Work Note 3, DM803, fall 2016

Exercises September 19

1. Sometimes we are interested in having a next operation which, given a reference to an element, returns a reference to the element with the next higher key. How could this be implemented in a tree structure such as red-black trees, for example. How would it be done for skip lists? What about previous?
2. Consider the implementation of meld and split for skip lists, and how maxLevel should be adjusted to be close to L(n). Importantly, in order to compute L(n), we need to know n. How can you add information to the skip list structure and maintain this such that n can always be determined in expected time O(1)? Could we do something reasonable without n?
3. Prove that one gets the best constant for the expected search time by choosing p=1/e as claimed in Table 1 of [P89]. Then explain all the other constants in the table. You need to manipulate the logarithmic expressions some to reverse engineer the table data.
4. Consider a standard binary heap. Is it possible to make an O(n) traversal of the heap and print all the priorities in non-decreasing order during that traversal? Argue your position.
5. Leftover problems from last exercise class.

Lecture September 22 [Changed to Sept. 23]

• Universal hashing, [CLRS09] section 11.3.3.
• Perfect hashing, [CLRS09] section 11.5.