讲解COMP 250、MATH 240辅导、java编程语言调试、java辅导
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COMP 250 Fall 2019
The teaching staff handling this assignment consists of Giulia Alberini (email), Sayantan Datta
(email), Anmoljeet Gill (email), Alexander Orzechowski (email), Kelly Rombough (email), Rajveer
Gandhi (email).
You can find their office hours on the google calendare shared with you on mycourses.
Learning Objectives
This assignment is meant for you to practice working with hashmaps and graphs. In particular,
you will construct a graph of vertices and edges using a breadth-first or depth-first search. You will
learn more in-depth about graph theory in MATH 240 and COMP 251. You will also implement
one of the O(n log(n)) sorting algorithms that you’ve learned about in class, such as MergeSort or
QuickSort.
General Instructions
• Submission instructions
– Late assignments will be accepted up to 2 days late and will be penalized by 10 points per
day. Note that submitting one minute late is the same as submitting 23 hours late. We
will deduct 10 points for any student who has to resubmit after the due date (i.e. late)
irrespective of the reason, be it wrong file submitted, wrong file format was submitted
or any other reason. This policy will hold regardless of whether or not the student can
provide proof that the assignment was indeed “done” on time.
– Don’t worry if you realize that you made a mistake after you submitted : you can submit
multiple times. We will grade all your submissions and the best score will be selected.
We encourage you to submit a first version a few days before the deadline (computer
crashes do happen and myCourses may be overloaded during rush hours).
– Please store all your files in a folder called “Assignment4”, zip the folder and submit it
to myCourses. Inside your zipped folder, there must be the following files.
∗ Sorting.java
∗ SearchEngine.java
Do not submit any other files, especially .class files. Any deviation from these
requirements may lead to lost marks.
1
• Starter code is provided for this assignment. Do not change any of the class names, file names,
method headers. You can add helper methods if you wish. Note also that for this assignment,
you are NOT allowed to import any other class (all import statements other than the one
provided in the starter code will be removed). In terms of package names, you can do whatever
you wish. Any failure to comply with these rules will give you an automatic 0.
• We included with these instructions a tester class, which is a mini version of the final tester
used to evaluate your assignment. If your code fails those tests, it means that there is a
mistake somewhere. Even if your code passes those tests, it may still contain some errors. We
will test your code on a much more challenging set of examples.
• You will automatically get 0 if your code does not compile.
• Failure to comply with any of these rules will be penalized. If anything is unclear, it is up to
you to clarify it by asking either directly a TA during office hours, or on the discussion board
on myCourses.
2
Introduction
A search engine is software, usually accessed on the Internet, designed to carry out searches in a
database of information according to the user’s query. The results provided are meant to best match
what the user is trying to find and they are usually displayed in order of importance. The first search
engine ever developed is considered to be Archie. It was named to resemble the word “archive”,
and it was created in the 1990 by Alan Emtage, Bill Heelan and J. Peter Deutsch, computer science
students at McGill. Today, there are many different search engines available on the Internet, each
with their own abilities and features. The most popular of them all being Google.
A search engine performs the following tasks:
• Crawling and Indexing
• Ranking
• Searching
Crawling and Indexing Crawling is the process during which a search engine use programs,
generally called spiders or crawlers, to collect information from web pages. By following links from
a web page to another, the spiders find new content. Once a page is crawled, the data contained
in the page is processed and indexed. Indexing is the process of storing and organizing the content
found during the crawling. It generally involve associating words and other definable tokens found
on web pages to a list of web pages that contain them. It also includes recording links to other
pages. These associations are all stored in a database which is then used for web search queries.
During this process the webgraph is created/updated. This is a directed graph where each node
represents a page, and an edge represents an hyperlink from one page to another. This is also
when what we refer to as the ‘word index’ is created. You can think of it as a mapping from single
words to a list of urls containing them. Note that for actual search engine, this task (crawling and
indexing) is never actually completed. Given the constantly changing nature of the Web, spiders
are always crawling to keep the database up to date. The Google Search index contains hundreds
of billions of webpages.
Ranking How useful a search engine is depends on how relevant are the results obtained when we
query it. There are millions of pages containing a specific word, but some pages are generally more
relevant, or authoritative than others. To determine which pages to show in the search results and in
what order, search engines use the data collected to perform a ranking. Different search engines use
different algorithms to rank pages. Developed by Larry Page and Sergey Brin in 1996, PageRank
is one of the algorithms used at Google to rank webpages based on their importance. Using the
webgraph, each webpage is assigned a numerical value that measure its relative importance. In
simple terms, the rank of a webpage depends on the webpages that link to it. The contribution to
the rank is higher if these linking webpages have a high rank themselves, while it’s lower if these
linking webpage also link to a lot of other webpages.
