cs 15: data structures

CS 15: Data Structures

Project 1: CalcYouLater

• We’ll cover two subjects important to Project 1:‣ Reverse Polish Notation (RPN)

‣ Exceptions

• Also recommended:‣ Go back and review lecture on stacks

‣ Especially: parsing parenthesized expressions

Background

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5 - 4 + 3 * 4 / 2 * 9

Conventional Arithmetic Notation

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5 - 4 + 3 * 4 / 2 * 9

• There is some ambiguity here. We can resolve it with parentheses:

5 - (4 + 3) * (4 / (2 * 9))


3

5 - 4 + 3 * 4 / 2 * 9


5 - (4 + 3) * (4 / (2 * 9))

• There is still ambiguity here: Which comes first, - or *?


3

5 - 4 + 3 * 4 / 2 * 9


5 - (4 + 3) * (4 / (2 * 9))


• We have developed conventions to address this:1. Parentheses

2. Exponents

3. Multiplication/Division

4. Addition/Subtraction


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PEMDAS

5 - 4 + 3 * 4 / 2 * 9


5 - (4 + 3) * (4 / (2 * 9))


• We have developed conventions to address this:1. Parentheses

2. Exponents

3. Multiplication/Division

4. Addition/Subtraction


3

PEMDAS

Notice: there is still ambiguity between mult/div and add/sub

5 - 4 + 3 * 4 / 2 * 9

• This is called infix notation: binary operators are placed between operands‣ Binary operator: operator that acts on two operands


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5 - 4 + 3 * 4 / 2 * 9

• This is called infix notation: binary operators are placed between operands‣ Binary operator: operator that acts on two operands

• Infix notation is fundamentally ambiguous: we’ll always need parentheses, special rules like PEMDAS


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• There is more than one way to write arithmetic expressions:

• infix: 3 + 4‣ Operator goes between operands

• prefix: + 3 4‣ Operator comes before operands

• postfix: 3 4 +‣ Operator comes after operands

Other Conventions

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Other Conventions

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Both prefix and postfix notation are ambiguity-free: no special rules needed, parentheses are optional





Other Conventions

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Other Conventions

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Invented by Polish logician Jan Łukasiewicz. Sometimes called Polish notation.





Other Conventions

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Other Conventions

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Invented by Polish logician Jan Łukasiewicz. Sometimes called Polish notation.

Sometimes called Reverse Polish notation (RPN)

What does this expression evaluate to:4 5 6 + *

RPN Example

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What does this expression evaluate to:4 5 6 + *Evaluates to 44

RPN Example

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What does this expression evaluate to:4 5 6 + *Evaluates to 44Read expression from left to right. Each time, we see an operator evaluate it on the previous two values:

RPN Example

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1. + acts on previous two values, 5 and 6. This gives us 11.‣ Think of 11 as “replacing” 5 6 + in our expression

RPN Example

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1. + acts on previous two values, 5 and 6. This gives us 11.‣ Think of 11 as “replacing” 5 6 + in our expression

2. * acts on previous two values, now 11 and 4. This gives us 44.

RPN Example

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What does this expression evaluate to:5 2 * 3 + 7 +

RPN Example 2

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What does this expression evaluate to:5 2 * 3 + 7 +Evaluates to 20

RPN Example 2

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What does this expression evaluate to:5 2 * 3 + 7 +Evaluates to 20

RPN Example 2

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What data structure can we use to evaluate RPN expressions?

• Answer: a stack!

• Stacks work well with RPN, since stacks let us access the most recently seen value(s)

RPN and Stacks

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RPN and Stacks

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Algorithm:



RPN and Stacks

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Algorithm:Read in input one item at a time.



RPN and Stacks

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Algorithm:Read in input one item at a time.Each time you see a number, push it on the stack.



RPN and Stacks

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Algorithm:Read in input one item at a time.Each time you see a number, push it on the stack.Each time you see a (binary) operator:

• Pop two numbers from the stack • Apply the operator to the two numbers • Push the result onto the stack.



RPN and Stacks

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Algorithm:Read in input one item at a time.Each time you see a number, push it on the stack.Each time you see a (binary) operator:


When you have read the entire input, the value left on the stack is the result.

RPN and Stacks

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Algorithm: Read in input. Each time you see a number, push it on the stack. Each time you see a (binary) operator:



Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackpush(6)

6

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackpush(5)

65

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackpush(2)

652

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackpush(3)

6523

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

6523

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackop1 = pop() (= 3)

6523

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackop1 = pop() (= 3) op2 = pop() (=2)

6523

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackop1 = pop() (= 3) op2 = pop() (=2) res = 3 + 2 (= 5)

6523

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackop1 = pop() (= 3) op2 = pop() (=2) res = 3 + 2 (= 5) push(res)

655

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

655

push(8)8

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

655

pop() twice (get 8 and 5) 8

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

655

pop() twice (get 8 and 5) res = 8 * 5 = 40 8

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

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pop() twice (get 8 and 5) res = 8 * 5 = 40 push(res) 40

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

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pop() twice (get 40 and 5)

40

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

65

pop() twice (get 40 and 5) res = 40 + 5 = 45

40

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

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pop() twice (get 40 and 5) res = 40 + 5 = 45 push(45)

45

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

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push(3)

453

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

6453


RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

6453

pop() twice (get 3 and 45) res = 3 + 45 = 48

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

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pop() twice (get 3 and 45) res = 3 + 45 = 48 push(48)

48

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

6


48

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

6

pop() twice (get 48 and 6) res = 48 * 6 = 288

48

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stackpop() twice (get 48 and 6) res = 48 * 6 = 288 push(288)

288

RPN and Stacks

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Evaluate:6 5 2 3 + 8 * + 3 + *

Stack

When done reading input, value left on stack is result. 288

• What do we do about operations like / and -, where order matters?

