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COMP226 Assignment 1
Continuous Assessment
Number
1 (of 2)
Weighting 10%
Assignment Circulated Monday 26 February 2024
Deadline 21:00 Friday 22 March 2024
Submission Mode Submit a single R file "solution.R" to the CodeGrade Assignment on Canvas
Learning Outcomes
Assessed
Have an understanding of market microstructure and its impact on trading.
Goal of Assignment Reconstruct a limit order book from order messages; compute quantities
based on the limit order book
Marking Criteria Pre-deadline visible CodeGrade tests of correctness of 6 functions (70%);
Post-deadline CodeGrade tests of 4 further functions (30%)
Submission necessary in
order to satisfy module
requirements
No
Late Submission Penalty Standard UoL policy; resubmissions after the deadline may not be
considered.
Expected time taken Roughly 8-12 hours
Warning
Submissions are automatically put through a plagiarism and collusion detection system. Students
found to have plagiarized or colluded will likely receive a mark of zero. Do not discuss or show your
work to others, and do not search for solutions to the assignment online. In previous years, students
have had their studies terminated and left without a degree because of plagiarism or collusion.
Rscript from Rstudio
In this assigment, we use Rscript (which is provided by R) to run our code, e.g.,
Rscript main.R template.R input/book_1.csv input/empty.txt
In R studio, you can call Rscript from the "terminal" tab (as opposed to the "console"). On Windows,
use Rscript.exe not Rscript:
Rscript.exe main.R template.R input/book_1.csv input/empty.txt
Distributed code and sample input and output data
As a first step, please download comp226_a1.zip via the assignment page on Canvas. Then unzip
comp226_a1.zip, which will yield the following contents in the directory comp226_a1:
comp226_a1
Brief summary
You are provided with three .R files, two complete, which should not be edited:
• main.R is the file you will run, e.g. with Rscript, by specifying several command line arguments
described below (see example above);
• common.R contains complete working functions that are used by main.R in conjunction with the
incomplete functions in template.R;
and one incomplete file:
• template.R is the file that you will edit -- the distributed version contains empty functions. It contains
10 empty functions.
If you run main.R using template.R as it is distributed, it runs without error, but does not produce the
desired output because the first 6 functions in template.R are provided empty. To get 70%, you will need to
correctly complete these 6 functions; if your answer is only partially correct you will get a mark less than 70%.
If you have correctly completed all these 6 functions, you can then -- and only then -- get marks for the 4
"extra" functions, which together account for 30% of the marks.
You should submit a single R file that contains your implementation of some or all of these 10 functions. Your
submission will be marked via extensive automated tests of correctness across a wide range of example
cases, with some chosen randomly at test time:
• The tests for the first 6 functions, which give up to 70% of the marks will run at the time of submission
and are fully visible pre-deadline on CodeGrade (detailed guidance on using CodeGrade to improve
your mark can be found at the end of this document);
• The tests for the final 3 functions will only run post-deadline, and only if you got full marks for the
first 6 functions.
You can (and if required should) submit multiple times to repeatedly use the CodeGrade pre-deadline tests;
for a correct solution CodeGrade will show that you pass all tests and have thus achieved the first 70% of
marks. It probably does not make sense for you to work much on the final 4 functions until you have achieved
this and submitted completely correct versions of the first 6 functions.
In addition to the visible pre-deadline tests on CodeGrade, for the first 6 functions, correct sample output is
provided so that you can check correctness of your implementations "offline" (without submitting to
CodeGrade), for example with a tool like diff (https://en.wikipedia.org/wiki/Diff) to compare the output that you
produce with the correct output. Offline testing is quick to do once you have set it up, and if you match all the
offline examples then chances are that you will also pass the CodeGrade tests (but make sure that you
check this to avoid nasty surprises).
template.R versus solution.R
Throughout the rest of this handout, we show example output from the incomplete template.R as well
as using a full solution file "solution.R" that contains a correct implementation of all the functions.
Obviously, you are not given the file solution.R, you need to create it from template.R.
The first 6 functions to implement
The first 6 functions, which are worth 70% of the marks, are broken down into two groups. The percentages
in square brackets show the breakdown of the marks by function.