Searching In this step, search engines receive a query from the user. Using the index created
while crawling and the ranking based on the webgraph, the engine outputs a list of relevant web
3
pages in order of importance. Do all search engines give the same results? Not necessarily. Search
engines use different spiders to crawl and use different proprietary algorithms to index the data.
Each index is therefore a search engine’s representation of how they see the web. Also the algorithms
to rank and search the data are different, so every search engine has its own approach to finding
what you’re trying to find. Finally, personalisation adapts the search to a specific computer/user.
The results may be based on your geographical location, what else you’ve searched for, and what
results were preferred by other users searching for the same thing, for example. Search engines might
use and weigh all these factors in a unique way, which will lead to different search results.
The starter code contains four classes:
• XmlParser: This class has two methods: one to read the content of the xml database (which
will serve as proxy for a web page for our assignment) given a url, and the other to extract
information from it.
– getContent(String url) returns an ArrayList of Strings corresponding to the set of
words in the web page located at the given url. You will need to use this method while
crawling in order to build your word index.
– getLinks(String url) returns an ArrayList of Strings containing all the hyperlinks
going out of the given url. You will need this method to build the graph representing
the data found while crawling.
You should NOT modify this class at all.
• MyWebGraph: This class implements the data structure needed to store the data collected
during the crawling phase. It has an inner class called WebVertex which is used to store data
related to a specific web page. You should NOT modify this class at all.
• SearchEngine: This is one of the classes in which you will have to add your code. You will
be implementing methods that performs the three tasks described above.
• Sorting: This is utility class containing methods that implement some sorting algorithm.
You will be asked to implement one of them.
Your task
In this assignment you are going to write a search engine program that will:
1. explore a portion of the web (which we’ll simulate through a database),
2. build an index of the words contained in each web page,
3. analyze the structure of the web graph to establish which web page should be ranked higher,
4. use this analysis to perform simple searches where a query word is entered by the user and a
sorted list of relevant web pages available is returned.
Although we are going to apply these tools using a local database, what you will develop is a
simplified version of the what is used at Google to answer actual web queries.
4
To be able to implement the search engine program described above, we need a graph data structure
that will allow us to store all the information related to the web pages. Such class is provided to
you.
[Provided] The class MyWebGraph is an implementation of a directed graph using adjacency lists.
You will be using this type of data structure to store the information your program collects
in the crawling phase. Each node in the graph will store a String corresponding to the url of
a webpage. The class has the following field:
• A HashMap storing all the vertices in the graph. Note that we’ll be labelling each vertex
with the String corresponding to the url of the webpage represented by this vertex.
The class has also the following public methods:
• The constructor MyWebGraph() which does not take any input and initializes the field
with an empty HashMap.
• An addVertex() method which takes as input a String representing a url. The method
adds the corresponding vertex to the graph.
• An addEdge() method which takes as input two Strings, i.e. two urls. The method
adds to the graph an edge from the vertex labelled with the first input to the vertex
labelled with the second input.
• A getNeighbors() method which takes as input a String representing a url and returns
an ArrayList of all the hyperlinks contained in the specified url.
• A getVertices() method which returns the list of all the urls represented in the graph.
• A getEdgesInto() method that takes as input a String representing a url. The method
returns an ArrayList of Strings of all the pages that contains an hyperlink to the
specified url.
• A getOutDegree() method which takes as input a String representing a url and returns
the number of hyperlinks in the specified page.
• A getPageRank() method which takes as input a String representing a url and returns
the rank of the specified page.
• A setPageRank() method which takes as input a String representing a url and a double
representing its rank. The method assigns the input number as the rank of the specified
page.
• A getVisited() method which takes as input a String representing a url and returns
whether or not the specified page has been visited.
• A setVisited() method which takes as input a String representing a url and a boolean.
The method assigns the input boolean to the visited field of the specified page.
[20 points] The Sorting class is a utility class that at the moment contains a static method
called slowSort(). This method takes as input a HashMap where the values are Comparable.
