Description
Introduction
Welcome to your third programming assignment of the Advanced Algorithms and Complexity class! In this
programming assignment, you will be practicing reducing hard real-world problems to SAT problem, which
can in turn often be solved efficiently in practice using specialized programs called SAT-solvers.
Learning Outcomes
Upon completing this programming assignment you will be able to:
1. Reduce real-world problems to instances of classical NP-complete problems.
2. Design and implement efficient algorithms to reduce the following computational problems to SAT:
(a) assigning frequencies to the cells of a GSM network;
(b) determine whether it is possible to leave no signs of a party in the apartment;
(c) determine whether there is a way to allocate advertising budget given a set of constraints.
Passing Criteria: 2 out of 3
Passing this programming assignment requires passing at least 2 out of 3 code problems from this assignment.
In turn, passing a code problem requires implementing a solution that passes all the tests for this problem
in the grader and does so under the time and memory limits specified in the problem statement.
1
Contents
1 Problem: Assign Frequencies to the Cells of a GSM Network 3
2 Problem: Cleaning the Apartment 6
3 Advanced Problem: Advertisement Budget Allocation 9
4 General Instructions and Recommendations on Solving Algorithmic Problems 13
4.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Frequently Asked Questions 16
5.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 I submit the solution only for one problem, but all the problems in the assignment are graded.
Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 16
5.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 17
5.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 17
5.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . . 18
5.7 My implementation always fails in the grader, though I already tested and stress tested it a
lot. Would not it be better if you give me a solution to this problem or at least the test cases
that you use? I will then be able to fix my code and will learn how to avoid making mistakes.
Otherwise, I do not feel that I learn anything from solving this problem. I am just stuck. . . . 18
2
1 Problem: Assign Frequencies to the Cells of a GSM Network
Problem Introduction
In this problem, you will learn to reduce the real-world problem about
assigning frequencies to the transmitting towers of the cells in a GSM
network to a problem of proper coloring a graph into 3 colors. Then you
will design and implement an algorithm to reduce this problem to an
instance of SAT.
Problem Description
Task. GSM network is a type of infrastructure used for communication via mobile phones. It includes
transmitting towers scattered around the area which operate in different frequencies. Typically there is
one tower in the center of each hexagon called “cell” on the grid above — hence the name “cell phone”.
A cell phone looks for towers in the neighborhood and decides which one to use based on strength
of signal and other properties. For a phone to distinguish among a few closest towers, the frequencies
of the neighboring towers must be different. You are working on a plan of GSM network for mobile,
and you have a restriction that you’ve only got 3 different frequencies from the government which you
can use in your towers. You know which pairs of the towers are neighbors, and for all such pairs the
towers in the pair must use different frequencies. You need to determine whether it is possible to assign
frequencies to towers and satisfy these restrictions.
This is equivalent to a classical graph coloring problem: in other words, you are given a graph, and
you need to color its vertices into 3 different colors, so that any two vertices connected by an edge
need to be of different colors. Colors correspond to frequencies, vertices correspond to cells, and edges
connect neighboring cells. Graph coloring is an NP-complete problem, so we don’t currently know an
efficient solution to it, and you need to reduce it to an instance of SAT problem which, although it is
NP-complete, can often be solved efficiently in practice using special programs called SAT-solvers.
Input Format. The first line of the input contains integers 𝑛 and 𝑚 — the number of vertices and edges in
the graph. The vertices are numbered from 1 through 𝑛. Each of the next 𝑚 lines contains two integers
𝑢 and 𝑣 — the numbers of vertices connected by an edge. It is guaranteed that a vertex cannot be
connected to itself by an edge.
Constraints. 2 ≤ 𝑛 ≤ 500; 1 ≤ 𝑚 ≤ 1000; 1 ≤ 𝑢, 𝑣 ≤ 𝑛; 𝑢 ̸= 𝑣.
Output Format. You need to output a boolean formula in the conjunctive normal form (CNF) in a specific
format. If it is possible to color the vertices of the input graph in 3 colors such that any two vertices
connected by an edge are of different colors, the formula must be satisfiable. Otherwise, the formula
must be unsatisfiable. The number of variables in the formula must be at least 1 and at most 3000.
The number of clauses must be at least 1 and at most 5000.
On the first line, output integers 𝐶 and 𝑉 — the number of clauses in the formula and the number of
variables respectively. On each of the next 𝐶 lines, output a description of a single clause. Each clause
has a form (𝑥4 𝑂𝑅 𝑥1 𝑂𝑅 𝑥8). For a clause with 𝑘 terms (in the example, 𝑘 = 3 for 𝑥4, 𝑥1 and 𝑥8), output
first those 𝑘 terms and then number 0 in the end (in the example, output “4 − 1 8 0”). Output each
term as integer number. Output variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers 1, 2, . . . , 𝑉 respectively. Output
3
negations of variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers −1, −2, . . . , −𝑉 respectively. Each number other than
the last one in each line must be a non-zero integer between −𝑉 and 𝑉 where 𝑉 is the total number
of variables specified in the first line of the output. Ensure that 1 ≤ 𝐶 ≤ 5000 and 1 ≤ 𝑉 ≤ 3000.
