Description
1 Minimum – Maximum, 10 points
You are given a weighted graph G and two designated vertices s and t. Your goal
is to find a path from s to t in which the minimum edge weight is maximized,
i.e. if there are two paths with weights 10− > 1− > 5 and 2− > 7− > 3 then
the second path is considered better since the minimum weight (2) is greater
than the minimum weight of the first (1). Describe an efficient greedy algorithm
to solve this problem and show its complexity. Prove its correctness. (Note:
This problem is exactly the same problem to Pr. 8 in HW1.)
2 Unique MST, 10 points
Suppose you are given a connected graph G, with edge costs that are all distinct.
Prove that G has a unique minimum spanning tree.
3 Near-tree, 10 points
Let us say that a graph G = (V, E) is a near-tree if it is connected and has
at most n + 8 edges, where n = |V |. Give an algorithm with running time
O(n) that takes a near-tree G with costs on its edges, and returns a minimum
spanning tree of G. You may assume that all the edge costs are distinct.
4 Subsequence, 10 points
You are trying to find whether a sequence of characters is a subsequence of
another sequence. A subsequence of a string is obtained by deleting zero or
more symbols of that string. For example,
RACECAR
is a subsequence of
QRAECCET CAURP
Give a linear algorithm that takes two sequences S
0 of length m and S of length
n and decides whether S
0
is a subsequence of S. Prove its correctness.
1
5 Winter in Canada, 10 points
Your friends are planning an expedition to a small town deep in the Canadian
north next winter break. They’ve researched all the travel options and have
drawn up a directed graph whose nodes represent intermediate destinations and
edges represent the roads between them.
In the course of this, they’ve also learned that extreme weather causes roads in
this part of the world to become quite slow in the winter and may cause large
travel delays. They’ve found an excellent travel Web site that can accurately
predict how fast they’ll be able to travel along the roads; however, the speed of
travel depends on the time of year. More precisely, the Web site answers queries
of the following form: given an edge e = (v, w) connecting two sites v and w,
and given a proposed starting time t from location v, the site will return a value
fe(t), the predicted arrival time at w. The Web site guarantees that fe(t) ≥ t
for all edges e and all times t (you can’t travel backward in time), and that
fe(t) is a monotone increasing function of t (that is, you do not arrive earlier
by starting later). Other than that, the functions fe(t) may be arbitrary. For
example, in areas where the travel time does not vary with the season, we would
have fe(t) = t + le, where le is the time needed to travel from the beginning to
the end of edge e.
Your friends want to use the Web site to determine the fastest way to travel
through the directed graph from their starting point to their intended destination. (You should assume that they start at time 0, and that all predictions
made by the Web site are completely correct.) Give a polynomial-time algorithm to do this, where we treat a single query to the Web site (based on a
specific edge e and a time t) as taking a single computational step.
6 Shortest Path, 10 points
Given a directed graph G = (V, E) with nonnegative edge weights and the
shortest path distances d(s, u) from a source vertex s to all other vertices in
G. However, you are not given the shortest path tree. Devise a linear time
algorithm, to find a shortest path from s to a given vertex t.
7 Worst Gas Mileage in the World, 10 points
Suppose you were to drive from USC to Santa Monica along I-10. Your gas
tank, when full, holds enough gas to go p miles, and you have a map that
contains the information on the distances between gas stations along the route.
Let d1 < d2 < . . . < dn be the locations of all the gas stations along the route
where di
is the distance from USC to the gas station. We assume that the
distance between neighboring gas stations is at most p miles. Your goal is to
make as few gas stops as possible along the way. Design a greedy algorithm to
determine at which gas stations you should stop and prove that your strategy
yields an optimal solution. Prove its correctness.
2
8 MST, 10 points
You are given a minimum spanning tree T in a graph G = (V, E). Suppose we
remove an edge from G creating a new graph G1. Assuming that G1 is still
connected, devise a linear time algorithm to find a MST in G1.
9 Recurrences, 10 points
Solve the following recurrences by the master theorem.
1. T(n) = 9T(n/3) + n + log n.
2. T(n) = 3T(n/4) + n log n.
3. T(n) = 8T(n/2) + (n + 1)2 − 10n.
4. T(n) = T(2n/3) + 1.
5. T(n) = √
7T(n/2) + n
√
3
.
10 More on Recurrences, 10 points
The recurrence T(n) = 7T(n/2)+n
2 describes the running time of an algorithm
ALG. A competing algorithm ALG0
has a running time of T
0
(n) = aT0
(n/4) +
n
2
log n. What is the largest value of a such that ALG0
is asymptotically faster
than ALG?
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