The method returns an ArrayList containing all the keys in the map, sorted in descending
5
order based on the values they map to. The time complexity of slowSort is O(n
2
), where
n is the number of pairs in the map. Your task is to implement a method called fastSort
which performs the same task as slowSort but with a time complexity of O(n · log(n)).
[80 points] The SearchEngine class has the following fields:
• A HashMap called wordIndex used to represent the mapping from single words to a list
of urls containing them.
• A MyWebGraph called internet which is used to store all relevant information regarding
the structure of the webgraph.
• An XmlParser called parser which you should use to retrieve information for the
database serving as a proxy for the web.
In this class, you should implement the following methods:
[35 points ] crawlAndIndex
This method takes a String as input representing a url. It will perform a graph traversal
of the web, starting at the given url. You can use either a depth-first search or a breadth-
first search, and you can make it either recursive or non-recursive. For each url visited,
the method should do the following:
• Update the web graph by adding the appropriate vertex and edges.
• Update the word index so that every word in the url just visited appear in the
mapping.
Note that, it is possible for the graph to contain cycles. For instance, if it possible that
two web pages link to each other. To prevent this method from getting stuck in an
infinite loop, you should make sure to keep track of which web page has already been
visited. This method should stop crawling once all web pages (in the database) have
been visited.
[35 points ] assignPageRanks and computeRanks
The method computeRanks is a helper method for assignPageRanks. We specifically
ask you to implement it so that our grader will be able to assign partial marks to your
implementation. The purpose of assignPageRanks is to assign a rank to each vertex in
the web graph. It will only be called after crawlAndIndex has been executed. Here are
the ideas we will be using to assign a rank to a web page:
• Good web pages are cited by many other pages. If we think about it in terms of the
webgraph, this means that we should prefer web pages (i.e. vertices) with a large
in-degree.
• Web pages that link to a large number of other pages are less valuable. In terms of
webgraph, this means that we will value less web pages (i.e. vertices) with a large
out-degree.
• A link from a web page is more valuable if the web page is itself a good one. In
graph terms, higher the rank of a web page (i.e. vertex), more valuable an in-edge
6
from it would be.
Note that the rank of a page depends solely on the structure of the graph we have created
while crawling. To represent the ideas just described, let
• the pr(w) be the page rank of a vertex w
• the out(w) be the out-degree of a vertex w
• w1, w2, ..., wk be all the vertices in the graph that have an out edge going into v.
Then, the following equation determines the page rank of a vertex v:
The constant d is called the damping factor and it is added for technical reasons to
account for the probability that an imaginary surfer who is randomly clicking on links
will eventually stop clicking. For this assignment, we will be using a damping factor of
0.5. As you may notice, pr(v) is defined as a function of pr(wi). If N is the total number
of vertices in the graph (v1, v2, . . . , vN ), then we have a system of N linear equations in
N variables. The unknown values, i.e. our variables, are the pr(vi). To determine their
values, we should solve the system of linear equations described above. To do so, we
could use linear algebra, but the implementation of gaussian elimination runs in O(n
3
).
To improve the efficiency, we can instead use an iterative algorithm that approximate
the result. The idea is the following:
• Start by initializing pr(vi) to 1 for all 0 ≤ i ≤ N
• Repeat the following until convergence:
– compute pr(vi) for all i using the formula above.
This single step is what the helper method computeRanks should be doing.
The method takes as input an ArrayList representing the urls in the
web graph and returns an ArrayList representing the newly computed
ranks for those urls. Note that the double in the output list is matched to the
url in the input list using their position in the list. That is, the page rank of the
url stored in position i in the input list, can be found in position i in the output
list.
Convergence is reached when |prk−1
(vi) − prk
(vi)| < for all i, where prj
(vi) represents
the computation of pr(vi) in the j-th iteration, and is the input to the method.
Note that, you can assume that the web graph does not contain self loops and every
vertex in the graph has at least one outgoing edge.
7
Consider the following web graph:
Suppose we’d like to compute the page rank assigned to each vertex with an equal to
0.01. We start by assigning value 1 to each of them:
• pr(A) = 1
• pr(B) = 1
• pr(C) = 1
• pr(D) = 1
Then we need to use the following formula (same as the formula above, but with damping
factor of 0.5) to recompute the page ranks for each vertex, until convergence.