Order book stats:
1. book.total_volume <- function(book) [5%]
2. book.best_prices <- function(book) [5%]
3. book.midprice <- function(book) [5%]
4. book.spread <- function(book) [5%]
These first 4 functions are intentionally very easy, and are meant to as get you used to the format of book.
The next 2 functions are more involved and relate to reconstructing the order book from an initial book and a
file of messages.
Reconstructing the limit order book:
5. book.reduce <- function(book, message) [15%]
6. book.add <- function(book, message) [35%]
Running main.R with template.R
An example of calling main.R with template.R is as follows.
Rscript main.R template.R input/book_1.csv input/empty.txt
As seen in this example, main.R takes as arguments the path to three input files (the order of the
arguments matters):
1. an R file with the functon implementations (template.R in the example)
2. an initial order book (input/book_1.csv in the example)
3. order messages to be processed (input/empty.txt in the example)
Let's see the source code of main.R and the output that it produces.
options(warn=-1)
args <- commandArgs(trailingOnly = TRUE); nargs = length(args)
log <- (nargs == 4) # TRUE is there are exactly 4 arguments
arg_format <- "<--log> <solution_path> <book_path> <messages_path>"
if (nargs < 3 || nargs > 4) # check that there are 3 or 4 arguments
 stop(paste("main.R has 3 required arguments and 1 optional flag:", arg_format))
if (nargs == 4 && args[1] != "--log") # if 4 check that --log is the first
 stop(paste("Bad arguments format, expected:", arg_format))
solution_path <- args[nargs-2]
book_path <- args[nargs-1]
messages_path <- args[nargs]
if (!all(file.exists(c(solution_path, book_path, messages_path))))
 stop("File does not exist at path provided.")
source(solution_path); source("common.R") # source common.R from pwd
book <- book.load(book_path)
book <- book.reconstruct(data.load(messages_path), init=book, log=log)
book.summarise(book)
ret <- book.extra3(book)
cat('ret', ret, 'n')
In short, main.R:
• checks that the command line arguments are ok
• assigns them to variables (solution_path, data_path, and book_path respectively)
• sources common.R and the file at solution_path; loads the initial book
• reconstructs the book according to the messages
• prints out the resulting book; prints out the book stats
Let's see the output for the example above:
$ Rscript main.R template.R input/book_1.csv input/empty.txt
$ask
 oid price size
1 a 105 100
$bid
 oid price size
1 b 95 100
Total volume:
Best prices:
Mid-price:
Spread:
Now let's see what the output would look like for a correct implementation:
$ Rscript main.R solution.R input/book_1.csv input/empty.txt
$ask
 oid price size
1 a 105 100
$bid
 oid price size
1 b 95 100
Total volume: 100 100
Best prices: 95 105
Mid-price: 100
Spread: 10
You will see that now the order book stats have been included in the output, because the four related
functions that are empty in template.R have been implemented in solution.R.
The initial order book
Here is the contents of input/book_1.csv, one of 3 provided initial book examples:
oid,side,price,size
a,S,105,100
b,B,95,100
Let's justify the columns to help parse this input:
oid side price size
a S 105 100
b B 95 100
The first row is a header row. Every subsequent row contains a limit order, which is described by the
following fields:
• oid (order id) is stored in the book and used to process (partial) cancellations of orders that arise in
"reduce" messages, described below;
• side identifies whether this is a bid ('B' for buy) or an ask ('S' for sell);
• price and size are self-explanatory.
Existing code in common.R will read in a file like input/book_1.csv and create the corresponding two
(possibly empty) orders book as two data frames that will be stored in the list book, a version of which will be
passed to all of the functions that you are required to implement.
Note that if we now change the message file to a non-empty one, template.R will produce the same output
(since it doesn't use the messages; you need to write the code for functions 5 and 6 to process them):
$ Rscript main.R template.R input/book_1.csv input/message_a.txt
$ask
 oid price size
1 a 105 100
$bid
 oid price size
1 b 95 100
Total volume:
Best prices:
Mid-price:
Spread:
If correct message parsing and book updating is implemented, book would be updated according to
input/message_a.txt to give the following output:
$ Rscript main.R solution.R input/book_1.csv input/message_a.txt
$ask
 oid price size
8 a 105 100
7 o 104 292
6 r 102 194
5 k 99 71
4 q 98 166
3 m 98 88
2 j 97 132
1 n 96 375
$bid
 oid price size
1 b 95 100
2 l 95 29
3 p 94 87
4 s 91 102
Total volume: 318 1418
Best prices: 95 96
Mid-price: 95.5
Spread: 1
Before we go into details on the message format and reconstructing the order book, let's discuss the first four
functions that compute the book stats.
