Description
1. Problem Statement:
In this project, you will use several of the data structures we’ve been studying so far, including vectors
and stacks, to construct a rudimentary mathematical expression calculator. You will write a program that
allows a user to enter an expression, such as (3.2 + 5) / 7 * 2.1, and then evaluates it. In this case, the
program would produce the result 2.46.
1.1 Infix, Prefix, and Postfix Expressions
Expressions like the one used in the example above are called “infix”, meaning operators (e.g. 3.2, 5, 7,
2.1) are usually placed on either side of an operand (e.g. +, -, *, /). Although this system is typically used
when humans perform arithmetic, it does have several important flaws that make working with them
difficult for computers.
One of the most important is that infix expressions are potentially ambiguous. For example, “5 + 3 * 4”,
could be interpreted as either “(5 + 3) * 4” or as “5 + (3 * 4)”. To avoid ambiguity, scholars and
mathematicians have collectively agreed to assign precedence levels to each operator. For example,
multiplication and division have higher precedence than addition and subtraction, and exponentiation
has a higher precedence than multiplication or division. These precedence rules allow us to correctly
interpret the expression, meaning “5 + (3 * 4)” would be chosen over “(5 + 3) * 4”, given no other
information. Of course, if we ever want (or need) to be more explicit, we can also use parentheses to
resolve ambiguity issues.
There do exist other notation systems besides infix. The next two most common are called “prefix” and
“postfix”. In the prefix system, operators are placed before the operands, and in the postfix system,
operators are placed after the operands. As a simple example, “3 + 5” becomes “+ 3 5” in the prefix
system and “3 5 +” in the postfix system.
Both the prefix and postfix systems have the nice property of being completely unambiguous. The infix
expression “5 + (3 * 2)” becomes “+ 5 * 3 2” in the prefix system and “5 3 2 * +” in the postfix system,
while the expression “(5 + 3) * 2” becomes “* + 5 3 2” and “5 3 + 2 *”, respectively. There is no need for
parentheses in the postfix expression because there is no possibility for misinterpretation. This property
makes evaluating prefix and postfix expressions considerably easier than evaluating infix ones. Our
general strategy, then, will be to convert the infix expression to an equivalent prefix or postfix
expression that can then be evaluated directly.
1.2 Evaluating Prefix and Postfix Expressions
Functionally, there is little difference between working with prefix expressions and working with postfix
ones. Historically, however, computer scientists have dealt more with postfix expressions, since the
associated algorithms are marginally simpler to implement. We will follow that tradition as well, so from
now on, we will limit our discussion to postfix expressions.
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Postfix expressions can be efficiently evaluated using a simple stack-based algorithm. To help
demonstrate how it works, we will use one of the expressions above as an illustrative example. The infix
expression “(5 + 3) * 2” corresponds to the postfix expression “5 3 + 2 *”. It should be apparent from the
infix expression that the result is 8 * 2 = 16.
We will iterate over the postfix expression from left to right. Every time we see a number, we will push
that number onto a stack. Every time we see an operator, we will pop two numbers from the stack,
perform the associated operation, and then push the result back on the stack. When we reach the end of
the expression, there should only be one number on the stack, which will be the result of the expression.
This is a detailed example:
Token Stack Notes
5 5 Numbers are pushed directly onto the stack
3 5 3 Numbers are pushed directly onto the stack
+ 5 3 1. Pop two numbers off the stack
5 3 2. Add the numbers together (because the token is a +)
5 3 8 3. Push the result onto the stack
2 8 2 Numbers are pushed directly onto the stack
* 8 2 1. Pop two numbers off the stack
8 2 2. Multiply the two numbers (because the token is a *)
8 2 16 3. Push the result onto the stack
16 DONE – The result is at the top of the stack.
It is quite easy to check if a given postfix expression is well-formed while this algorithm is executing, as
there are just two possible erroneous conditions we might encounter:
1. If at any point, we try to pop from an empty stack, the expression must have been invalid.
2. If the stack is either empty or contains more than 1 element after the algorithm has completed, the
expression is also invalid.