After the first iteration we get:
• pr(A) = 1/2 + 1/2 ∗ (1/3 + 1) = 7/6 = 1.166
• pr(B) = 1/2 + 1/2 ∗ (1/2) = 3/4 = 0.75
• pr(C) = 1/2 + 1/2 ∗ (1/2 + 1/3 + 1) = 17/12 = 1.416
• pr(D) = 1/2 + 1/2 ∗ (1/3) = 4/6 = 0.666
After the second iteration we get:
• pr(A) = 1/2 + 1/2 ∗ (0.75/3 + 1.416) = 1.333
• pr(B) = 1/2 + 1/2 ∗ (1.166/2) = 0.791
• pr(C) = 1/2 + 1/2 ∗ (1.166/2 + 0.75/3 + 0.666) = 1.25
• pr(D) = 1/2 + 1/2 ∗ (0.75/3) = 0.625
8
After 5 iterations we reach convergence since the difference between the rank values in
the 4th iteration and the rank values in the fifth iteration is smaller than . The ranks
at the end are:
• pr(A) = 1.270
• pr(B) = 0.819
• pr(C) = 1.274
• pr(D) = 0.636
[10 points ] getResults
This method takes as input a String representing a single-word query. It then returns
an ArrayList of urls that contain such word, ordered based on their rank from most
relevant to least relevant. This method will be called only after both crawlAndIndex
and assignPageRanks have been executed. This method should use one of the sorting
algorithms from the Sorting class. Ideally, it should use fastSort(), but if you did not
implement it, you can use slowSort().
Testing
To test your code we provided you with two XML files that act as a proxy for the internet. The XML
files contains proxy web-pages indicated by the tag name webpage. Each web-page contains multiple
hyperlinks to other web-pages indicated by the tag link. The contents of a web-page is marked using
the tag content. To help you with testing, we also provide the expected ranks of the web-pages
computed using the algorithm described in the previous section. Note that rank is not part of an
actual web-page, we simply provide them to help you with debugging. Page rank is indicated by
tag rank and can be accessed in your code using the function XmlParser::getPageRank() which
takes as input the name of the page.
• testAcyclic.xml - The provided tester use this database to test your code. Note that this
database does not have any circular chains - making it easy for you to test and debug.
• test.xml - We do not provide any test cases for this database but is more general than the
one above. We encourage you to create your tests with this database that will help you find
more intricate issues with your code.
Finally, you should create your databases that are much larger as we will test on a much bigger
database in the final grading.
9
COMP 250 Fall 2019
The teaching staff handling this assignment consists of Giulia Alberini (email), Sayantan Datta
(email), Anmoljeet Gill (email), Alexander Orzechowski (email), Kelly Rombough (email), Rajveer
Gandhi (email).
You can find their office hours on the google calendare shared with you on mycourses.
Learning Objectives
This assignment is meant for you to practice working with hashmaps and graphs. In particular,
you will construct a graph of vertices and edges using a breadth-first or depth-first search. You will
learn more in-depth about graph theory in MATH 240 and COMP 251. You will also implement
one of the O(n log(n)) sorting algorithms that you’ve learned about in class, such as MergeSort or
QuickSort.
General Instructions
• Submission instructions
– Late assignments will be accepted up to 2 days late and will be penalized by 10 points per
day. Note that submitting one minute late is the same as submitting 23 hours late. We
will deduct 10 points for any student who has to resubmit after the due date (i.e. late)
irrespective of the reason, be it wrong file submitted, wrong file format was submitted
or any other reason. This policy will hold regardless of whether or not the student can
provide proof that the assignment was indeed “done” on time.
– Don’t worry if you realize that you made a mistake after you submitted : you can submit
multiple times. We will grade all your submissions and the best score will be selected.
We encourage you to submit a first version a few days before the deadline (computer
crashes do happen and myCourses may be overloaded during rush hours).
– Please store all your files in a folder called “Assignment4”, zip the folder and submit it
to myCourses. Inside your zipped folder, there must be the following files.
∗ Sorting.java
∗ SearchEngine.java
Do not submit any other files, especially .class files. Any deviation from these
requirements may lead to lost marks.
1
• Starter code is provided for this assignment. Do not change any of the class names, file names,
method headers. You can add helper methods if you wish. Note also that for this assignment,
you are NOT allowed to import any other class (all import statements other than the one
provided in the starter code will be removed). In terms of package names, you can do whatever
you wish. Any failure to comply with these rules will give you an automatic 0.
• We included with these instructions a tester class, which is a mini version of the final tester
used to evaluate your assignment. If your code fails those tests, it means that there is a
mistake somewhere. Even if your code passes those tests, it may still contain some errors. We
will test your code on a much more challenging set of examples.
