Algorithms Homework #3 solved


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1. Exercise 16.2-5 (page 428)
Describe an efficient algorithm that, given a set ሼxଵ, xଶ,…,x୬ሽ of points on the real line,
determines the smallest set of unit-length closed intervals that contains all of the given points.
Argue that your algorithm is correct.
2. Exercise 16.3-3 (page 436)
What is an optimal Huffman code for the following set of frequencies, based on the first 8
Fibonacci numbers?
a: 1 b: 1 c: 2 d: 3 e: 5 f: 8 g: 13 h: 21
Can you generalize your answer to find the optimal code when the frequencies are the
first 𝑛 Fibonacci numbers?
3. Modified Problem 16-1 (pages 446-447)
Consider the problem of making change for 𝑛 dollars using the fewest number of coins. Assume
that each coin’s value is an integer.
a. Suppose that the available coins are in the denominations that are powers of 𝑐, i.e., the
denominations are 𝑐଴, cଵ,…,𝑐௞ for some integers 𝑐൐1 and 𝑘൒1. Describe a greedy
algorithm that solves the problem.
b. Give a set of coin denominations for which the greedy algorithm does not yield an optimal
solution. Your set should include a one dollar so that there is a solution for every value of 𝑛.
c. Give an 𝑂ሺ𝑛𝑘ሻ-time algorithm that makes change for any set of 𝑘 different coin
denominations, assuming that one of the coins is one dollar.
4. Two players called P1 and P2 are playing a card game. In the game, there are 𝑚 lines of cards
on the table. (Each line of cards may have different numbers of cards.) On each card, there is a
positive integer score on it. Each card is faced up and the score on the card is visible.
The players take turns choosing one card. (Assume P1 always play first.) In P1’s turn he takes a
card from the left of any non-empty line, and in P2’s turn he takes a card from the right of any
non-empty line. Players want to maximize the sum of scores on the cards he took. The game ends
when all lines become empty. Figure 1 is an example when m is 3.
Please give an Oሺ𝑛 ൅ 𝑚 log 𝑚ሻ algorithm to find out the sum of scores that P1 gets if both
players play optimally (𝑛 is the total number of cards in the game). Briefly justify that your
algorithm is correct.

Figure 1: The example shows a case when m is three. Line 1, Line 2 and Line 3 have 4 cards, 3
cards and 4 cards respectively. In this case, if both players play optimally, P1 will get all the
cards with the black outer frame. So, the sum of scores that P1 gets is 35.
5. Given an array consists of n positive integers 𝑎ଵ, 𝑎ଶ,…,𝑎୬ with the condition
𝑎௜ ൑ 𝑎௜ାଵ ൑2൉𝑎௜ for any positive integer 𝑖 (0 ൏ 𝑖 ൏  𝑛).
For each number in the array, it is allowed to remain unchanged or turn negative. The task is to
add negative sign on some numbers in the array so that the summation of all values in the array
meets the limit absሺ∑௜ୀଵ~௡ 𝑎௜ሻ ൑ 𝑎ଵ . Please design an algorithm with O(n)-time that solve the
problem. Briefly justify that your algorithm is correct.
6. Exercise 22.2-7 (page 602).
There are two types of professional wrestlers: “babyfaces” and “heels”. Between any pair of
professional wrestlers, there may or may not be a rivalry. Suppose we have n professional
wrestlers and we have a list of r pairs of wrestlers for which there are rivalries. Give an
O(n+r)-time algorithm that determines whether it is possible to designate some of the wrestlers
as babyfaces and the remainder as heels such that each rivalry is between a babyface and a heel.
If it is possible to perform such a designation, your algorithm should produce it.
7. Exercise 22.3-12 (page 612).
Show that we can use a depth-first search of an undirected graph G to identify the connected
components of G, and that the depth-first forest contains as many trees as G has connected
components. More precisely, show how to modify depth-first search so that it assigns to each
vertex v an integer label between 1 and k, where k is the number of connected components of
G, such that = if and only if u and v are in the same connected component.
8. Exercise 22.4-4 (page 615)
Prove or disprove: If a directed graph G contains cycles, then TOPOLOGICAL-SORT(G)
produces a vertex ordering that minimizes the number of “bad” edges that are inconsistent with
the ordering produced.
9. Given a directed graph G consisting of V vertices and E edges. In graph G, every node has an
out-degree at least n. Please design an algorithm that finds a simple cycle of length of at
least n + 1 in G with O(E)-time complexity. Briefly justify that your algorithm is correct.
(A simple cycle of length m is a sequence of nodes 𝑣ଵ,  𝑣ଶ, …, 𝑣௠, 𝑣௠ାଵ in which the only
repeated vertices are the first and last vertices, and for any integer 𝑖 (1  ൑  𝑖  ൑  𝑚) there exist an
edge incident from vi to vi + 1.)
10. Given an undirected graph G with V vertices and a sequence S which represents the BFS traversal
order of a graph. We assume in graph G there is exactly one simple path between any two
vertices. Please write an algorithm to determine whether S is a valid BFS traversal sequence of G
with time complexity Oሺ𝑛 log 𝑛ሻ. Briefly justify that your algorithm is correct.
(In the BFS algorithm, the traversal order of the vertices depends on the order of choosing edges,
so there may be multiple valid BFS traversal sequence.)