1. 3 Dijkstra’s Shunting Yard
We’ve seen that postfix expressions are very simple to evaluate, but we still need to handle the
conversion from infix to postfix in the first place. To accomplish that, we need an algorithm called
Dijkstra’s Shunting Yard. The image below1 demonstrates the basic process.
We envision our input expression as a series of train cars, one car per token, lying on a railroad track.
Each car has the option of either moving forward on the same track or being rerouted to an auxiliary
track below. The auxiliary track leads to a dead-end, but it also connects back to the original track, acting
as a physically-inspired stack. Shunting yards allow train operators to adjust the position of cars within
the train, but we will use the same concept to adjust the position of operators and operands in a
mathematical expression.
1 https://en.wikipedia.org/wiki/Shunting-yard_algorithm
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This is the algorithm in detail:
We will process each token from the original infix expression one a time, using a stack to simulate the
auxiliary shunting yard track. Our logic is specialized based on each type of token.
• Numbers – Numbers pass through the yard without modification. They go directly to the output.
• Operators – Operators are rerouted through the auxiliary track, but we must deal with the fact
that the operators have different precedence levels. All operators with precedence ≥ the incoming
operator must first be pushed from the auxiliary track onto the output before the incoming
operator can be placed in the auxiliary track.
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• Left parentheses – Parentheses allow us to manually intervene when the traditional precedence
rules do not have the desired effect. Left parentheses are pushed directly onto the auxiliary track.
• Right parentheses – Right parentheses have a slightly more complicated logic associated with
them. We must continually pop cars from the auxiliary track onto the output until we find the
matching left parenthesis in the auxiliary track. Once we find it, we can remove that car
completely. (Remember, postfix expressions do not have parentheses at all because they are
completely unambiguous.)
After every token in the input expression has been processed, any remaining cars in the auxiliary track
(items on the stack) are pushed onto the output.
As with the postfix evaluation algorithm, it is possible for the Shunting Yard algorithm to fail. We
primarily concern ourselves with mismatched parentheses. It is possible for an expression to be
missing either a left parenthesis or a right one, so we must address both possibilities.
We can detect a missing left parenthesis when the current token is a right parenthesis and we are
popping operators from the stack. If the stack becomes empty before a match is found, the expression
was invalid. Similarly, we can detect a missing right parenthesis by examining the operators that remain
in the stack as they are being popped onto the output. If any of them are a left parenthesis, the expression
was also invalid.
A detailed example of an execution of the Shunting Yard algorithm is provided below, for the expression:
“3 * (4 / 5) + 2”:
Token Output Stack Notes
3 3 Numbers pass straight through the yard
* 3 * Operators are pushed onto the stack
( 3 * ( Left parentheses are pushed onto the stack
4 3 4 * ( Numbers pass straight through the yard
/ 3 4 * ( / Operators are pushed onto the stack
5 3 4 5 * ( / Numbers pass straight through the yard
) 3 4 5 / * Pop from the stack until the matching ‘(‘ is found. Then pop the ‘(‘.
+ 3 4 5 / * 1. ‘*’ has a higher precedence than ‘+’, so it must be popped first
3 4 5 / * + 2. ‘+’ can safely be pushed onto the stack
2 3 4 5 / * 2 + Numbers pass straight through the yard
3 4 5 / * 2 + The stack is emptied onto the output.
3 4 5 / * 2 + DONE!
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2. Design:
There are three fundamental design issues: the driver, the shunting yard algorithm, and the postfix
evaluation algorithm.
2.1 Driver
Our driver will be very simple. It should:
1. Ask the user to enter an infix expression.
2. Print the program’s interpretation of the infix expression.
3. Attempt to convert the infix expression to an equivalent postfix expression. (If the expression is
invalid, inform the user and exit the program.)
4. Attempt to evaluate the postfix expression. (If the postfix expression is invalid, inform the user
and exit the program.)
