# 2.7 Discussion and Exercises

Most of the data structures described in this chapter are folklore. They can be found in implementations dating back over 30 years. For example, implementations of stacks, queues, and deques, which generalize easily to the , and structures described here, are discussed by Knuth [46, Section 2.2.2].

Brodnik et al.  seem to have been the first to describe the and prove a lower-bound like that in Section 2.6.2. They also present a different structure that uses a more sophisticated choice of block sizes in order to avoid computing square roots in the method. Within their scheme, the block containing is block , which is simply the index of the leading 1 bit in the binary representation of . Some computer architectures provide an instruction for computing the index of the leading 1-bit in an integer.

A structure related to the is the two-level tiered-vector of Goodrich and Kloss . This structure supports the and operations in constant time and and in time. These running times are similar to what can be achieved with the more careful implementation of a discussed in Exercise 2.10.

Exercise 2..1   The method inserts all elements of the  into the list at position . (The method is a special case where .) Explain why, for the data structures in this chapter, it is not efficient to implement by repeated calls to . Design and implement a more efficient implementation.

Exercise 2..2   Design and implement a . This is an implementation of the interface in which the operation removes an element that is chosen uniformly at random among all the elements currently in the queue. (Think of a as a bag in which we can add elements or reach in and blindly remove some random element.) The and operations in a should run in constant time per operation.

Exercise 2..3   Design and implement a (triple-ended queue). This is a implementation in which and run in constant time and and run in time In other words, modifications are fast if they are near either end or near the middle of the list.

Exercise 2..4   Implement a method that rotates'' the array so that moves to , for all .

Exercise 2..5   Implement a method that rotates'' a so that list item becomes list item . When run on an , or a , should run in time.

Exercise 2..6   Modify the implementation so that the shifting done by , , and is done using the faster method.

Exercise 2..7   Modify the implementation so that it does not use the operator (which is expensive on some systems). Instead, it should make use of the fact that, if is a power of 2, then (Here, is the bitwise-and operator.)

Exercise 2..8   Design and implement a variant of that does not do any modular arithmetic at all. Instead, all the data sits in a consecutive block, in order, inside an array. When the data overruns the beginning or the end of this array, a modified operation is performed. The amortized cost of all operations should be the same as in an .

Hint: Getting this to work is really all about how you implement the operation. You would like to put the data structure into a state where the data cannot run off either end until at least operations have been performed.

Test the performance of your implementation against the . Optimize your implementation (by using ) and see if you can get it to outperform the implementation.

Exercise 2..9   Design and implement a version of a that has only wasted space, but that can perform and operations in time.

Exercise 2..10   Design and implement a version of a that has only wasted space, but that can perform and operations in time. (For an idea on how to do this, see Section 3.3.)

Exercise 2..11   Design and implement a version of a that has only wasted space, but that can perform and operations in time. (See Section 3.3 for ideas on how to achieve this.)

Exercise 2..12   Design and implement a . This three level structure implements the interface using wasted space. In this structure, and take constant time; while and take amortized time.

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