See the examples below for further clarification of the output format.
If there are many different formulas that satisfy the requirements above, you can output any one of
them.
Note that your formula will be checked internally by the grader using a SAT-solver.
Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot
be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we
use under a reasonable time limit, the grader will timeout, and the problem won’t pass.
We guarantee that there are solutions of this problem that output formulas which are
resolved almost instantly by the SAT-solver used. However, don’t try to intentionally
break the system by submitting very complex SAT instances, because the problem still
won’t pass, and you will violate Coursera Honor Code by doing that.
Time Limits.
language C C++ Java Python C# Haskell JavaScript Ruby Scala
time (sec) 1 1 1.5 5 1.5 2 5 5 3
Memory Limit. 512MB.
Sample 1.
Input:
3 3
1 2
2 3
1 3
Output:
1 1
1 -1 0
Explanation:
1 2
3
The input graph has just 3 vertices, so of course they all can be colored in different colors using only
3 colors. That’s why we need to output a satisfiable formula. The formula in the output uses just 1
variable 𝑥1 and 1 clause, and the only clause is (𝑥1 𝑂𝑅 𝑥1) which is, of course, satisfiable: for any value
of 𝑥1, the boolean value of the formula is true. Note that you could output another satisfiable formula,
like 𝑥1 or (𝑥1 𝑂𝑅 𝑥2 𝑂𝑅 𝑥3) 𝐴𝑁𝐷 (𝑥1 𝑂𝑅 𝑥2), or one of many others.
4
Sample 2.
Input:
4 6
1 2
1 3
1 4
2 3
2 4
3 4
Output:
2 1
1 0
-1 0
Explanation:
1 2
4 3
The input graph has 4 vertices, and each pair of them is connected by an edge. In a proper coloring, all
these vertices must be of different colors, but we have only 3 different colors, so it is impossible. Thus,
we need to output an unsatisfiable formula. The formula in the output has 2 clauses with one variable,
it is (𝑥1)𝐴𝑁𝐷(𝑥1). Note that you could output another formula, like (𝑥1 𝑂𝑅 𝑥2) 𝐴𝑁𝐷 (𝑥1) 𝐴𝑁𝐷 (𝑥2),
or one of many other unsatisfiable formulas.
Starter Files
The starter solutions for this problem read the data from the input, pass it to a procedure that outputs a
fixed satisfiable formula. You need to change the main procedure to implement some reduction of the graph
coloring problem to SAT problem if you’re using C++, Java, or Python3. For other programming languages,
you need to implement a solution from scratch. Filename: gsm_network
What To Do
Create a separate variable 𝑥𝑖𝑗 for each vertex 𝑖 of the initial graph and each possible color 𝑗, 1 ≤ 𝑗 ≤ 3
which means “vertex 𝑖 has color 𝑗”. Think how to write down conditions like “each vertex has to be colored
by some color” and “vertices connected by an edge must have different colors” with clauses of CNF using
these variables.
Need Help?
Ask a question or see the questions asked by other learners at this forum thread.
5
2 Problem: Cleaning the Apartment
Problem Introduction
In this problem, you will learn to determine whether it is possible to
clean an apartment after a party without leaving any traces of the party.
You will learn how to reduce it to the classic Hamiltonian Path problem,
and then you will design and implement an efficient algorithm to reduce
it to SAT.
Problem Description
Task. You’ve just had a huge party in your parents’ house, and they are returning tomorrow. You need
to not only clean the apartment, but leave no trace of the party. To do that, you need to clean all
the rooms in some order. After finishing a thorough cleaning of some room, you cannot return to it
anymore: you are afraid you’ll ruin everything accidentally and will have to start over. So, you need to
move from room to room, visit each room exactly once and clean it. You can only move from a room
to the neighboring rooms. You want to determine whether this is possible at all.
This can be reduced to a classic Hamiltonian Path problem: given a graph, determine whether there is
a route visiting each vertex exactly once. Rooms are vertices of the graph, and neighboring rooms are
connected by edges. Hamiltonian Path problem is NP-complete, so we don’t know an efficient algorithm
to solve it. You need to reduce it to SAT, so that it can be solved efficiently by a SAT-solver.
Input Format. The first line contains two integers 𝑛 and 𝑚 — the number of rooms and the number of
corridors connecting the rooms respectively. Each of the next 𝑚 lines contains two integers 𝑢 and 𝑣
describing the corridor going from room 𝑢 to room 𝑣. The corridors are two-way, that is, you can go
both from 𝑢 to 𝑣 and from 𝑣 to 𝑢. No two corridors have a common part, that is, every corridor only
allows you to go from one room to one other room. Of course, no corridor connects a room to itself.
