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程序代写案例-CS 124

时间：2021-04-21

CS 124 Homework 6: Spring 2021

Your name:

Collaborators:

No. of late days used on previous psets:

No. of late days used after including this pset:

Homework is due Wednesday 2021-04-21 at 11:59pm ET. You are allowed up to twelve (college)/forty (ex-

tension school) late days through the semester, but the number of late days you take on each assignment must be a

nonnegative integer at most two (college)/four (extension school).

Try to make your answers as clear and concise as possible; style will count in your grades. Be sure to read and

know the collaboration policy in the course syllabus. Assignments must be submitted in pdf format on Gradescope.

If you do assignments by hand, you will need to scan in your results to turn them in.

For all homework problems where you are asked to design or give an algorithm, you must prove the correctness

of your algorithm and prove the best upper bound that you can give for the running time. Generally better running

times will get better credit; generally exponential time algorithms (unless specifically asked for) will receive no or

little credit. You should always write a clear informal description of your algorithm in English. You may also write

pseudocode if you feel your informal explanation requires more precision and detail, but keep in mind pseudocode

does NOT substitute for an explanation. Answers that consist solely of pseudocode will receive little or not credit.

Again, try to make your answers clear and concise.

1. (15 points) My (Professor Mitzenmacher’s) RSA public key (n,e) is:

(46947848749720430529628739081,37267486263679235062064536973).

Convert the message

I want an A

into a number, using ASCII in the natural way. (So for “A b”: in ASCII, A = 65, space = 32, and b =

98; translating each number into 8 bits gives “A b” = 010000010010000001100010 in binary.) Encode the

message as though you were sending it to me using my RSA key, and write for me the corresponding encoded

message in decimal.

2. (15 points) We have considered, in the context of a randomized algorithm for 2SAT, a random walk with a

completely reflecting boundary at 0—that is, whenever position 0 is reached, with probability 1 we move to

position 1 at the next turn. Consider now a random walk with a partially reflecting boundary at 0– whenever

position 0 is reached, with probability 1/4 we move to position 1 at the next turn, and with probability 3/4 we

stay at 0. Everywhere else the random walk either moves up or down 1, each with probability 1/2.

Find the expected number of moves to reach n starting from position i using a random walk with a partially

reflecting boundary. (You may assume there is a unique solution to this problem, as a function of n and i; you

should prove that your solution satisfies the appropriate recurrence.)

3. (0 points, optional)1 You have been given a square plot of land that has been divided into n rows and columns,

yielding n2 square subplots. Some of these subplots have rocky ground and cannot support plant growth, while

others have soil and can support growing a palm tree. You would like to plant palm trees on a subset of the

1We won’t use this question for grades. Try it if you’re interested. It may be used for recommendations/TF hiring.

1

square subplots (at most one palm tree per subplot) so that every row and every column has exactly the same

number p of palm trees. Furthermore, you would like to do this so that p is as large as possible. Devise an

efficient algorithm to determine how to accomplish this. (You may give the running time in terms of the time

to solve a suitable flow problem.)

4. Show how to reduce the following problems to linear programming.

• (15 points) Find the maximum flow from a vertex s to a vertex t in a directed graph with edge capacities—

except that, at each vertex, half the flow into the vertex is lost (or kept) at the vertex, and the other half

flows out. The goal is to maximize the flow that reaches the destination t.

• (15 points) Find the maximum flow from a vertex s to a vertex t in a directed graph with edge capacities—

except that, for each edge e, there is also a fixed cost ce for each unit of flow through the edge. We need

to find the maximum flow with the minimum cost. That is, there may be many possible flows that achieve

the maximum flow; if there is more than one such flow, find the one of minimum cost. (Hint: you may

need to use more than one linear program!)

5. (15 points) Consider the two-player zero-sum game given by the following matrix. (A positive payoff goes to

the row player: e.g. if the row player picks the first row and the column player picks the first column, the row

player scores 3 and the column player scores -3.)

3 1 0 −4

6 −2 −2 0

−3 2 3 −3

−7 4 −5 7

• Write down a linear program to determine the row player strategy that maximizes the value of the game

to the row player, in terms of the probabilities c1, c2, c3, c4 that the column player picks column 1, 2, 3,

4. Do the same for the column player.

• Write down a linear program to determine the the row player strategy that maximizes the value of the

game to the row player, assuming the column player plays optimally. Do the same for the column player.

Solve these linear programs, and give the proper strategies for both players. (You’ll probably want to

find an existing LP solver.)

• What is the value of the game? That is, should the column player pay the row player to play, or vice

versa, and how much should one player pay the other to make the game fair?

6. (15 points) For the Harvard Tutoring Club, there are n high school students who need tutors, and n available

tutors in the club to tutor them. Each student needs a tutor, and each tutor can work with at most one student.

Each high school student looks at the tutors and decides on a subset of them that they would want to work

with. Suppose that for any subset S of the students, the collective set of tutors T (S) that they are willing to

work with satisfies the condition |T (S)| ≥ |S|. Prove that there is a way to assign every student a tutor that

they are willing to work with. Hint: think of this as a flow problem.