• You will automatically get 0 if your code does not compile.
• Failure to comply with any of these rules will be penalized. If anything is unclear, it is up to
you to clarify it by asking either directly a TA during office hours, or on the discussion board
on myCourses.
2
Introduction
A search engine is software, usually accessed on the Internet, designed to carry out searches in a
database of information according to the user’s query. The results provided are meant to best match
what the user is trying to find and they are usually displayed in order of importance. The first search
engine ever developed is considered to be Archie. It was named to resemble the word “archive”,
and it was created in the 1990 by Alan Emtage, Bill Heelan and J. Peter Deutsch, computer science
students at McGill. Today, there are many different search engines available on the Internet, each
with their own abilities and features. The most popular of them all being Google.
A search engine performs the following tasks:
• Crawling and Indexing
• Ranking
• Searching
Crawling and Indexing Crawling is the process during which a search engine use programs,
generally called spiders or crawlers, to collect information from web pages. By following links from
a web page to another, the spiders find new content. Once a page is crawled, the data contained
in the page is processed and indexed. Indexing is the process of storing and organizing the content
found during the crawling. It generally involve associating words and other definable tokens found
on web pages to a list of web pages that contain them. It also includes recording links to other
pages. These associations are all stored in a database which is then used for web search queries.
During this process the webgraph is created/updated. This is a directed graph where each node
represents a page, and an edge represents an hyperlink from one page to another. This is also
when what we refer to as the ‘word index’ is created. You can think of it as a mapping from single
words to a list of urls containing them. Note that for actual search engine, this task (crawling and
indexing) is never actually completed. Given the constantly changing nature of the Web, spiders
are always crawling to keep the database up to date. The Google Search index contains hundreds
of billions of webpages.
Ranking How useful a search engine is depends on how relevant are the results obtained when we
query it. There are millions of pages containing a specific word, but some pages are generally more
relevant, or authoritative than others. To determine which pages to show in the search results and in
what order, search engines use the data collected to perform a ranking. Different search engines use
different algorithms to rank pages. Developed by Larry Page and Sergey Brin in 1996, PageRank
is one of the algorithms used at Google to rank webpages based on their importance. Using the
webgraph, each webpage is assigned a numerical value that measure its relative importance. In
simple terms, the rank of a webpage depends on the webpages that link to it. The contribution to
the rank is higher if these linking webpages have a high rank themselves, while it’s lower if these
linking webpage also link to a lot of other webpages.
Searching In this step, search engines receive a query from the user. Using the index created
while crawling and the ranking based on the webgraph, the engine outputs a list of relevant web
3
pages in order of importance. Do all search engines give the same results? Not necessarily. Search
engines use different spiders to crawl and use different proprietary algorithms to index the data.
Each index is therefore a search engine’s representation of how they see the web. Also the algorithms
to rank and search the data are different, so every search engine has its own approach to finding
what you’re trying to find. Finally, personalisation adapts the search to a specific computer/user.
The results may be based on your geographical location, what else you’ve searched for, and what
results were preferred by other users searching for the same thing, for example. Search engines might
use and weigh all these factors in a unique way, which will lead to different search results.
The starter code contains four classes:
• XmlParser: This class has two methods: one to read the content of the xml database (which
will serve as proxy for a web page for our assignment) given a url, and the other to extract
information from it.
– getContent(String url) returns an ArrayList of Strings corresponding to the set of
words in the web page located at the given url. You will need to use this method while
crawling in order to build your word index.
– getLinks(String url) returns an ArrayList of Strings containing all the hyperlinks
going out of the given url. You will need this method to build the graph representing
the data found while crawling.
You should NOT modify this class at all.
• MyWebGraph: This class implements the data structure needed to store the data collected
during the crawling phase. It has an inner class called WebVertex which is used to store data
related to a specific web page. You should NOT modify this class at all.
• SearchEngine: This is one of the classes in which you will have to add your code. You will
be implementing methods that performs the three tasks described above.
• Sorting: This is utility class containing methods that implement some sorting algorithm.
You will be asked to implement one of them.
Your task
In this assignment you are going to write a search engine program that will:
1. explore a portion of the web (which we’ll simulate through a database),
2. build an index of the words contained in each web page,
3. analyze the structure of the web graph to establish which web page should be ranked higher,
4. use this analysis to perform simple searches where a query word is entered by the user and a
sorted list of relevant web pages available is returned.