Computing limit order book stats
The first four of the functions that you need to implement compute limit order book stats, and can be
developed and tested without parsing the order messages at all. In particular, you can develop and test the
first four functions using an empty message file, input/empty.txt, as in the first example above.
The return values of the four functions should be as follows (where, as usual in R, single numbers are
actually numeric vectors of length 1):
• book.total_volumes should return a list with two named elements, bid, the total volume in the bid
book, and ask, the total volume in the ask book;
• book.best_prices should return a list with two named elements, bid, the best bid price, and ask,
the best ask price;
• book.midprice should return the midprice of the book;
• book.spread should return the spread of the book.
Check that the example outputs above are what you expect them to be.
Reconstructing the order book from messages
We now move on to the reconstructing the order book from the messages in the input message file. You do
not need to look into the details of the (fully implemented) functions book.reconstruct or book.handle
in common.R that manage the reconstruction of the book from the starting initial book according to the
messages (but you can if you want).
In the next section, we describe the two types of message, "Add" messages and "Reduce" messages. All
you need to know to complete the assignment is that messages in the input file are processed in order, i.e.,
line by line, with "Add" messages passed to book.add and "Reduce" messages passed to book.reduce,
along with the current book in both cases.
Message Format
The message file contains one message per line (terminated by a single linefeed character, 'n'), and each
message is a series of fields separated by spaces. Here's an example, which contains an "Add" message
followed by a "Reduce" message:
A c S 97 36
R a 50
An "Add" message looks like this:
'A' oid side price size
• 'A': fixed string identifying this as an "Add" message;
• oid: "order id" used by subsequent "Reduce" messages;
• side: 'B' for a buy order (a bid), and an 'S' for a sell order (an ask);
• price: limit price of this order;
• size: size of this order.
A "Reduce" message looks like this:
'R' oid size
• 'R': fixed string identifying this as a "Reduce" message;
• oid: "order id" identifies the order to be reduced;
• size: amount by which to reduce the size of the order (not its new size); if size is equal to or greater
than the existing size, the order is removed from the book.
Processing messages
A "Reduce" message affects at most one existing order. An "Add" message will either:
• not cross the spread and then add a single row to the book (orders at the same price are stored
separately to preserve their distinct "oid"s);
• cross the spread and in that case can affect any number of orders on the other side of the book (and
may or may not result in a remaining limit order for residual volume).
The provided example message files are split into cases that include crosses and those that don't. This
allows you to develop your code incrementally and test it on inputs of differing difficulty. We do an example of
each case, one by one. In each example we start from input/book_1.csv; we only show this initial book in
the first case.
Example of processing a reduce message
$ Rscript main.R solution.R input/book_1.csv input/empty.txt
$ask
 oid price size
1 a 105 100
$bid
 oid price size
1 b 95 100
Total volume: 100 100
Best prices: 95 105
Mid-price: 100
Spread: 10
$ cat input/message_ex_reduce.txt
R a 50
$ Rscript main.R solution.R input/book_1.csv input/message_ex_reduce.txt
$ask
 oid price size
1 a 105 50
$bid
 oid price size
1 b 95 100
Total volume: 100 50
Best prices: 95 105
Mid-price: 100
Spread: 10
Example of processing an add (non-crossing) message
$ cat input/message_ex_add.txt
A c S 97 36
$ Rscript main.R solution.R input/book_1.csv input/message_ex_add.txt
$ask
 oid price size
2 a 105 100
1 c 97 36
$bid
 oid price size
1 b 95 100
Total volume: 100 136
Best prices: 95 97
Mid-price: 96
Spread: 2
Example of processing a crossing add message
$ cat input/message_ex_cross.txt
A c B 106 101
$ Rscript main.R solution.R input/book_1.csv input/message_ex_cross.txt
$ask
[1] oid price size
<0 rows> (or 0-length row.names)
$bid
 oid price size
1 c 106 1
2 b 95 100
Total volume: 101 0
Best prices: 106 NA
Mid-price: NA
Spread: NA
The NAs in the last example are the expected output for certain fields when at least one side of the book is
empty and the corresponding quantity is undefined.
