Work Note 12, DM803, fall 2014

Lecture November 25

• Level Ancestors [BF03].

Exercises November 26

1. This is an open exercise problem; not necessarily in the sense that it is unknown in general, but I do not have the answer: How unbalanced can you make a splay tree? You could answer this for small n; possible by writing a program and performing some sort of exhaustive search. Can one arrange imbalance for families of access sequences for growing n? Note that this is starting with an empty tree and applying insertions and accesses, and that a newly inserted node is also splayed to the top.
2. Show that the splay operations can be formulated as at most two single rotations.
3. Compute the potential of a perfectly balanced tree as well as the potential of a tree which has deteriorated to a linked list.
4. At the lecture, we did the amortized analysis of only one of the splay operations; the one where x was the left child of a left child. Complete the analysis of the other operations as well.
5. We argued briefly in class that a single rotation preserves the search tree invariant. Argue more broadly that this is also the case for the following: Consider any connected subgraph of nodes in a search tree (for the single rotation, the subgraph has two nodes). Now, remove all trees hanging off nodes in this subgraph in an in-order fashion and, separately, all the keys of nodes of the subgraph in an in-order fashion. Then rearrange the subgraph into any other connected subgraph. Add the keys to the nodes in-order and attach the trees again, also inorder. Argue that the tree is a search tree.
6. For Y-Fast Tries, we had to keep the sizes of the small red-black trees in the range [(log M) / 4, 2 log M]. Thus, for deletion, when a tree got too small, we would merge it with a neighbor. If this merge was then above or too close to the high end of the allowed interval, we would split into two even-sized trees (±1). Choose when to split with the aim of maximizing the worst-case distance to one of the extreme points of the allowed interval for any of the up to two trees involved that will be created. (In the lecture, we arbitrarily chose to split if the size exceeded (3/2) log M.)

Announcements

• Due to the cancellations last week, this weekly note is identical to the previous one; with the obvious exceptions... ツ
• Please see the job announcement if you are interested in becoming a teaching assistant ("instruktor") in the Spring. The application deadline is November 28, 2014.