Note that a corridor from 𝑢 to 𝑣 can be listed several times, and there can be listed both a corridor
from 𝑢 to 𝑣 and a corridor from 𝑣 to 𝑢.
Constraints. 1 ≤ 𝑛 ≤ 30; 0 ≤ 𝑚 ≤
𝑛(𝑛−1)
2
; 1 ≤ 𝑢, 𝑣 ≤ 𝑛.
Output Format. You need to output a boolean formula in the CNF form in a specific format. If it is
possible to go through all the rooms and visit each one exactly once to clean it, the formula must be
satisfiable. Otherwise, the formula must be unsatisfiable. The sum of the numbers of variables used in
each clause of the formula must not exceed 120 000.
On the first line, output integers 𝐶 and 𝑉 — the number of clauses in the formula and the number of
variables respectively. On each of the next 𝐶 lines, output a description of a single clause. Each clause
has a form (𝑥4 𝑂𝑅 𝑥1 𝑂𝑅 𝑥8). For a clause with 𝑘 terms (in the example, 𝑘 = 3 for 𝑥4, 𝑥1 and 𝑥8), output
first those 𝑘 terms and then number 0 in the end (in the example, output “4 − 1 8 0”). Output each
term as integer number. Output variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers 1, 2, . . . , 𝑉 respectively. Output
negations of variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers −1, −2, . . . , −𝑉 respectively. Each number other than
the last one in each line must be a non-zero integer between −𝑉 and 𝑉 where 𝑉 is the total number
of variables specified in the first line of the output. Ensure that the total number of non-zero integers
in the 𝐶 lines describing the clauses is at most 120 000.
See the examples below for further clarification of the output format.
If there are many different formulas that satisfy the requirements above, you can output any one of
them.
6
Note that your formula will be checked internally by the grader using a SAT-solver.
Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot
be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we
use under a reasonable time limit, the grader will timeout, and the problem won’t pass.
We guarantee that there are solutions of this problem that output formulas which are
resolved almost instantly by the SAT-solver used. However, don’t try to intentionally
break the system by submitting very complex SAT instances, because the problem still
won’t pass, and you will violate Coursera Honor Code by doing that.
Time Limits.
language C C++ Java Python C# Haskell JavaScript Ruby Scala
time (sec) 2 2 3 10 3 4 10 10 6
Memory Limit. 512MB.
Sample 1.
Input:
5 4
1 2
2 3
3 5
4 5
Output:
1 1
1 -1 0
Explanation:
1 2 3 5 4
There is a Hamiltonian path 1 − 2 − 3 − 5 − 4, so we need to output a satisfiable formula. The formula
in the output uses just 1 variable 𝑥1 and 1 clause, and the only clause is (𝑥1 𝑂𝑅 𝑥1) which is, of course,
satisfiable: for any value of 𝑥1, the boolean value of the formula is true. Note that you could output
another satisfiable formula, like 𝑥1 or (𝑥1 𝑂𝑅 𝑥2 𝑂𝑅 𝑥3) 𝐴𝑁𝐷 (𝑥1 𝑂𝑅 𝑥2), or one of many others.
Sample 2.
Input:
4 3
1 2
1 3
1 4
Output:
2 1
1 0
-1 0
7
Explanation:
1 2
3 4
There is no way to visit each room exactly once: either we don’t visit one of the rooms 2, 3 or 4,
or we visit room 1 at least twice. Thus, we need to output an unsatisfiable formula. The formula in
the output has 2 clauses with one variable, it is (𝑥1)𝐴𝑁𝐷(𝑥1). Note that you could output another
formula, like (𝑥1 𝑂𝑅 𝑥2) 𝐴𝑁𝐷 (𝑥1) 𝐴𝑁𝐷 (𝑥2), or one of many other unsatisfiable formulas.
Starter Files
The starter solutions for this problem read the data from the input, pass it to a procedure that outputs a fixed
satisfiable formula. You need to change the main procedure to implement some reduction of the Hamiltonian
path problem to SAT if you are using C++, Java, or Python3. For other programming languages, you need
to implement a solution from scratch. Filename: cleaning_apartment
What To Do
Create a separate variable 𝑥𝑖𝑗 for each vertex 𝑖 and each position in the Hamiltonian path 𝑗. 𝑥𝑖𝑗 is true if
vertex 𝑖 is at the position 𝑗 of Hamiltonian path. Think how to express all the restrictions in terms of these
variables: all vertices must be on the path, each vertex must be visited exactly once, there is only one vertex
on each position in the path, two successive vertices must be connected by an edge.
Need Help?
Ask a question or see the questions asked by other learners at this forum thread.
