
insertAtExternal is performed, within the execution of an insert
operation, may be nonconsecutive on the path from the newly inserted node to
the root.
C-10.10
Show that at most one node in an AVL tree becomes unbalanced after operation
removeExternal is performed within the execution of a remove
dictionary operation.
C-10.11
Show that at most one trinode restructuring operation is needed to restore
balance after any insertion in an AVL tree.
C-10.12
Let T and U be (2,4) trees storing n and m entries, respectively, such that all the
entries in T have keys less than the keys of all the entries in U. Describe an
O(logn + logm) time method for joining Tand U into a single tree that stores all
the entries in T and U.
C-10.13
Repeat the previous problem for red-black trees T and U.
C-10.14
Justify Proposition 10.7.
C-10.15
The Boolean indicator used to mark nodes in a red-black tree as being "red" or
"black" is not strictly needed when we have distinct keys. Describe a scheme for
implementing a red-black tree without adding any extra space to standard binary
search tree nodes. How does your scheme affect the search and update times?
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