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Homework 4: hash tables and sorting algorithms
Homework 4
This assignment consists of two parts. The first part consists of problems that will give you
more practice working with hash tables and sorting algorithms. The second part consists of a coding
exercise that asks you to implement a shortest path algorithm for a graph. You will also get some
practice working with multiple source files and Java applets.
Part I
1. Hash tables (5 points)
(a) Consider an open-address hash table with uniform hashing. Give upper bounds on the
expected number of probes in an unsuccessful search and on the expected number of
probes in a successful search when the load factor is 3/4 and when it is 7/8. Consider all
three schemes discussed in class namely: linear probing, quadratic probing, and double
hashing. Tabulate your results for comparison.
(b) Suppose that in the separate chaining scheme, we also keep the lists in sorted order.
With this modification, what is the running time for successful searches, unsuccessful
searches, insertions, and deletions? State your answers in terms of the load factor α.
2. (5 points)
This question deals with the heapsort algorithm shown below.
function Heapsort(arr )
Heapify array arr;
for i = n − 1 down to 1 do
swap arr[i] with arr[0];
restore heap property for the tree arr[0], . . . , arr[i − 1] by percolating down the root;
end
end
• Identify a suitable loop invariant that will help you show the correctness of heapsort.
• Show that your loop invariant holds by showing the initialization and maintenance steps.
You can assume that the percolate down operation is correct.
• Use your loop invariant to show the partial correctness of heapsort.
3. (7 points)
This question deals with the quick sort algorithm.
(a) Suppose that we modify the partitioning algorithm so that it always partitions an input
array of length n into two partitions in such a way that the length of the left partition
is n − K and the length of the right partition is K − 1 (for some constant K, where
K 0). Let us refer to this partitioning algorithm as KPartition.
Now consider the following variation of quick sort called KQuickSort:
1
CPSC 331: Spring 2018 HW4
function KQuickSort(int[] A, int low, int high)
n = high - low + 1;
if n < K then
insertionSort(A, low, high) ;
else
q = KPartition(A, low, high) ;
KQuickSort (A, low, q-1) ;
KQuickSort (A, q+1, high) ;
end
end
Let T(n) denote the worst-case number of steps to run KQuickSort on input arrays of
size n. Complete the following recurrence relation for T(n). Assume that KPartition
takes Θ(n) steps to partition an array of size n.
T(n) ≤
(
n < K,
n ≥ K.
(b) Use the substitution method to find an upper bound on T(n). For simplicity, assume
that n is a multiple of K, i.e. n = m · K (where m is a non-negative integer).
(c) Using the Big-O notation, give an asymptotic expression for the worst-case running time
of KQuickSort. A proof is not required.
(d) How does the worst-case asymptotic running time of KQuickSort compare with that of
quick sort?
4. (3 points)
Prove that a tree with n vertices has n − 1 edges.
Part II
In this coding exercise, you will implement a shortest path algorithm that finds the shortest
path between two nodes (if any) in a maze. You can either use a breadth first search (BFS) to find
the shortest unweighted path, or (for an added bonus) use Dijkstra’s shortest path algorithm to
determine both weighted and unweighted shortest paths.
Introduction
You are given a maze that is represented as a graph consisting of an n × n grid of vertices.
Vertices in the maze are connected via weighted undirected edges to form paths as shown in the
figure below.
CPSC 331: Spring 2018 HW4
The vertices are numbered from 1 to n
2
in a column-major fashion as shown above. The connectivity
in the graph is represented in a text file. The first line of this text file consists of the number
n and the rest of the lines consist of three integer entries per line in the format:
<fromVertex <toVertex <weight
In other words, each line after the first corresponds to an edge in the graph and there are as
many entries as the number of edges. The edges are bidirectional, i.e. each edge appears twice in
the file.
For the example graph pictured above, please see the file maze.txt. The first few lines from
this file are:
3
CPSC 331: Spring 2018 HW4
4
1 2 11
1 5 11
2 1 11
2 3 8
3 2 8
3 4 8
3 7 8
4 3 8
5 1 11
...