8
3 Advanced Problem: Advertisement Budget Allocation
We strongly recommend you start solving advanced problems only when you are done with the basic problems
(for some advanced problems, algorithms are not covered in the video lectures and require additional ideas
to be solved; for some other advanced problems, algorithms are covered in the lectures, but implementing
them is a more challenging task than for other problems).
Problem Introduction
In the previous programming assignment, you worked for an online advertisement system. In this programming assignment, you’ll work for a
big company that uses advertising to promote itself. You will need to
determine whether it is possible to allocate advertising budget and satisfy all the constraints. You will learn how to reduce this problem to a
particular type of Integer Linear Programming problem. Then you will
design and implement an efficient algorithm to reduce this type of Integer
Linear Programming to SAT.
Problem Description
Task. The marketing department of your big company has many subdepartments which control advertising
on TV, radio, web search, contextual advertising, mobile advertising, etc. Each of them has prepared
their advertising campaign plan, and of course you don’t have enough budget to cover all of their
proposals. You don’t have enough time to go thoroughly through each subdepartment’s proposals and
cut them, because you need to set the budget for the next year tomorrow. You decide that you will
either approve or decline each of the proposals as a whole.
There is a bunch of constraints you face. For example, your total advertising budget is limited. Also,
you have some contracts with advertising agencies for some of the advertisement types that oblige
you to spend at least some fixed budget on that kind of advertising, or you’ll see huge penalties, so
you’d better spend it. Also, there are different company policies that can be of the form that you
spend at least 10% of your total advertising spend on mobile advertising to promote yourself in this
new channel, or that you spend at least $1M a month on TV advertisement, so that people always
remember your brand. All of these constraints can be rewritten as an Integer Linear Programming: for
each subdepartment 𝑖, denote by 𝑥𝑖 boolean variable that corresponds to whether you will accept or
decline the proposal of that subdepartment. Then each constraint can be written as a linear inequality.
For example, ∑︀𝑛
𝑖=1
spend𝑖
· 𝑥𝑖 ≤ TotalBudget is the inequality to ensure your total budget is enough to
accept all the selected proposals. And ∑︀𝑛
𝑖=1
spend𝑖
· 𝑥𝑖 ≤ 10 · mobile corresponds to the fact that mobile
advertisement budget is at least 10% of the total spending.
You will be given the final Integer Linear Programming problem in the input, and you will need to
reduce it to SAT. It is guaranteed that there will be at most 3 different variables with
non-zero coefficients in each inequality of this Integer Linear Programming problem.
Input Format. The first line contains two integers 𝑛 and 𝑚 — the number of inequalities and the number
of variables. The next 𝑛 lines contain the description of 𝑛×𝑚 matrix 𝐴 with coefficients of inequalities
(each of the 𝑛 lines contains 𝑚 integers, and at most 3 of them are non-zero), and the last line contains
the description of the vector 𝑏 (𝑛 integers) for the system of inequalities 𝐴𝑥 ≤ 𝑏. You need to determine
whether there exists a binary vector 𝑥 satisfying all those inequalities.
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Constraints. 1 ≤ 𝑛, 𝑚 ≤ 500; −100 ≤ 𝐴𝑖𝑗 ≤ 100; −1 000 000 ≤ 𝑏𝑖 ≤ 1 000 000.
Output Format. You need to output a boolean formula in the CNF form in a specific format. If it is
possible to accept some of the proposals and decline all the others while satisfying all the constraints,
the formula must be satisfiable. Otherwise, the formula must be unsatisfiable. The number of variables
in the formula must not exceed 3000, and the number of clauses must not exceed 5000.
On the first line, output integers 𝐶 and 𝑉 — the number of clauses in the formula and the number of
variables respectively. On each of the next 𝐶 lines, output a description of a single clause. Each clause
has a form (𝑥4 𝑂𝑅 𝑥1 𝑂𝑅 𝑥8). For a clause with 𝑘 terms (in the example, 𝑘 = 3 for 𝑥4, 𝑥1 and 𝑥8), output
first those 𝑘 terms and then number 0 in the end (in the example, output “4 − 1 8 0”). Output each
term as integer number. Output variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers 1, 2, . . . , 𝑉 respectively. Output
negations of variables 𝑥1, 𝑥2, . . . , 𝑥𝑉 as numbers −1, −2, . . . , −𝑉 respectively. Each number other than
the last one in each line must be a non-zero integer between −𝑉 and 𝑉 where 𝑉 is the total number
of variables specified in the first line of the output. Ensure that 1 ≤ 𝐶 ≤ 5000 and 1 ≤ 𝑉 ≤ 3000.
See the examples below for further clarification of the output format.
If there are many different formulas that satisfy the requirements above, you can output any one of
them.
Note that your formula will be checked internally by the grader using a SAT-solver.
Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot
be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we
use under a reasonable time limit, the grader will timeout, and the problem won’t pass.
