1 All Links before Finds
Suppose that we use the disjoint-set union data structure for a sequence of m Make-Set, Find-Set
and Link operations, but we are told that all of the Link operations are performed before any Find-
Set operation. (As usual, assume that there are n Make-Set operations.) Furthermore, suppose
that we implement the data structure using both Union-by-Rank and Path Compression. Show
that the total actual running time for all m operations is O(m).
Note: Recall that the Link operation is given the two roots of the trees to be merged. Thus, you
do not need to perform any Find-Set operations to execute a Link. (This is just an exercise. It
would be unclear how an application using this data structure would know which items are roots.)
Hint: Think about paying for the path compression during a Find-Set.
Note: There is no limit on the number of Find-Set operations. While we do know that m is the
total number of operations (including Find-Sets), you cannot assume a relationship between m and
the number of items n. I.e., the number of Find-Set operations could greatly exceed the number
of items n.
2 Offline Minimum
Do Problem 19-1 in CLRS (Problem 21-1 in the third edition), except in part a use the following
sequence of operations:
4 6 E 3 1 2 E 7 E 5 E 9 E 8 E E
Note: The premise of this problem is that you can turn the extract-min question around when the
problem is offline. Instead of asking
Which item will this call to Extract-Min return?
you ask
Which Extract-Min call will return this item?
For example, if the item in question is 1 (the smallest item), then it will be returned by some call
to Extract-Min (unless 1 is inserted after the last call to Extract-Min). So, the question is which
call will return 1.
