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Programming assignment 2 (150 points)
You should make sure your C program returns EXIT_SUCCESS or 0 indicating successful program
termination.
To use the autograder script to test your program, you can type:
./autograder
or
python3 autograder.py
We will be automatically building, testing, and grading your assignment. Make sure that we can build
your assignment by just using the Makefile and that we can run the program by invoking our autograder.
Make sure to follow the required output format; any extraneous output such as debugging statements
will confuse the grading program.
2. edgelist: Loading, representing, and printing an
undirected unweighted graph (easy) (20 points)
Graphs are fundamental data structures. A graph consists of nodes and edges connecting pairs of
nodes. A basic kind of graph is an undirected, unweighted graph, meaning that the edges are not
directional, and each edge doesn't have any additional properties. Here is an example of an undirected,
unweighted graph G=(V,E), V={0,1,2,3}, E={(0,1),(0,2),(0,3),(1,3)} of four nodes and four edges:
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Programming assignment 2 (150 points)
There are several important ways to represent graphs.
The first graph representation is an adjacency matrix
(https://en.wikipedia.org/wiki/Adjacency_matrix)
. The adjacency matrix of the above graph is:
0
1
1
1
1
0
0
1
1
0
0
0
1
1
0
0
The 0, 1, 1, 1 in the first row of the matrix indicates the 0th node is connected to the 1st, 2nd, and 3rd
nodes, and so on.
The second graph representation is an adjacency list
(https://en.wikipedia.org/wiki/Adjacency_list) .
For a graph consisting of N nodes, the adjacency list data structure is an array of N separate linked lists
for each node p, where each link in the linked list records a node q if the edge (p,q) exists. For example,
the adjacency list representation of the above graph is:
0->1->2->3->/
1->0->3->/
2->0->/
3->0->1->/
The ->/ indicates a null pointer terminating the linked list.
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Programming assignment 2 (150 points)
The third graph representation is by listing the edges of the graph. For example, the edge list of the
above graph is:
0
0
0
1
2
3
1
3
In this part of the assignment, you will write a program that:
1. Loads an adjacency matrix representation of an undirected unweighted graph,
2. Holds that graph representation as a adjacency list data structure,
3. Prints out the edge list representation of the graph.
An important C header file, graphutils.h, is provided to you in 2023_0s_211/pa2/graphutils.h. This file
offers ready-to-use functions for loading a adjacency matrix, creating an adjacency list data structure,
and freeing the adjacency list. You should call these functions to simplify your code.
Input format
Your program should take a single command line argument specifying the path to an input file. Test
cases for your program are in the tests/ directory. In each test case, the first line records the number of
nodes N in the graph. Then, the adjacency matrix is recorded in the subsequent N rows of the file.
Output format
Expected outputs from your program for each test case are in the answers/ directory. You should print
one line for each edge of the graph; each line should list the pair of nodes (separated by a space)
constituting a graph edge.
This is an undirected graph, so the order of the nodes does not matter. The autograder will recognize reordering of the nodes as correct. The ordering of which edges are printed first also does not matter. The
autograder will recognize re-ordering of the edges as correct.
3. isTree: Determining whether an undirected graph is a
tree using depth-first search (medium) (20 points)
An undirected graph is a tree if and only if the graph contains no cycles. For example, this is a tree
because it contains no cycles:
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Programming assignment 2 (150 points)
While this graph contains cycles and therefore is not a tree:
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Programming assignment 2 (150 points)
In this part of the assignment, you will write a depth-first search through a graph to determine whether
the graph contains cycle. A cycle is detected when depth-first search find a graph node that was already
visited.
Input format
Your program should take a single command line argument specifying the path to an input file. Test
cases for your program are in the tests/ directory. In each test case, the first line records the number of
nodes N in the graph. Then, the adjacency matrix is recorded in the subsequent N rows of the file.
Output format
You should print "yes" if the graph is a tree, "no" if the graph is not a tree. Expected outputs from your
program for each test case are in the answers/ directory.
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Programming assignment 2 (150 points)
4. solveMaze: Finding the shortest path through a maze
with cycles using breadth-first search (medium) (20
points)
In this third part of the assignment, you will write a program to find a shortest path in a graph from a
source node to a destination node using breadth-first search. The graph representing the maze may
contain cycles, so it is important avoid revisiting nodes that have already been visited.
Many important problems in artificial intelligence, robotics motion planning, and self-driving cars boil
down to solving mazes on graphs. In classes such as AI and robotics, you will learn about advanced
algorithms for solving mazes using heuristics (or informed guesses) that minimize search time.