Shortest Unweighted Path
For this part, you can ignore the edge weights.
Your task is to determine the shortest path from a source vertex to a target vertex using a
breadth first traversal. Source and target vertex pairs will be specified via an input query file that
consists of one pair per line. For each pair, please determine the shortest path from the source to
the target and use the the supplied MazeVisualizer to visualize the result. If there is no path
between a given pair, please indicate that by printing out an appropriate message.
Each line in the input query file has the following format:
<source vertex <target vertex
For example, for the source and target vertices 1 and 12 in the figure above, the input line is:
1 12
and the result is pictured below.
Please write a Java program that performs the following tasks:
• Reads the maze information from an input file and uses an appropriate data structure to store
this information.
4
CPSC 331: Spring 2018 HW4
• Reads source and target vertex pairs from an input query file. For each pair, perform a
BFS to find the shortest path (ignoring edge weights) from the source to the target. Adjacent
vertices should be enqueued in ascending order of vertex number. Alternatively, if you
are implementing the bonus portion (see below), your program should implement this using
Dijkstra’s algorithm with unit edge weights.
• Visualizes the paths using the supplied MazeVisualizer. MazeVisualizer is a Java applet
that provides methods to add edges and paths to be visualized. Multiple paths can be visualized
simultaneously. The main method in MazeVisualizer includes relevant calls that
demonstrate its use.
Your program will be invoked from the command line as follows:
java HW4 maze-file query-file
Please use the supplied maze file and the figure above to test your program. Additional test
files and associated output will be made available closer to the due date. Your program should be
able to infer the number of vertices in the maze from the maze file. Please do not hard code this
information as you will be required to test with larger mazes.
Unlike previous assignments, if you need to make use of auxiliary data structures, please feel
free to make use of Java collection classes. The implementation of the shortest path algorithm must
be your own. The use of any other code that is not your own is not allowed.
Bonus: Shortest Path using Dijkstra’s Algorithm
Optionally, please implement Dijkstra’s shortest path algorithm to find the shortest weighted
path between the source and target vertices supplied in the query file. Visualize these paths using
the supplied MazeVisualizer. Your program will be invoked from the command line as follows:
java HW4 [option] maze-file query-file
The command-line argument [option] can take one of two values:
--unweighted: determine the shortest unweighted path.
--weighted: determine the shortest weighted path.
To help you test your program, the shortest path using Dijkstra’s algorithm for the pair 1 and
12 (for the example above) is pictured below.
5
CPSC 331: Spring 2018 HW4
Additional test files and output will be supplied closer to the due date.
If you need to make use of auxiliary data structures, please feel free to make use of Java collection
classes. The implementation of Dijkstra’s shortest path algorithm must be your own. The use of
any other code that is not your own is not allowed.
Grading
You will be graded based on the functionality of your program as demonstrated by test input files
and the corresponding visual output. The following high level grading scheme will be used:
• Program compiles (3 points).
• Functional implementation of unweighted shortest paths as demonstrated by test files (12
points). Partial credit will be awarded based on the level of functionality achieved:
– File I/O (2 points).
– Graph representation (3 points).
– Implementation of shortest path algorithm (5 points for BFS, 10 points for Dijkstra’s).
– Demo (2 points).
If your program is not working, please let your TA know (during your demo) which parts
you have implemented for partial credit.
Submission
Part I
Please use D2L to submit a PDF file called part1.pdf (scanned or typeset) that is packaged
with the rest of the files for Part II. If you are scanning your solution, please submit the scanned
pages as a single PDF file.
Part II
Please package all Java source files needed to compile your program in a compressed archive
called hw4.zip. You do not need to include any maze files or query files. Please make sure that your
name appears at the beginning of each source file that you authored. In your archive, please also
include a README file that includes instructions on how to compile and run your program. Please
use the README file to cite any sources that you used. Please do not place the files in different
folders. Please also include the file part1.pdf in your archive.