Although we are going to apply these tools using a local database, what you will develop is a
simplified version of the what is used at Google to answer actual web queries.
4
To be able to implement the search engine program described above, we need a graph data structure
that will allow us to store all the information related to the web pages. Such class is provided to
you.
[Provided] The class MyWebGraph is an implementation of a directed graph using adjacency lists.
You will be using this type of data structure to store the information your program collects
in the crawling phase. Each node in the graph will store a String corresponding to the url of
a webpage. The class has the following field:
• A HashMap storing all the vertices in the graph. Note that we’ll be labelling each vertex
with the String corresponding to the url of the webpage represented by this vertex.
The class has also the following public methods:
• The constructor MyWebGraph() which does not take any input and initializes the field
with an empty HashMap.
• An addVertex() method which takes as input a String representing a url. The method
adds the corresponding vertex to the graph.
• An addEdge() method which takes as input two Strings, i.e. two urls. The method
adds to the graph an edge from the vertex labelled with the first input to the vertex
labelled with the second input.
• A getNeighbors() method which takes as input a String representing a url and returns
an ArrayList of all the hyperlinks contained in the specified url.
• A getVertices() method which returns the list of all the urls represented in the graph.
• A getEdgesInto() method that takes as input a String representing a url. The method
returns an ArrayList of Strings of all the pages that contains an hyperlink to the
specified url.
• A getOutDegree() method which takes as input a String representing a url and returns
the number of hyperlinks in the specified page.
• A getPageRank() method which takes as input a String representing a url and returns
the rank of the specified page.
• A setPageRank() method which takes as input a String representing a url and a double
representing its rank. The method assigns the input number as the rank of the specified
page.
• A getVisited() method which takes as input a String representing a url and returns
whether or not the specified page has been visited.
• A setVisited() method which takes as input a String representing a url and a boolean.
The method assigns the input boolean to the visited field of the specified page.
[20 points] The Sorting class is a utility class that at the moment contains a static method
called slowSort(). This method takes as input a HashMap where the values are Comparable.
The method returns an ArrayList containing all the keys in the map, sorted in descending
5
order based on the values they map to. The time complexity of slowSort is O(n
2
), where
n is the number of pairs in the map. Your task is to implement a method called fastSort
which performs the same task as slowSort but with a time complexity of O(n · log(n)).
[80 points] The SearchEngine class has the following fields:
• A HashMap called wordIndex used to represent the mapping from single words to a list
of urls containing them.
• A MyWebGraph called internet which is used to store all relevant information regarding
the structure of the webgraph.
• An XmlParser called parser which you should use to retrieve information for the
database serving as a proxy for the web.
In this class, you should implement the following methods:
[35 points ] crawlAndIndex
This method takes a String as input representing a url. It will perform a graph traversal
of the web, starting at the given url. You can use either a depth-first search or a breadth-
first search, and you can make it either recursive or non-recursive. For each url visited,
the method should do the following:
• Update the web graph by adding the appropriate vertex and edges.
• Update the word index so that every word in the url just visited appear in the
mapping.
Note that, it is possible for the graph to contain cycles. For instance, if it possible that
two web pages link to each other. To prevent this method from getting stuck in an
infinite loop, you should make sure to keep track of which web page has already been
visited. This method should stop crawling once all web pages (in the database) have
been visited.
[35 points ] assignPageRanks and computeRanks
The method computeRanks is a helper method for assignPageRanks. We specifically
ask you to implement it so that our grader will be able to assign partial marks to your
implementation. The purpose of assignPageRanks is to assign a rank to each vertex in
the web graph. It will only be called after crawlAndIndex has been executed. Here are
the ideas we will be using to assign a rank to a web page:
• Good web pages are cited by many other pages. If we think about it in terms of the
webgraph, this means that we should prefer web pages (i.e. vertices) with a large
in-degree.
• Web pages that link to a large number of other pages are less valuable. In terms of
webgraph, this means that we will value less web pages (i.e. vertices) with a large
out-degree.
• A link from a web page is more valuable if the web page is itself a good one. In
graph terms, higher the rank of a web page (i.e. vertex), more valuable an in-edge
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from it would be.
Note that the rank of a page depends solely on the structure of the graph we have created
while crawling. To represent the ideas just described, let
• the pr(w) be the page rank of a vertex w
• the out(w) be the out-degree of a vertex w
• w1, w2, ..., wk be all the vertices in the graph that have an out edge going into v.