Input format
Your program should take TWO command line arguments. The first argument specifies the path to an
input file describing a graph like previous portions of this assignment. The second argument specifies
the path to an input file describing a query. The first line of the query file specifies the source node where
you begin your search. The second line of the query file specifies the target node you want to reach.
Output format
You should print a list of edges that, taken together, connect the source node to the target node in the
graph. Again, the ordering of the nodes in each edge does not matter. The ordering of the edges does
not matter. The autograder will check to see if you give a minimal set of edges that connect the source
and target nodes.
5. mst: Finding the minimum spanning tree of a
undirected weighted graph (medium) (20 points)
A weighted graph is a graph that has a numerical weight property on each edge. The minimum spanning
tree (MST) of an undirected weighted graph is a tree that connects all nodes in the graph, and at the
same time minimizing the sum of the weights of the tree's edges. Many important problems in computer
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Programming assignment 2 (150 points)
networking and operations research boil down to finding MSTs on graphs. As an example, this is a
undirected weighted graph:
And this is its MST:
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Programming assignment 2 (150 points)
The edges (0,1) and (1,2) connects all nodes in the graph, and picking these edges minimizes the total
weight of the tree. If all the weights in an undirected weighted graph are unique, then the MST is also
unique, meaning everyone will find the same MST for a given graph.
In this part of the assignment, you will write a program implementing another example of a greedy
algorithm to find the MST. Several algorithms solve this problem, but Prim's algorithm
(https://en.wikipedia.org/wiki/Prim%27s_algorithm) is likely the easiest to implement.
Input format
Your program should take a single command line argument specifying the path to an input file. Test
cases for your program are in the tests/ directory. In each test case, the first line records the number of
nodes N in the graph. Then, the adjacency matrix is recorded in the subsequent N rows of the file. This
time, the adjacency matrix contains floating point numbers. 0.0 indicates no edge between two nodes.
Any other value indicates an edge with the given value as the edge weight.
Output format
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Programming assignment 2 (150 points)
Expected outputs from your program for each test case are in the answers/ directory. You should print a
list of edges that, taken together, form the MST of the input graph. Again, the ordering of the nodes in
each edge does not matter. The ordering of the edges does not matter.
6. findCycle: Finding a cycle in a directed graph using
depth-first search (hard) (25 points)
A directed graph is a graph where edges are directional; that is, edges (p,q) and (q,p) are distinct. An
important class of directed graphs are directed acyclic graphs (DAGs), which have broad applications in
programming languages and compilers. A DAG is any directed graph with no cycles. For example, this is
a directed graph:
The above graph is not a DAG because it contains cycles. The cycles are:
1 2
4 7
4 5 7
By extension, these rotated versions are also valid cycles of the above graph (and they are all such
possible rotations):
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2
7
5
7
Programming assignment 2 (150 points)
1
4
7 4
4 5
In this final part of the assignment, you will bring together ideas you have used throughout this
assignment to find and print a cycle in a directed graph. If no cycles are found, your program will report
that the graph is a DAG. You can use any algorithm for this task; either the DFS or the BFS approaches
you have used in this assignment so far can be useful.
Input format
Your program should take a single command line argument specifying the path to an input file. Test
cases for your program are in the tests/ directory. In each test case, the first line records the number of
nodes N in the graph. Then, the adjacency matrix is recorded in the subsequent N rows of the file. This
time, the adjacency matrix represents a directed graph.
Output format
You should print a single line of nodes (separated by spaces) that forms a cycle in the input directed
graph. For example, for the example directed graph above you can print any one of the seven cycles
listed above. This time, the ordering of the nodes does matter as this is a directed graph. If no cycles
were found, your program should print "DAG", meaning the graph is acyclic. The known cycles for each
test case are in the answers/ directory. You can print out rotated versions of the known cycles; the
autograder will see that rotated cycles are equivalent.
Hint
Suppose you enter the graph from 0, and find a cycle by following the path
0->7->4->5->7
Upon seeing 7 again, you know you have detected a cycle. You have to carefully determine where the
cycle begins and ends in the path you have traversed.
7. matChainMul: Multiply a chain of matrices with an
optimal number of multiplication operations (hard) (25
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Programming assignment 2 (150 points)
points)
Dynamic programming algorithms are an important class of algorithms where a complex optimization
problem is divided into subproblems, and the subproblems are tackled recursively. They have many
applications throughout computer science such as computer networks and also in genome sequencing.
Here we will study how dynamic programming is used to optimally multiply a chain of matrices
such that the fewest number of multiplication operations are needed.