This assignment consists of two parts. The first part consists of problems that will give you
more practice working with hash tables and sorting algorithms. The second part consists of a coding
exercise that asks you to implement a shortest path algorithm for a graph. You will also get some
practice working with multiple source files and Java applets.
Part I
1. Hash tables (5 points)
(a) Consider an open-address hash table with uniform hashing. Give upper bounds on the
expected number of probes in an unsuccessful search and on the expected number of
probes in a successful search when the load factor is 3/4 and when it is 7/8. Consider all
three schemes discussed in class namely: linear probing, quadratic probing, and double
hashing. Tabulate your results for comparison.
(b) Suppose that in the separate chaining scheme, we also keep the lists in sorted order.
With this modification, what is the running time for successful searches, unsuccessful
searches, insertions, and deletions? State your answers in terms of the load factor α.
2. (5 points)
This question deals with the heapsort algorithm shown below.
function Heapsort(arr )
Heapify array arr;
for i = n − 1 down to 1 do
swap arr[i] with arr[0];
restore heap property for the tree arr[0], . . . , arr[i − 1] by percolating down the root;
end
end
• Identify a suitable loop invariant that will help you show the correctness of heapsort.
• Show that your loop invariant holds by showing the initialization and maintenance steps.
You can assume that the percolate down operation is correct.
• Use your loop invariant to show the partial correctness of heapsort.
3. (7 points)
This question deals with the quick sort algorithm.
(a) Suppose that we modify the partitioning algorithm so that it always partitions an input
array of length n into two partitions in such a way that the length of the left partition
is n − K and the length of the right partition is K − 1 (for some constant K, where
K 0). Let us refer to this partitioning algorithm as KPartition.
Now consider the following variation of quick sort called KQuickSort:
1
CPSC 331: Spring 2018 HW4
function KQuickSort(int[] A, int low, int high)
n = high - low + 1;
if n < K then
insertionSort(A, low, high) ;
else
q = KPartition(A, low, high) ;
KQuickSort (A, low, q-1) ;
KQuickSort (A, q+1, high) ;
end
end
Let T(n) denote the worst-case number of steps to run KQuickSort on input arrays of
size n. Complete the following recurrence relation for T(n). Assume that KPartition
takes Θ(n) steps to partition an array of size n.
T(n) ≤
(
n < K,
n ≥ K.
(b) Use the substitution method to find an upper bound on T(n). For simplicity, assume
that n is a multiple of K, i.e. n = m · K (where m is a non-negative integer).
(c) Using the Big-O notation, give an asymptotic expression for the worst-case running time
of KQuickSort. A proof is not required.
(d) How does the worst-case asymptotic running time of KQuickSort compare with that of
quick sort?
4. (3 points)
Prove that a tree with n vertices has n − 1 edges.
Part II
In this coding exercise, you will implement a shortest path algorithm that finds the shortest
path between two nodes (if any) in a maze. You can either use a breadth first search (BFS) to find
the shortest unweighted path, or (for an added bonus) use Dijkstra’s shortest path algorithm to
determine both weighted and unweighted shortest paths.
Introduction
You are given a maze that is represented as a graph consisting of an n × n grid of vertices.
Vertices in the maze are connected via weighted undirected edges to form paths as shown in the
figure below.
CPSC 331: Spring 2018 HW4
The vertices are numbered from 1 to n
2
in a column-major fashion as shown above. The connectivity
in the graph is represented in a text file. The first line of this text file consists of the number
n and the rest of the lines consist of three integer entries per line in the format:
<fromVertex <toVertex <weight
In other words, each line after the first corresponds to an edge in the graph and there are as
many entries as the number of edges. The edges are bidirectional, i.e. each edge appears twice in
the file.
For the example graph pictured above, please see the file maze.txt. The first few lines from
this file are:
3
CPSC 331: Spring 2018 HW4
4
1 2 11
1 5 11
2 1 11
2 3 8
3 2 8
3 4 8
3 7 8
4 3 8
5 1 11
...