5. Display the result of the evaluation.
2.2 Converting Infix to Postfix
To convert the infix expression given by the user to a postfix expression, you will need to implement
Dijkstra’s Shunting Yard algorithm. Doing so becomes complicated as more operators are added, so we
will try to reduce that complexity by limiting ourselves to just 4:
Operator Symbol Priority
Addition + 1
Subtraction – 1
Multiplication * 2
Division / 2
Furthermore, we will allow users to enter parentheses to explicitly indicate in which order certain
operations should be performed.
Interestingly, neither the actual value of the operands nor the meaning behind the operator
symbols is relevant when implementing the Shunting Yard algorithm. We just need to know
whether a certain string represents an operator and if so, what priority that operator has. In that regard,
the Shunting Yard algorithm can be viewed as a simple transformation routine—it rearranges an
expression, but it does not actually perform any mathematical operations.
2.3 Evaluating a Postfix Expression
If the Shunting Yard algorithm succeeds in converting the user’s infix expression to an equivalent postfix
expression, the next step will be to evaluate that postfix expression using the algorithm outlined in the
last section.
Unlike the Shunting Yard implementation, the evaluation implementation DOES need to be aware
of the intended mathematical interpretation of a given symbol (e.g. “+” means addition, etc.), and
the actual value of the operand is important. It’s important that you understand conceptually the
difference between the two algorithms. The first transforms one expression into another expression,
while the second transforms an expression into a result (a number). Both use stacks as part of their
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implementation, and both need to be able to decide whether a given string represents an operator, but
when writing the evaluation implementation, you will need to translate a string into an action or into
a number.
You have the knowledge necessary to use templates in the implementation of the evaluation routine (so
we could work with integers, doubles, floats, etc.), but doing so is not likely to be particularly helpful.
Most calculators generally assume that all operands are real numbers, so it is alright for us to assume that
all numbers are doubles in this assignment.
3. Implementation:
Driver
We will use vectors of strings to represent individual expressions (both infix and prefix). Each element of
the vector will represent a single token. For example, the expression “3.2 * (4.0 / 5.1) + 2” would be
represented as a vector containing {“3.2”, “*”, “(“, “4.0”, “/”, “5.1”, “)”, “+”, “2”}.
One of the first issues you must address is how to read the input from the user into an empty vector.
Generally, parsing expressions that may or may not have spaces in them is a difficult problem, and doing
so is not the purpose of this assignment, so we will take a simpler approach and assume that the user
inserts spaces between each term of the expression. For example, we can assume the user enters “3.2 * (
4.0 / 5.1 ) + 2” (with the spaces) instead of “3.2*(4.0/5.1)+2” (without spaces). If that is the case, the
expression can easily be parsed using the >> operator:
Vector expression;
string token;
while (cin >> token)
{
expression.pushBack(token);
}
Remember the >> operator skips over spaces automatically and returns false when it fails. With cin, the
stream operator will fail if the user enters either Ctrl + D or Ctrl + Z, depending on your operating
system.
Shunting Yard Implementation
The next issue is how to implement the Shunting Yard algorithm. For this part, you will need to
implement the following function prototype:
bool shuntingYard(const Vector& expression, Vector& postfix)
This function will be given the infix expression provided by the user and will attempt to convert it to an
equivalent postfix expression, that will be stored in ‘postfix’. You may assume that ‘postfix’ is initially
empty. This function will return true if we were able to perform the conversion and false if the expression
is malformed. Remember, the Shunting Yard algorithm will only fail if the parentheses are mismatched. It
is possible for the expression to be invalid, but still pass through the shunting yard.
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The only requirement for this function is that you must use a Stack in the implementation. All other
details will be left to you. As a suggestion, it might be a good idea to write a function that returns true if a
given string represents an operator and another to return the precedence of a given string.