(https://en.wikipedia.org/wiki/Matrix_chain_multiplication) The matrix chain multiplication problem is to
multiply at least three matrices, such as A*B*C*D. Because matrix multiplication is associative, one can
perform this multiplication in one of several orders of operation:
1. A(B(CD))
2. A((BC)D)
3. (AB)(CD)
4. (A(BC))D
5. ((AB)C)D
Depending on the dimensions of the matrices A, B, C, and D, the number of multiplications needed may
vary greatly depending on how you parenthesize the matrix chain multiplication problem.
The dynamic programming approach to find the parenthesization that needs the fewest multiplication
operations is to break the problem into subproblems, recursively find the cost of each subproblem, and
then select among the subproblems for the most optimal approach. To be more concrete, suppose we
want to perform that matrix chain multiplication of ABCD. For the first decision we would select among
three ways to partition the problem:
1. A(BCD)
2. (AB)(CD)
3. (ABC)D
Now, the minimum cost of calculating BCD cannot be determined directly, so we have to recursively
determine the cost of calculating B(CD) vs. (BC)D in order to determine the true cost of option 1.
Likewise, the minimum cost of calculating ABC cannot be determined directly, so we have to recursively
determine the cost of calculating A(BC) vs. (AB)C in order to determine the true cost of option 3.
Input format
In the matChainMul/tests/ directory you have six given input test cases. In each test case file the format
is as follows:
1. The first line is the number matrices in the chain.
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Programming assignment 2 (150 points)
2. The second line is the row and column dimensions of the first matrix, separated by a space.
3. The subsequent several lines are the contents of the first matrix, one row per line, and each column
element separated by a space in each line.
4. After the first matrix is complete, the second matrix begins with a single line describing the row and
column dimensions of the second matrix, and so on.
For example, this input:
3
3
0
0
1
2
1
1
1
2
2
2
2
1
1
2
3
Describes the matrix chain multiplication:
Your program
You should write a C program that takes as a command line input the path to the test case file:
./matChainMul tests/test0.txt
Then, you should use a dynamic programming algorithm to calculate the product while using as few
multiplication operations as possible.
You should use as a reference the provided example codes for dynamic programming and for matrix
(non-chain) multiplication.
You are provided two key pieces of provided code for this assignment.
The matChainMul_provided.c code
In 2023_0s_211/pa2/matChainMul/matChainMul_provided.c you are given a working example of a C
program that calculates the optimal cost of performing matrix chain multiplication. This code does not
perform the actual matrix multiplication to find the product. Your task is to complete the program in
matChainMul/ to calculate the matrix chain multiplication product.
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Programming assignment 2 (150 points)
The pa2/matMul/matMul.h code
In 2023_0s_211/pa2/matMul/matMul.h you are given an implementation of matrix multiplication that calls
a special function mul() to perform multiplication. The function also tallies all the scalar multiplications
performed as part of matrix multiplication. Upon program exit, the program writes a file called
mul_op_count.txt which reports the number of scalar multiplications that took place as part of program
execution.
Expected output format
In the answers/ directory you have a file which records the expected number of operations and also the
expected product matrix. You should print the product matrix only, one row per line, each column
element separated by a space. For example, the answer.txt to the example above is:
12
4 6
4 6
4 6
Which means that 12 multiplications are needed at minimum, and the matrix product is:
You should print the matrix product. As a side of effect of using the using the matMul function in
matMul.h, your program will also write a file called mul_op_count.txt which reports the number of scalar
multiplications invoked to multiply the matrix chain. The autograder will check to see you used the
expected number of scalar multiplication operations.
How to submit
From the pa2/ directory, you can check on the outputs of our autograder script.
./assignment_autograder.py
or
python3 assignment_autograder.py
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Programming assignment 2 (150 points)
When you are ready to submit, from the 2023_0s_211/pa2/ directory, you should run these
commands to make sure you are in the correct working directory:
pwd
should report you are in the directory 2023_0s_211/pa2.
ls
should list at least these files and
subdirectories: assignment_autograder.py, quickselect/, editDistance/, and balanced/. Once you are
sure you are in the right directory, run:
tar cvf pa2.tar ./
The assignment_autograder can check and grade the generated tar file:
./assignment_autograder.py pa2.tar
Make sure the tar file passes this test and returns the grade you expect. We will not accept or
regrade empty tar files or tar files containing the wrong organization. Upload the file pa2.tar here on
Canvas. By submitting your assignment, you are agreeing to the Rutgers Honor Pledge: “On my
honor, I have neither received nor given any unauthorized assistance on this assignment.” We will
not be accepting late assignments. The Canvas submission site will close to enforce the deadline,
so be sure to submit early. If anything is not clear, reach out to your classmates and the instructors
on the class Piazza!
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