We guarantee that there are solutions of this problem that output formulas which are
resolved almost instantly by the SAT-solver used. However, don’t try to intentionally
break the system by submitting very complex SAT instances, because the problem still
won’t pass, and you will violate Coursera Honor Code by doing that.
Time Limits.
language C C++ Java Python C# Haskell JavaScript Ruby Scala
time (sec) 1 1 1.5 5 1.5 2 5 5 3
Memory Limit. 512MB.
Sample 1.
Input:
2 3
5 2 3
-1 -1 -1
6 -2
Output:
1 1
1 -1 0
Explanation:
Here we have two inequalities: 5 · 𝑥1 + 2 · 𝑥2 + 3 · 𝑥3 ≤ 6 and (−1) · 𝑥1 + (−1) · 𝑥2 + (−1) · 𝑥3 ≤ −2.
The binary vector 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 1 satisfies all the inequalities, so we need to output a satisfiable
formula. The formula in the output uses just one variable 𝑥1 and one clause, and the only clause is
(𝑥1 𝑂𝑅 𝑥1) which is, of course, satisfiable: for any value of 𝑥1, the boolean value of the formula is true.
Note that you could output another satisfiable formula, like 𝑥1 or (𝑥1 𝑂𝑅 𝑥2 𝑂𝑅 𝑥3) 𝐴𝑁𝐷 (𝑥1 𝑂𝑅 𝑥2),
or one of many others. You do not have to make sure that the formula you output is satisfied only
by the binary vectors which satisfy the given system of inequalities, but the intended solution ensures
that.
10
Sample 2.
Input:
3 3
1 0 0
0 1 0
0 0 1
1 1 1
Output:
1 1
1 -1 0
Explanation:
There are three inequalities of the form 𝑥𝑖 ≤ 1. Of course, any binary vector satisfies all those inequalities, so we need to output a satisfiable formula. The formula in the output is the same as in the
previous example.
Sample 3.
Input:
2 1
1
-1
0 -1
Output:
2 1
1 0
-1 0
Explanation:
There are two inequalities 𝑥1 ≤ 0 and −𝑥1 ≤ −1 ⇒ 𝑥1 ≥ 1, but 𝑥1 cannot be less than 0 and more
than 1 simultaneously, so we need to output an unsatisfiable formula. The formula in the output has
2 clauses with one variable, it is (𝑥1)𝐴𝑁𝐷(𝑥1). Note that you could output another formula, like
(𝑥1 𝑂𝑅 𝑥2) 𝐴𝑁𝐷 (𝑥1) 𝐴𝑁𝐷 (𝑥2), or one of many other unsatisfiable formulas.
Sample 4.
Input:
2 3
1 1 0
0 -1 -1
0 -2
Output:
2 1
1 0
-1 0
Explanation:
There are two inequalities: 𝑥1 + 𝑥2 ≤ 0 and (−1) · 𝑥2 + (−1) · 𝑥3 ≤ −2 ⇒ 𝑥2 + 𝑥3 ≥ 2. From the first
one, it follows for a binary vector 𝑥 that 𝑥1 = 𝑥2 = 0. From the second one, it follows for a binary
vector 𝑥 that 𝑥2 = 𝑥3 = 1. But 𝑥2 cannot be 0 and 1 simultaneously, so there is no binary vector 𝑥
satisfying all the inequalities, and so we need to output an unsatisfiable formula. The formula in the
output is the same as in the previous example.
11
Starter Files
The starter solutions for this problem read the data from the input, pass it to procedure that outputs a fixed
satisfiable formula. You need to change the main procedure to implement some reduction of Integer Linear
Programming problem to SAT if you are using C++, Java, or Python3. For other programming languages,
you need to implement a solution from scratch. Filename: budget_allocation
What To Do
In this case, you can use the binary vector 𝑥 that you are looking for itself as the basis for the CNF formula.
Think how to use the fact that there can be at most 3 non-zero coefficients in each inequality. Try thinking
about assignments of all the corresponding variables that invalidate a particular inequality — how many
different such assignments are there at most? You know that none of those assignments should hold if the
binary vector 𝑥 satisfies all the inequalities. Think how to write this condition compactly in the CNF form,
then join those clauses for each inequality.
Need Help?
Ask a question or see the questions asked by other learners at this forum thread.
12
4 General Instructions and Recommendations on Solving Algorithmic Problems
Your main goal in an algorithmic problem is to implement a program that solves a given computational
problem in just few seconds even on massive datasets. Your program should read a dataset from the standard
input and write an answer to the standard output.
Below we provide general instructions and recommendations on solving such problems. Before reading
them, go through readings and screencasts in the first module that show a step by step process of solving
two algorithmic problems: link.
4.1 Reading the Problem Statement
You start by reading the problem statement that contains the description of a particular computational task
as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems
your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems
you first need to come up with an algorithm yourself.
