Work Note 2, DM803, fall 2016

Exercises September 12

1. Show that there is at most a logarithmic number of nodes on the right-most path in a leftist heap with n element.
   Hint: Show that a node of rank r has exponentially many nodes in its subtree. Then consider how ranks behave on the right-most path.

2. Show that two standard heaps (as known from DM507) with 2^p-1 elements each plus one extra element x can be combined into one heap in time O(p).
   Show that a standard heap can be build out of n=2^p-1 elements in time O(n) by a bottom-up application of the result above.
   Given n elements, how can a leftist heap be constructed in time O(n)?

3. How can the operations decreasekey, delete, and update be implemented to run in time O(log n) for leftist heaps? It is assumed that you are given a reference to the element and you may introduce additional references in the structure.
   Hint: Some operations can be reduced to others.

4. [CLRS09] 17.3-3, 17.3-6, 17.3-7.
   Hint: in 17.3-7, you may assume that the median of n elements can be found in time O(n).

5. Come up with an implementation of a queue (not priority queue), where, in addition to the usual operations enqueue and dequeue, there is an operation findmin that reports the minimum element in the queue (assuming we have a total ordering on the elements). All operations should be O_A(1).

6. Show the claim from the lecture that there are at most a logarithmic number of light nodes on the right-most path of a skew heap.

7. For skew heaps, what if we define heavy as |T_r(x)| ≥ ¾ |T_x|? Or |T_r(x)| ≥ ¼ |T_x|? How does that affect the analysis?

8. One can probably easily imagine scenarios where leftist heaps are preferable to skew heaps, but what if you are just interested in the final result of running an algorithm on n elements and during the course of the algorithm, one has to merge priority queues n times. Which priority queue implementation do you think would be best?

9. Here is an alternative to leftist heaps: As for leftist heaps, we consider binary heap-ordered trees. However, instead of rank, each node has a size attribute, storing the size of the subtree in which it is a root. Just as for leftist heaps, we define meld, and then all other operations follow from that.
   The meld operation is defined as follows. Assume that we want to meld X and Y, and assume that X has the smallest key k in the root (otherwise we just switch the two arguments) and that its two subtrees are X_left and X_right. Now sort the trees X_left, X_right, and Y in non-decreasing order of size and name them A, B, and C, i.e., A.size ≤ B.size ≤ C.size. Make a node that has k in the root and the two subtrees meld(A,B) and C. Consider correctness and complexity.

Lecture September 15

• Scapegoat Trees [GR93].

Announcements

• In general, the exercises are not ordered according to difficulty. Thus, harder exercises may be followed by easier ones. In preparation for exercis classes, go through all of them, think about them, and try to solve them. But if you don't manage to solve an exercise, just proceed to the next one. It's important to go through the process of problem solving, thereby fully understanding the problems and where it becomes difficult before seeing the answers in class.