2

学霸联盟

Your name:

Collaborators:

No. of late days used on previous psets:

No. of late days used after including this pset:

Homework is due Wednesday 2021-04-21 at 11:59pm ET. You are allowed up to twelve (college)/forty (ex-

tension school) late days through the semester, but the number of late days you take on each assignment must be a

nonnegative integer at most two (college)/four (extension school).

Try to make your answers as clear and concise as possible; style will count in your grades. Be sure to read and

know the collaboration policy in the course syllabus. Assignments must be submitted in pdf format on Gradescope.

If you do assignments by hand, you will need to scan in your results to turn them in.

For all homework problems where you are asked to design or give an algorithm, you must prove the correctness

of your algorithm and prove the best upper bound that you can give for the running time. Generally better running

times will get better credit; generally exponential time algorithms (unless specifically asked for) will receive no or

little credit. You should always write a clear informal description of your algorithm in English. You may also write

pseudocode if you feel your informal explanation requires more precision and detail, but keep in mind pseudocode

does NOT substitute for an explanation. Answers that consist solely of pseudocode will receive little or not credit.

Again, try to make your answers clear and concise.

1. (15 points) My (Professor Mitzenmacher’s) RSA public key (n,e) is:

(46947848749720430529628739081,37267486263679235062064536973).

Convert the message

I want an A

into a number, using ASCII in the natural way. (So for “A b”: in ASCII, A = 65, space = 32, and b =

98; translating each number into 8 bits gives “A b” = 010000010010000001100010 in binary.) Encode the

message as though you were sending it to me using my RSA key, and write for me the corresponding encoded

message in decimal.

2. (15 points) We have considered, in the context of a randomized algorithm for 2SAT, a random walk with a

completely reflecting boundary at 0—that is, whenever position 0 is reached, with probability 1 we move to

position 1 at the next turn. Consider now a random walk with a partially reflecting boundary at 0– whenever

position 0 is reached, with probability 1/4 we move to position 1 at the next turn, and with probability 3/4 we

stay at 0. Everywhere else the random walk either moves up or down 1, each with probability 1/2.

Find the expected number of moves to reach n starting from position i using a random walk with a partially

reflecting boundary. (You may assume there is a unique solution to this problem, as a function of n and i; you

should prove that your solution satisfies the appropriate recurrence.)

3. (0 points, optional)1 You have been given a square plot of land that has been divided into n rows and columns,

yielding n2 square subplots. Some of these subplots have rocky ground and cannot support plant growth, while

others have soil and can support growing a palm tree. You would like to plant palm trees on a subset of the

1We won’t use this question for grades. Try it if you’re interested. It may be used for recommendations/TF hiring.

1

square subplots (at most one palm tree per subplot) so that every row and every column has exactly the same

number p of palm trees. Furthermore, you would like to do this so that p is as large as possible. Devise an

efficient algorithm to determine how to accomplish this. (You may give the running time in terms of the time

to solve a suitable flow problem.)

4. Show how to reduce the following problems to linear programming.

• (15 points) Find the maximum flow from a vertex s to a vertex t in a directed graph with edge capacities—

except that, at each vertex, half the flow into the vertex is lost (or kept) at the vertex, and the other half

flows out. The goal is to maximize the flow that reaches the destination t.

• (15 points) Find the maximum flow from a vertex s to a vertex t in a directed graph with edge capacities—

except that, for each edge e, there is also a fixed cost ce for each unit of flow through the edge. We need

to find the maximum flow with the minimum cost. That is, there may be many possible flows that achieve

the maximum flow; if there is more than one such flow, find the one of minimum cost. (Hint: you may

need to use more than one linear program!)

5. (15 points) Consider the two-player zero-sum game given by the following matrix. (A positive payoff goes to

the row player: e.g. if the row player picks the first row and the column player picks the first column, the row

player scores 3 and the column player scores -3.)

3 1 0 −4

6 −2 −2 0

−3 2 3 −3

−7 4 −5 7

• Write down a linear program to determine the row player strategy that maximizes the value of the game

to the row player, in terms of the probabilities c1, c2, c3, c4 that the column player picks column 1, 2, 3,

4. Do the same for the column player.

• Write down a linear program to determine the the row player strategy that maximizes the value of the

game to the row player, assuming the column player plays optimally. Do the same for the column player.

Solve these linear programs, and give the proper strategies for both players. (You’ll probably want to

find an existing LP solver.)

• What is the value of the game? That is, should the column player pay the row player to play, or vice

versa, and how much should one player pay the other to make the game fair?

6. (15 points) For the Harvard Tutoring Club, there are n high school students who need tutors, and n available

tutors in the club to tutor them. Each student needs a tutor, and each tutor can work with at most one student.

Each high school student looks at the tutors and decides on a subset of them that they would want to work

with. Suppose that for any subset S of the students, the collective set of tutors T (S) that they are willing to

work with satisfies the condition |T (S)| ≥ |S|. Prove that there is a way to assign every student a tutor that

they are willing to work with. Hint: think of this as a flow problem.

2

学霸联盟