Then, the following equation determines the page rank of a vertex v:
The constant d is called the damping factor and it is added for technical reasons to
account for the probability that an imaginary surfer who is randomly clicking on links
will eventually stop clicking. For this assignment, we will be using a damping factor of
0.5. As you may notice, pr(v) is defined as a function of pr(wi). If N is the total number
of vertices in the graph (v1, v2, . . . , vN ), then we have a system of N linear equations in
N variables. The unknown values, i.e. our variables, are the pr(vi). To determine their
values, we should solve the system of linear equations described above. To do so, we
could use linear algebra, but the implementation of gaussian elimination runs in O(n
3
).
To improve the efficiency, we can instead use an iterative algorithm that approximate
the result. The idea is the following:
• Start by initializing pr(vi) to 1 for all 0 ≤ i ≤ N
• Repeat the following until convergence:
– compute pr(vi) for all i using the formula above.
This single step is what the helper method computeRanks should be doing.
The method takes as input an ArrayList
web graph and returns an ArrayList
ranks for those urls. Note that the double in the output list is matched to the
url in the input list using their position in the list. That is, the page rank of the
url stored in position i in the input list, can be found in position i in the output
list.
Convergence is reached when |prk−1
(vi) − prk
(vi)| < for all i, where prj
(vi) represents
the computation of pr(vi) in the j-th iteration, and is the input to the method.
Note that, you can assume that the web graph does not contain self loops and every
vertex in the graph has at least one outgoing edge.
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Consider the following web graph:
Suppose we’d like to compute the page rank assigned to each vertex with an equal to
0.01. We start by assigning value 1 to each of them:
• pr(A) = 1
• pr(B) = 1
• pr(C) = 1
• pr(D) = 1
Then we need to use the following formula (same as the formula above, but with damping
factor of 0.5) to recompute the page ranks for each vertex, until convergence.
After the first iteration we get:
• pr(A) = 1/2 + 1/2 ∗ (1/3 + 1) = 7/6 = 1.166
• pr(B) = 1/2 + 1/2 ∗ (1/2) = 3/4 = 0.75
• pr(C) = 1/2 + 1/2 ∗ (1/2 + 1/3 + 1) = 17/12 = 1.416
• pr(D) = 1/2 + 1/2 ∗ (1/3) = 4/6 = 0.666
After the second iteration we get:
• pr(A) = 1/2 + 1/2 ∗ (0.75/3 + 1.416) = 1.333
• pr(B) = 1/2 + 1/2 ∗ (1.166/2) = 0.791
• pr(C) = 1/2 + 1/2 ∗ (1.166/2 + 0.75/3 + 0.666) = 1.25
• pr(D) = 1/2 + 1/2 ∗ (0.75/3) = 0.625
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After 5 iterations we reach convergence since the difference between the rank values in
the 4th iteration and the rank values in the fifth iteration is smaller than . The ranks
at the end are:
• pr(A) = 1.270
• pr(B) = 0.819
• pr(C) = 1.274
• pr(D) = 0.636
[10 points ] getResults
This method takes as input a String representing a single-word query. It then returns
an ArrayList of urls that contain such word, ordered based on their rank from most
relevant to least relevant. This method will be called only after both crawlAndIndex
and assignPageRanks have been executed. This method should use one of the sorting
algorithms from the Sorting class. Ideally, it should use fastSort(), but if you did not
implement it, you can use slowSort().
Testing
To test your code we provided you with two XML files that act as a proxy for the internet. The XML
files contains proxy web-pages indicated by the tag name webpage. Each web-page contains multiple
hyperlinks to other web-pages indicated by the tag link. The contents of a web-page is marked using
the tag content. To help you with testing, we also provide the expected ranks of the web-pages
computed using the algorithm described in the previous section. Note that rank is not part of an
actual web-page, we simply provide them to help you with debugging. Page rank is indicated by
tag rank and can be accessed in your code using the function XmlParser::getPageRank() which
takes as input the name of the page.
• testAcyclic.xml - The provided tester use this database to test your code. Note that this
database does not have any circular chains - making it easy for you to test and debug.
• test.xml - We do not provide any test cases for this database but is more general than the
one above. We encourage you to create your tests with this database that will help you find
more intricate issues with your code.
Finally, you should create your databases that are much larger as we will test on a much bigger
database in the final grading.
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