Evaluate Postfix Implementation
You will also need to implement the postfix evaluation algorithm. For this part, you will need to
implement the following function prototype:
bool evaluatePostfix(const Vector& postfix, double& result)
This function will be given the postfix expression produced by the Shunting Yard algorithm as input. It
will attempt to evaluate the expression, returning true if evaluation was successful. If evaluation fails
(because the postfix expression was invalid), the function will return false instead. The result will be
saved in ‘result’ if the function returns true.
As for shuntingYard(), evaluatePostfix() MUST use a Stack in the implementation. All other details
will be left to you. You may write as many auxiliary functions as you’d like, as long as you do not modify
the evaluatePostfix() function prototype (or shuntingYard()) prototypes.
General Advice
Before you begin coding, it would be wise to try running through each of the algorithms on your own. If
you cannot execute the algorithm by hand, it is much more difficult to implement it in C++. You should
also try converting several simple infix expressions to postfix to make sure you’re familiar with the
notation. The attached file contains several examples of infix expressions, their corresponding postfix
expressions, and the results, so you can use it to validate both your intuition and your code.
Once you feel comfortable with each of the algorithms, it would be a good idea to start working on the
evaluatePostfix() function, as it is considerably simpler than shuntingYard(). It will also give you
good practice with working with the Stack class. Get the function working with well-formed expressions
before adding error checking. Test this function thoroughly to make sure it does consistently produce the
correct answers for valid expressions and that it consistently fails on invalid ones.
After evaluatePostfix() has been completed, you should start working on shuntingYard(). It would
be best to start with simple expressions without parentheses. Once you get the algorithm working with
those simple expressions, add the logic for dealing with parentheses. The last addition should be the
error checking to deal with mismatched parentheses. As with evaluatePostfix(), test
shuntingYard() thoroughly to make sure it does consistently produce the correct postfix expressions.
As always, do everything you can to make your code “bullet-proof”. Try providing inputs that are
egregiously wrong to see what your code does, for example. Stress tests like these are helpful for finding
weaknesses and vulnerabilities. By eliminating them from the start, your code will be more robust and
reliable.
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4. Style
Make sure your code adheres to the guidelines provided in the Style Guide (available on Moodle). Your
goal is to create code that is concise, descriptive, and easy for other humans to read. Avoid typos, spelling
mistakes, or anything else that degrades the aesthetic of your code. Your final submission should be
work that you are proud to call your own.
5. Testing:
Test your program to check that it operates correctly for all of the requirements listed above. Also check
for the error handling capabilities of the code. Try your program with several input values, and save your
testing output in text files for inclusion in your project report.
6. Documentation:
When you have completed your C++ program, write a short report using the project report template
describing what the objectives were, what you did, and the status of the program. Does it work properly
for all test cases? Are there any known problems? Save this report to be submitted electronically.
7. Project Submission:
In this class, we will be using electronic project submission to make sure that all students hand their
programming projects and labs on time, and to perform automatic plagiarism analysis of all programs
that are submitted.
When you have completed the tasks above go to Moodle to upload your documentation (a single .pdf file),
and all C++ program files (.h and .cpp). Make sure your proof of testing is included in the documentation
or is submitted as a separate file. Do NOT upload an executable version of your program.
The dates on your electronic submission will be used to verify that you met the due date above. Late
projects will receive NO credit. You will receive partial credit for all programs that compile even if they
do not meet all program requirements, so make sure to submit something before the due date, even if the
project is incomplete.
8. Academic Honesty Statement:
Students are expected to submit their own work on all programming projects, unless group projects have
been explicitly assigned. Students are NOT allowed to distribute code to each other, or copy code from
another individual or website. Students ARE allowed to use any materials on the class website, or in the
textbook, or ask the instructor for assistance.
This course will be using highly effective program comparison software to calculate the similarity of all
programs to each other, and to homework assignments from previous semesters. Please do not be
tempted to plagiarize from another student.
Violations of the policies above will be reported to the Provost’s office and may result in a ZERO on the
programming project, an F in the class, or suspension from the university, depending on the severity of
the violation and any history of prior violations.