4.2 Designing an Algorithm
If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected
running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the
subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second.
So, if the maximum size of a dataset in the problem description is 𝑛 = 105
, then most probably an algorithm
with quadratic running time is not going to fit into time limit (since for 𝑛 = 105
, 𝑛
2 = 1010) while a solution
with running time 𝑂(𝑛 log 𝑛) will fit. However, an 𝑂(𝑛
2
) solution will fit if 𝑛 is up to 103 = 1000, and if
𝑛 is at most 100, even 𝑂(𝑛
3
) solutions will fit. In some cases, the problem is so hard that we do not know
a polynomial solution. But for 𝑛 up to 18, a solution with 𝑂(2𝑛𝑛
2
) running time will probably fit into the
time limit.
To design an algorithm with the expected running time, you will of course need to use the ideas covered
in the lectures. Also, make sure to carefully go through sample tests in the problem description.
4.3 Implementing Your Algorithm
When you have an algorithm in mind, you start implementing it. Currently, you can use the following
programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript,
Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and
Python3. If you are going to use one of these programming languages, use these starter files. For other
programming languages, you need to implement a solution from scratch.
4.4 Compiling Your Program
For solving programming assignments, you can use any of the following programming languages: C, C++,
C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing
starter solution files for C++, Java, and Python3. The programming language of your submission is detected
automatically, based on the extension of your submission.
We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given
restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit.
You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have
no guarantee that a correct solution for a particular problem running under the given time and memory
constraints exists in any of those other languages.
Your solution will be compiled as follows. We recommend that when testing your solution locally, you
use the same compiler flags for compiling. This will increase the chances that your program behaves in the
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same way on your machine and on the testing machine (note that a buggy program may behave differently
when compiled by different compilers, or even by the same compiler with different flags).
∙ C (gcc 5.2.1). File extensions: .c. Flags:
gcc – pipe – O2 – std = c11 < filename > – lm
∙ C++ (g++ 5.2.1). File extensions: .cc, .cpp. Flags:
g ++ – pipe – O2 – std = c ++14 < filename > – lm
If your C/C++ compiler does not recognize -std=c++14 flag, try replacing it with -std=c++0x flag
or compiling without this flag at all (all starter solutions can be compiled without it). On Linux
and MacOS, you most probably have the required compiler. On Windows, you may use your favorite
compiler or install, e.g., cygwin.
∙ C# (mono 3.2.8). File extensions: .cs. Flags:
mcs
∙ Haskell (ghc 7.8.4). File extensions: .hs. Flags:
ghc – O2
∙ Java (Open JDK 8). File extensions: .java. Flags:
javac – encoding UTF -8
java – Xmx1024m
∙ JavaScript (Node v6.3.0). File extensions: .js. Flags:
nodejs
∙ Python 2 (CPython 2.7). File extensions: .py2 or .py (a file ending in .py needs to have a first line
which is a comment containing “python2”). No flags:
python2
∙ Python 3 (CPython 3.4). File extensions: .py3 or .py (a file ending in .py needs to have a first line
which is a comment containing “python3”). No flags:
python3
∙ Ruby (Ruby 2.1.5). File extensions: .rb.
ruby
∙ Scala (Scala 2.11.6). File extensions: .scala.
scalac
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4.5 Testing Your Program
When your program is ready, you start testing it. It makes sense to start with small datasets (for example,
sample tests provided in the problem description). Ensure that your program produces a correct result.
You then proceed to checking how long does it take your program to process a massive dataset. For
this, it makes sense to implement your algorithm as a function like solve(dataset) and then implement an
additional procedure generate() that produces a large dataset. For example, if an input to a problem is a
sequence of integers of length 1 ≤ 𝑛 ≤ 105
, then generate a sequence of length exactly 105
, pass it to your
solve() function, and ensure that the program outputs the result quickly.
Also, check the boundary values. Ensure that your program processes correctly sequences of size 𝑛 =
1, 2, 105
. If a sequence of integers from 0 to, say, 106
is given as an input, check how your program behaves
when it is given a sequence 0, 0, . . . , 0 or a sequence 106
, 106
, . . . , 106
. Check also on randomly generated
data. For each such test check that you program produces a correct result (or at least a reasonably looking
result).
In the end, we encourage you to stress test your program to make sure it passes in the system at the first
attempt. See the readings and screencasts from the first week to learn about testing and stress testing: link.
4.6 Submitting Your Program to the Grading System
When you are done with testing, you submit your program to the grading system. For this, you go the
submission page, create a new submission, and upload a file with your program. The grading system then
compiles your program (detecting the programming language based on your file extension, see Subsection 4.4)
and runs it on a set of carefully constructed tests to check that your program always outputs a correct result
and that it always fits into the given time and memory limits. The grading usually takes no more than a
minute, but in rare cases when the servers are overloaded it might take longer. Please be patient. You can
safely leave the page when your solution is uploaded.
