Description
1. (25 pt) Given an undirected weighted graph G with n nodes and m edges, and we have used Prim algorithm
to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed
from w to w
0
, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm
should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are
distinct.
2. (21 pt) Give the time complexity for the following divide-and-conquer algorithms:
• Algorithm A solves the problem by dividing it into 9 subproblems of one third the size, recursively
solves each subproblem, and then combines the solutions in quadratic time.
• Algorithm B solves the problem of size n by recursively solving two subproblems of size n − 1, and
then combine the solutions in constant time.
• Algorithm C solves the problem by dividing it into 5 subproblems of half the size, recursively solves
each subproblem, and then combines the solutions in linear time.
3. (25pt) Consider an n-node complete binary tree T, where n = 2
d −1 for some d. Each node v of T is labaled
with a real number xv
. You may assume that the real numbers labeling the nodes are all distinct. A node
v of T is a local minimum if the label xv
is less than the label xw for all nodes w that are joined to v by an
edge.
You are given such a complete binary tree T, but the labeling is only specified in the following implicit way:
for each node v, you can determine the value xv by probing the node v. Show how to find a local minimum
of T using only O(log n) probes to the nodes of T. Explain why your algorithm is correct.
4. (29pt) Given a list of intervals [si
, f
i
] for i = 1, . . . , n, please design a divide-and-conquer algorithm that
returns the length of the largest overlap among all pairs of intervals. The algorithm should run in O(n log n)
time. For example, if n = 3, the intervals are {[1, 6], [2, 7], [6, 8]}, then the length of the largest overlap is
4. (Hint: sort the intervals by si values, and divide them into two parts based on their si values).
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manner. Sloppy answers are expected to receiver fewer points.
Æ Unless specified, you should justify your algorithm with proof of correctness and time complexity.