Shortest Unweighted Path
For this part, you can ignore the edge weights.
Your task is to determine the shortest path from a source vertex to a target vertex using a
breadth first traversal. Source and target vertex pairs will be specified via an input query file that
consists of one pair per line. For each pair, please determine the shortest path from the source to
the target and use the the supplied MazeVisualizer to visualize the result. If there is no path
between a given pair, please indicate that by printing out an appropriate message.
Each line in the input query file has the following format:
<source vertex <target vertex
For example, for the source and target vertices 1 and 12 in the figure above, the input line is:
1 12
and the result is pictured below.
Please write a Java program that performs the following tasks:
• Reads the maze information from an input file and uses an appropriate data structure to store
this information.
4
CPSC 331: Spring 2018 HW4
• Reads source and target vertex pairs from an input query file. For each pair, perform a
BFS to find the shortest path (ignoring edge weights) from the source to the target. Adjacent
vertices should be enqueued in ascending order of vertex number. Alternatively, if you
are implementing the bonus portion (see below), your program should implement this using
Dijkstra’s algorithm with unit edge weights.
• Visualizes the paths using the supplied MazeVisualizer. MazeVisualizer is a Java applet
that provides methods to add edges and paths to be visualized. Multiple paths can be visualized
simultaneously. The main method in MazeVisualizer includes relevant calls that
demonstrate its use.
Your program will be invoked from the command line as follows:
java HW4 maze-file query-file
Please use the supplied maze file and the figure above to test your program. Additional test
files and associated output will be made available closer to the due date. Your program should be
able to infer the number of vertices in the maze from the maze file. Please do not hard code this
information as you will be required to test with larger mazes.
Unlike previous assignments, if you need to make use of auxiliary data structures, please feel
free to make use of Java collection classes. The implementation of the shortest path algorithm must
be your own. The use of any other code that is not your own is not allowed.
Bonus: Shortest Path using Dijkstra’s Algorithm
Optionally, please implement Dijkstra’s shortest path algorithm to find the shortest weighted
path between the source and target vertices supplied in the query file. Visualize these paths using
the supplied MazeVisualizer. Your program will be invoked from the command line as follows:
java HW4 [option] maze-file query-file
The command-line argument [option] can take one of two values:
--unweighted: determine the shortest unweighted path.
--weighted: determine the shortest weighted path.
To help you test your program, the shortest path using Dijkstra’s algorithm for the pair 1 and
12 (for the example above) is pictured below.
5
CPSC 331: Spring 2018 HW4
Additional test files and output will be supplied closer to the due date.
If you need to make use of auxiliary data structures, please feel free to make use of Java collection
classes. The implementation of Dijkstra’s shortest path algorithm must be your own. The use of
any other code that is not your own is not allowed.
Grading
You will be graded based on the functionality of your program as demonstrated by test input files
and the corresponding visual output. The following high level grading scheme will be used:
• Program compiles (3 points).
• Functional implementation of unweighted shortest paths as demonstrated by test files (12
points). Partial credit will be awarded based on the level of functionality achieved:
– File I/O (2 points).
– Graph representation (3 points).
– Implementation of shortest path algorithm (5 points for BFS, 10 points for Dijkstra’s).
– Demo (2 points).
If your program is not working, please let your TA know (during your demo) which parts
you have implemented for partial credit.
Submission
Part I
Please use D2L to submit a PDF file called part1.pdf (scanned or typeset) that is packaged
with the rest of the files for Part II. If you are scanning your solution, please submit the scanned
pages as a single PDF file.
Part II
Please package all Java source files needed to compile your program in a compressed archive
called hw4.zip. You do not need to include any maze files or query files. Please make sure that your
name appears at the beginning of each source file that you authored. In your archive, please also
include a README file that includes instructions on how to compile and run your program. Please
use the README file to cite any sources that you used. Please do not place the files in different
folders. Please also include the file part1.pdf in your archive.
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