As a result, you get a feedback message from the grading system. The feedback message that you will love
to see is: Good job! This means that your program has passed all the tests. On the other hand, the three
messages Wrong answer, Time limit exceeded, Memory limit exceeded notify you that your program
failed due to one these three reasons. Note that the grader will not show you the actual test you program
have failed on (though it does show you the test if your program have failed on one of the first few tests;
this is done to help you to get the input/output format right).
4.7 Debugging and Stress Testing Your Program
If your program failed, you will need to debug it. Most probably, you didn’t follow some of our suggestions
from the section 4.5. See the readings and screencasts from the first week to learn about debugging your
program: link.
You are almost guaranteed to find a bug in your program using stress testing, because the way these
programming assignments and tests for them are prepared follows the same process: small manual tests,
tests for edge cases, tests for large numbers and integer overflow, big tests for time limit and memory limit
checking, random test generation. Also, implementation of wrong solutions which we expect to see and stress
testing against them to add tests specifically against those wrong solutions.
Go ahead, and we hope you pass the assignment soon!
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5 Frequently Asked Questions
5.1 I submit the program, but nothing happens. Why?
You need to create submission and upload the file with your solution in one of the programming languages C,
C++, Java, or Python (see Subsections 4.3 and 4.4). Make sure that after uploading the file with your solution
you press on the blue “Submit” button in the bottom. After that, the grading starts, and the submission
being graded is enclosed in an orange rectangle. After the testing is finished, the rectangle disappears, and
the results of the testing of all problems is shown to you.
5.2 I submit the solution only for one problem, but all the problems in the
assignment are graded. Why?
Each time you submit any solution, the last uploaded solution for each problem is tested. Don’t worry: this
doesn’t affect your score even if the submissions for the other problems are wrong. As soon as you pass the
sufficient number of problems in the assignment (see in the pdf with instructions), you pass the assignment.
After that, you can improve your result if you successfully pass more problems from the assignment. We
recommend working on one problem at a time, checking whether your solution for any given problem passes
in the system as soon as you are confident in it. However, it is better to test it first, please refer to the
reading about stress testing: link.
5.3 What are the possible grading outcomes, and how to read them?
Your solution may either pass or not. To pass, it must work without crashing and return the correct answers
on all the test cases we prepared for you, and do so under the time limit and memory limit constraints
specified in the problem statement. If your solution passes, you get the corresponding feedback “Good job!”
and get a point for the problem. If your solution fails, it can be because it crashes, returns wrong answer,
works for too long or uses too much memory for some test case. The feedback will contain the number of
the test case on which your solution fails and the total number of test cases in the system. The tests for the
problem are numbered from 1 to the total number of test cases for the problem, and the program is always
tested on all the tests in the order from the test number 1 to the test with the biggest number.
Here are the possible outcomes:
Good job! Hurrah! Your solution passed, and you get a point!
Wrong answer. Your solution has output incorrect answer for some test case. If it is a sample test case from
the problem statement, or if you are solving Programming Assignment 1, you will also see the input
data, the output of your program and the correct answer. Otherwise, you won’t know the input, the
output, and the correct answer. Check that you consider all the cases correctly, avoid integer overflow,
output the required white space, output the floating point numbers with the required precision, don’t
output anything in addition to what you are asked to output in the output specification of the problem
statement. See this reading on testing: link.
Time limit exceeded. Your solution worked longer than the allowed time limit for some test case. If it
is a sample test case from the problem statement, or if you are solving Programming Assignment 1,
you will also see the input data and the correct answer. Otherwise, you won’t know the input and the
correct answer. Check again that your algorithm has good enough running time estimate. Test your
program locally on the test of maximum size allowed by the problem statement and see how long it
works. Check that your program doesn’t wait for some input from the user which makes it to wait
forever. See this reading on testing: link.
Memory limit exceeded. Your solution used more than the allowed memory limit for some test case. If it
is a sample test case from the problem statement, or if you are solving Programming Assignment 1,
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you will also see the input data and the correct answer. Otherwise, you won’t know the input and the
correct answer. Estimate the amount of memory that your program is going to use in the worst case
and check that it is less than the memory limit. Check that you don’t create too large arrays or data
structures. Check that you don’t create large arrays or lists or vectors consisting of empty arrays or
empty strings, since those in some cases still eat up memory. Test your program locally on the test of
maximum size allowed by the problem statement and look at its memory consumption in the system.
Cannot check answer. Perhaps output format is wrong. This happens when you output something
completely different than expected. For example, you are required to output word “Yes” or “No”, but
you output number 1 or 0, or vice versa. Or your program has empty output. Or your program outputs
not only the correct answer, but also some additional information (this is not allowed, so please follow
exactly the output format specified in the problem statement). Maybe your program doesn’t output
anything, because it crashes.