• For your project: Use the “deeper” item on the stack as the first operand

RPN and Stacks

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RPN and Stacks

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Evaluate:6 2 /

Stack



RPN and Stacks

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Evaluate:6 2 /

Stackpush(6)

6



RPN and Stacks

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Evaluate:6 2 /

Stackpush(2)

62



RPN and Stacks

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Evaluate:6 2 /

Stack

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RPN and Stacks

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Evaluate:6 2 /

Stackop2 = pop() (= 2)

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RPN and Stacks

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Evaluate:6 2 /

Stackop2 = pop() (= 2) op1 = pop() (= 6)

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RPN and Stacks

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Evaluate:6 2 /

Stackop2 = pop() (= 2) op1 = pop() (= 6) res = op1 / op2 = 3 6

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Evaluate the following RPN expression. Use a stack to do so: write items in the stack for push() operations, and cross them out for pop() operations.3 4 2 / 5 * + 9 2 * 3 / -

RPN and Stacks Example

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Evaluate the following RPN expression. Use a stack to do so: write items in the stack for push() operations, and cross them out for pop() operations.3 4 2 / 5 * + 9 2 * 3 / -

Answer: 7

RPN and Stacks Example

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• The absence of ambiguity makes RPN very appealing for computers!

• A number of computers, calculators have historically used RPN‣ See, e.g., dc: A simple calculator program provided by Unix

RPN

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3 + ((4 / 2) * 5) - ((9 * 2) / 3)3 4 2 / 5 * + 9 2 * 3 / - RPN: infix:

• The shunting algorithm developed by Edgar Dijkstra, converts expressions in infix notation to RPN

• It uses a stack and a queue

• Can you figure it out? Try it at home!

JFFE

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• You will implemented an RPN calculator that performs basic arithmetic, and more:‣ Booleans and logical operations

‣ if statements

‣ Storing expressions to be executed later

‣ Reading from a file

‣ …and more!

Your Project

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Exceptions

• When programming, an exception is like an error: it is an indicator that something has gone wrong

• In C++ (and other languages) we throw exceptions‣ Sometimes called “raising” an exception

• When an exception is thrown, either:‣ the program terminates, or

‣ we handle or catch the exception, and the program continues

Exceptions

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• If we #include the C++ library stdexcept, we get access to many kinds of common exceptions:‣ runtime_error

‣ range_error

‣ system_error

‣ …

Kinds of Exceptions

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• If we #include the C++ library stdexcept, we get access to many kinds of common exceptions:‣ runtime_error

‣ range_error

‣ system_error

‣ …

• Today we’ll use range_error as an example

• For your project you’ll use runtime_errors, but using them follows identical steps!

Kinds of Exceptions

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• When we throw an exception, we typically provide an associated error message:

Throwing an Exception

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int attempt_access(int arr[], int size, int index) { if (index < 0 or index >= size) throw range_error("Oh no! Given index was out of range.\n");

return arr[index]; }

• Once we throw an exception, if we do not have any special handling for it, the program will terminate

• In C++, we handle exceptions using try…catch… statements:

Handling Exceptions

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try { …some code you want to run which may throw an exception… }catch(type_of_exc) { …code to execute in the case that a type_of_exc exception occurs…}

int second_attempt_access(int arr[], int size, int index) { try { int result = attempt_access(arr, size, index); cout << "Successfully accessed element!" << endl; return result; } catch (range_error) { cout << "Terrible! Got a range error! " << "Going to return 0 by default." << endl; return 0; } }

Handling Exceptions

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int second_attempt_access(int arr[], int size, int index) { try { int result = attempt_access(arr, size, index); cout << "Successfully accessed element!" << endl; return result; } catch (range_error) { cout << "Terrible! Got a range error! " << "Going to return 0 by default." << endl; return 0; } }

• By catching a range_error exception, our program can continue running even though an exception was thrown

Handling Exceptions

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• Sometimes we want to catch an exception, but still print the error message associated with it

• We can do this using a const reference along with the C++ what() method

Handling Exceptions

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Catch and Print Exception

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int third_attempt_access(int arr[], int size, int index) { try { int result = attempt_access(arr, size, index); cout << "Successfully accessed element!" << endl; return result; } catch (const range_error &e) { cout << "We just got the following error message:\n"; cout << e.what(); return 0; } }

• Let’s take a look at the full example: exceptions.cpp

Exceptions

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cs 15: data structures

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