Problem Set #3: Heaps, Sorting, and MST
Handed out on Tues, Mar 4, due on Tues, Mar 11 at the beginning of class. Remember: write your
own answers and use English or pseudocode when algorithms are requested. Late homeworks will
not be accepted (turn in whatever you have).
Problem 0. Consider the two heaps shown below.
2
8 12
17 13 27
21
23
50
0
11 10
18 15 4032
4122 57 33
Figure 1: Heaps for problem 3.
1. Label each of the above heaps with their npl values.
2. Show the result of merging these two heaps assuming they are leftist heaps.
3. Show the result of merging them assuming they are skew heaps.
Problem 1. Describe a data structure that will allow you to delete any item in O(log n) time
given only its key, while still supporting extract min and insert in O(log n) time.
Problem 2. Explain why it is not possible to design a data structure that is guaranteed to be able
to perform a series of nmakeheap,insert,extract min operations in amortized constant time.
In other words, the total time for the noperations cannot be bounded by O(n).
Problem 3. If eis some maximum-weight edge on a cycle of G, argue why it is always possible to
construct a minimum spanning tree that does not contain edge e.
Extra Credit Problem. Explain how to modify AVL trees to implement a List ADT that
supports the following operations. Each operation should run in O(log n) time, where nis the
number of elements currently in the list. There are no “gaps” allowed in the list: if it contains an
item at index ithen it contains items at indices 0 . . . i −1.
a. access (L, i): return the ith item in the list.
b. insert(L, i, k): insert value kat the ith position in the list, shifting items at indices i+ 1
and above up by one.
c. delete(L, i): remove the ith item in the list and shift all items at indices i+ 1 and above
down by one.
You solution should consist of an explanation (English or pseudocode) for how to implement each
of the above operations.
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