Unknown signal 6 (or 7, or 8, or 11, or some other). This happens when your program crashes. It
can be because of division by zero, accessing memory outside of the array bounds, using uninitialized
variables, too deep recursion that triggers stack overflow, sorting with contradictory comparator, removing elements from an empty data structure, trying to allocate too much memory, and many other
reasons. Look at your code and think about all those possibilities. Make sure that you use the same
compilers and the same compiler options as we do. Try different testing techniques from this reading:
link.
Internal error: exception… Most probably, you submitted a compiled program instead of a source
code.
Grading failed. Something very wrong happened with the system. Contact Coursera for help or write in
the forums to let us know.
5.4 How to understand why my program fails and to fix it?
If your program works incorrectly, it gets a feedback from the grader. For the Programming Assignment 1,
when your solution fails, you will see the input data, the correct answer and the output of your program
in case it didn’t crash, finished under the time limit and memory limit constraints. If the program crashed,
worked too long or used too much memory, the system stops it, so you won’t see the output of your program
or will see just part of the whole output. We show you all this information so that you get used to the
algorithmic problems in general and get some experience debugging your programs while knowing exactly
on which tests they fail.
However, in the following Programming Assignments throughout the Specialization you will only get so
much information for the test cases from the problem statement. For the next tests you will only get the
result: passed, time limit exceeded, memory limit exceeded, wrong answer, wrong output format or some
form of crash. We hide the test cases, because it is crucial for you to learn to test and fix your program
even without knowing exactly the test on which it fails. In the real life, often there will be no or only partial
information about the failure of your program or service. You will need to find the failing test case yourself.
Stress testing is one powerful technique that allows you to do that. You should apply it after using the other
testing techniques covered in this reading.
5.5 Why do you hide the test on which my program fails?
Often beginner programmers think by default that their programs work. Experienced programmers know,
however, that their programs almost never work initially. Everyone who wants to become a better programmer
needs to go through this realization.
When you are sure that your program works by default, you just throw a few random test cases against
it, and if the answers look reasonable, you consider your work done. However, mostly this is not enough. To
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make one’s programs work, one must test them really well. Sometimes, the programs still don’t work although
you tried really hard to test them, and you need to be both skilled and creative to fix your bugs. Solutions
to algorithmic problems are one of the hardest to implement correctly. That’s why in this Specialization you
will gain this important experience which will be invaluable in the future when you write programs which
you really need to get right.
It is crucial for you to learn to test and fix your programs yourself. In the real life, often there will be no
or only partial information about the failure of your program or service. Still, you will have to reproduce the
failure to fix it (or just guess what it is, but that’s rare, and you will still need to reproduce the failure to
make sure you have really fixed it). When you solve algorithmic problems, it is very frequent to make subtle
mistakes. That’s why you should apply the testing techniques described in this reading to find the failing
test case and fix your program.
5.6 My solution does not pass the tests? May I post it in the forum and ask
for a help?
No, please do not post any solutions in the forum or anywhere on the web, even if a solution does not
pass the tests (as in this case you are still revealing parts of a correct solution). Recall the third item
of the Coursera Honor Code: “I will not make solutions to homework, quizzes, exams, projects, and other
assignments available to anyone else (except to the extent an assignment explicitly permits sharing solutions).
This includes both solutions written by me, as well as any solutions provided by the course staff or others”
(link).
5.7 My implementation always fails in the grader, though I already tested and
stress tested it a lot. Would not it be better if you give me a solution to
this problem or at least the test cases that you use? I will then be able to
fix my code and will learn how to avoid making mistakes. Otherwise, I do
not feel that I learn anything from solving this problem. I am just stuck.
First of all, you always learn from your mistakes.
The process of trying to invent new test cases that might fail your program and proving them wrong
is often enlightening. This thinking about the invariants which you expect your loops, ifs, etc. to keep and
proving them wrong (or right) makes you understand what happens inside your program and in the general
algorithm you’re studying much more.
Also, it is important to be able to find a bug in your implementation without knowing a test case and
without having a reference solution. Assume that you designed an application and an annoyed user reports
that it crashed. Most probably, the user will not tell you the exact sequence of operations that led to a crash.
Moreover, there will be no reference application. Hence, once again, it is important to be able to locate a
bug in your implementation yourself, without a magic oracle giving you either a test case that your program
fails or a reference solution. We encourage you to use programming assignments in this class as a way of
practicing this important skill.
If you have already tested a lot (considered all corner cases that you can imagine, constructed a set of
manual test cases, applied stress testing), but your program still fails and you are stuck, try to ask for help
on the forum. We encourage you to do this by first explaining what kind of corner cases you have already
considered (it may happen that when writing such a post you will realize that you missed some corner cases!)
and only then asking other learners to give you more ideas for tests cases.
References
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