Sample output
We provide sample output for 9 cases, namely all combinations of the 3 initial books (book_1.csv,
book_2.csv, book_3.csv) and 3 message files, all found in the input subdirectory. The 3 message files are
called:
file
messages_a.txt add messages only, i.e., requires book.add but not book.reduce; for all three
initial books, none of the messages cross the spreed
messages_ar.txt add and reduce messages, but for the initial book book_3.csv, no add message
crosses the spread
messages_arc.txt add and reduce messages, with some adds that cross the spread for all three
initial books
The 9 output files can be found in the output subdirectory of the comp226_a1 directory.
output
0 directories, 9 files
Hints for order book stats
For book.spread and book.midprice a nice implementation would use book.best_prices.
Hints for book.add and book.reduce
A possible way to implement book.add and book.reduce that makes use of the different example
message files is the following:
• First, do a partial implementation of book.add, namely implement add messages that do not cross.
Check your implementation with message_a.txt.
• Next, implement book.reduce fully. Check your combined (partial) implementation of book.add and
book.reduce with message_ar.txt and book_3.csv (only this combination with
message_ar.txt has no crosses).
• Finally, complete the implementation of book.add to deal with crosses. Check your implementation
with message_arc.txt and any initial book or with message_ar.txt and book_1.csv or
book_2.csv.
Hint on book.sort
$ Rscript main.R solution.R input/book_1.csv input/message_ex_same_price.txt
$ask
 oid price size
2 j 105 132
1 a 105 100
$bid
 oid price size
1 b 95 100
2 k 95 71
Total volume: 171 232
Best prices: 95 105
Mid-price: 100
Spread: 10
Note that earlier messages are closer to the top of the book. This is due to price-time precedence,
according to which orders are executed:
• Best price first, but when two orders have the same price, the earlier one is executed first
We provide book.sort that respects price-time precedence. It relies on the fact that the order ids increase
as follows:
a < k < ab < ba
where < is indicating "comes before" in the message files.
book.sort <- function(book, sort_bid=T, sort_ask=T) {
 if (sort_ask && nrow(book$ask) >= 1) {
 book$ask <- book$ask[order(book$ask$price,
 nchar(book$ask$oid),
 book$ask$oid,
 decreasing=F),]
 row.names(book$ask) <- 1:nrow(book$ask)
 }
 if (sort_bid && nrow(book$bid) >= 1) {
 book$bid <- book$bid[order(-book$bid$price,
 nchar(book$bid$oid),
 book$bid$oid,
 decreasing=F),]
 row.names(book$bid) <- 1:nrow(book$bid)
 }
 book
}
book.init <- function() {
 book <- list(
 ask=data.frame(matrix(ncol=3, nrow=0)),
 bid=data.frame(matrix(ncol=3, nrow=0))
 )
 colnames(book$ask) <- c("oid", "price", "size")
 colnames(book$bid) <- c("oid", "price", "size")
 return(book)
}
This method will ensure that limit orders are sorted first by price and second by time of arrival (so that for two
orders at the same price, the older one is nearer the top of the book). You are encouraged to use
book.sort in your own implementations. In particular, by using it you can avoid having to find exactly where
to place an order in the book.
Hint on using logging in book.reconstruct
In common.R a logging option has been added to book.reconstruct:
book.reconstruct <- function(data, init=NULL, log=F) {
 if (is.null(init)) init <- book.init()
 if (nrow(data) == 0) return(init)
 book <- Reduce(
 function(b, i) {
 new_book <- book.handle(b, data[i,])
 if (log) {
 cat("Step", i, "nn")
 book.summarise(new_book, with_stats=F)
 cat("====================nn")
 }
 new_book
 },
 1:nrow(data), init,
 )
 book.sort(book)
}
If you want to use this for debugging, you can turn it on with the --log flag, e.g.:
Rscript main.R --log solution.R input/book_1.csv input/message_arc.txt
Then book.summarise is used to give intermediate output after each message is processed.
Hint on stringsAsFactors=FALSE
Notice the use of stringsAsFactors=FALSE in book.load (and data.load) in common.R.
book.load <- function(path) {
 df <- read.table(
 path, fill=NA, stringsAsFactors=FALSE, header=TRUE, sep=','
 )
 book.sort(list(
 ask=df[df$side == "S", c("oid", "price", "size")],
 bid=df[df$side == "B", c("oid", "price", "size")]
 ))
}
Its use here is not optional, it is necessary and what ensures that the oid columns of book$bid and
book$ask have type character.
It is crucial that you ensure that the types of your oid columns in your books remain character rather than
factors. The following examples will explain the use of stringsAsFactors and help you to achieve this.
First we introduce a function that will check the type of this column on different data frames that we will
construct:
check <- function(df) {
 checks <- c("is.character(df$oid)",
 "is.factor(df$oid)")
 for (check in checks)
 cat(sprintf("%20s: %5s", check, eval(parse(text=check))), 'n')
}
Now we use this function to explore different cases. First we look at reading in a csv file.
> check(read.csv('input/book_1.csv'))
is.character(df$oid): FALSE
 is.factor(df$oid): TRUE
> check(read.csv('input/book_1.csv', stringsAsFactors=FALSE))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
What about creating a data.frame?
> check(data.frame(oid="a", price=1))
is.character(df$oid): FALSE
 is.factor(df$oid): TRUE
> check(data.frame(oid="a", price=1, stringsAsFactors=FALSE))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
What about using rbind?
> empty_df <- data.frame(oid=character(0), price=numeric(0))
> non_empty_df <- data.frame(oid="a", price=1, stringsAsFactors=FALSE)
> check(rbind(empty_df, data.frame(oid="a", price=1)))
is.character(df$oid): FALSE
 is.factor(df$oid): TRUE
> check(rbind(empty_df, non_empty_df))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
> check(rbind(non_empty_df, data.frame(oid="a", price=1)))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
Note: when a data.frame becomes empty the type of the oid column is malleable and it is crucial to use
stringsAsFactors=FALSE. We see this issue if we rbind a list with a data.frame.
> check(rbind(empty_df, list(oid="a", price=1)))
is.character(df$oid): FALSE
 is.factor(df$oid): TRUE
> check(rbind(empty_df, list(oid="a", price=1), stringsAsFactors=FALSE))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
> check(rbind(non_empty_df, list(oid="a", price=1)))
is.character(df$oid): TRUE
 is.factor(df$oid): FALSE
So it is crucial to use stringsAsFactors=FALSE when the data.frame is empty; I suggest to use it in
every case.
Extra problems for final 30%
Warning
The final 30% of marks are only available if CodeGrade gives you the full first 70% of marks. Please
only focus on the extra problems once you have achieved this.
For these final 30%, there are four functions that you are asked to implement. You can get marks for any
one of these independently. The marks available are as follows:
1. book.extra1 <- function(book,size) [6%]
2. book.extra2 <- function(book,size) [6%]
3. book.extra3 <- function(book) [6%]
4. book.extra4 <- function(book,k) [12%]
The functions are defined by a full specification, given below, but we intentionally do not give you explicit test
cases (you need to create them for yourself if you want) and you cannot find out your mark before the
deadline. We will also not offer detailed help on solving these parts, as they are meant to be more
challenging, with part of that challenge being to solve them on your own.
If you have achieved the full first 70%, then you can get extra marks for fully or partially correct
implementations of any of the 4 extra functions, independent of each other (e.g., you can complete
book.extra3 and not the other extra functions and still get marks).
The first three require you to compute an expectation of a discrete random variable. If you need a refresher,
go back to COMP111 Introduction to Artificial Intelligence. For our applicaton such an expectation is just the
average (since the probability distribution is uniform) over all the possible values of the random variable.
Assumption
For the extra tests, you can assume that bid book is not empty, so that the starting mid-price is only
NA if the ask book has no orders in it.
Extra problem 1
book.extra1 has two parameters, book (the order book), and size, which is an integer size between 1
and M, where M is the total volume in the ask book (you can assume that your function will only be tested on
such values of size).
The function should return the expected value of the midprice after execution of a buy limit order with size
size and price drawn uniformly at random from the set of prices in book$ask.
Extra problem 2
book.extra2 has exactly the same two parameters as book.extra1. The function should return the
expected value of the midprice after execution of a buy limit order with size size and price drawn uniformly
at random from the integer prices (the tick size is 1) between the best ask price and the highest ask price in
the book.
Hint: For the first 2 extra problems, if size is equal to M then the correct answer is NA.
Extra problem 3
book.extra3 only takes book as an argument. The function should return the expected value of the
midprice after execution of a buy market order with size s is executed, assuming that s is drawn uniformly at
random from the set {1,....,(M-1)}, where M is the total volume in the ask book.
Hint: For first 3 extra problems, one unified approach is to simulate the relevant orders using functions that
you implemented for book.reconstruct.
Extra problem 4
book.extra4 has two parameters, book (the order book), and k a non-negative number that will be
interpreted as a percentage, e.g., if k=0.4 then k corresponds to 0.4%.
The function should return: 0 if the ask book has no orders in it; otherwise the largest amount of buy volume
v such that a buy market order with size v causes the mid-price to increase by no more than k % (so an
order with size v+1 would either cause the midprice to increase by more than k % or would leave no asks left
in the book which means that the mid-price is NA).
Note: the return value should be an integer between 0 and the total ask volume in book.
Hint: For all 4 extra problems, to increase the chance that your implementations are correct, do one or two
examples where you compute the correct expectation by hand on and then check that your code produces
the correct answer.
Submission
Submission is via CodeGrade on Canvas. Remember to call the file that you submit "solution.R", as this is
the only thing that will be accepted by CodeGrade:
Using CodeGrade to improve your solution
For the first 70% of marks, after you submit the tests runs and then:
1. a provisional mark is visible;
2. any tests that you fail are visible on CodeGrade and can be used to help you improve your solution.
For example, I have edited the correct solution of book.best_prices to give the wrong answer 10% of the
time at random and submitted this as a test student. After waiting a short while for all the tests to run, one can
then see the result of the tests. Here they are for "Order book stats", for the first 4 functions:
Not only is book.best_prices failing some tests, but so is book.spread and book.midprice because
these happen to use book.best_prices in this implementation. Let's click on book.best_prices to
inspect the individual tests:
Note that if you find that the area of the interface that shows the test results is too small, it may be because
you are showing the rubric, which can be hidden (see the "Hide rubric" arrow in the bottom right of the
screenshot):
Let's zoom in on one of the failed tests for book.best_prices, which will allow us to see which inputs were
used:
In this case, book_2.csv was used; the first few tests in each category use the examples books that you
have been given, which makes it easy to investigate these failed tests.
When you have got all of the "Order book stats" functions correct and all the tests pass, it would look like this:
Let's do one more example, using book.add, where there is a message input as well as the book:
Finally, there is a second way that a test can fail. The above examples showed failures as "red crosses",
meaning that output was generated but that it didn't match the expected output. A second way that a test can
fail is with a "red exclamation mark":
There are two tabs, "Results" and "Output". As a default, CodeGrade starts on the results tab, which is used
in the examples above to see the details of tests that failed with a red cross:
To see any error messages that occured, which is relevant for tests that failed with a red exclamation mark,
switch to the output tab:
Finally, a comment on the post-deadline tests. You will get similar test output for the extra functions, provided
that you got the first 6 functions totally correct, but only after the deadline (this is completely intentional):
Tests
The first few tests in each category intentionally use book_1.csv, book_2.csv, and book_3.csv, so that
it is easy for you investigate if a test fails (since you have these csv files, and can use the empty
message file or create a simple message file yourself for testing).
Many of the tests use more complicated csv files, which are not available to you as csvs directly, but
you could create csvs if you wanted. That should not be necessary in most cases though since if your
code works on the example input/output files it should pass all the tests on CodeGrade.
Finally, please note that some tests are randomly generated. This is required to reduce the appeal of
hardcoding answers (if there was a small number of non-random tests, students could just hardcode
the expected answers rather than solve the general problem). This means that you may experience a
small amount of variance in the mark you see when you resubmit the same wrong code (correct code
will get full marks every time).
If you need help with the assignment we expect you to have submitted to CodeGrade. That way we can both
see how your code does on the tests, which will help us to help you, and we can enter comments directly on
the submitted source code, as in he following example:
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